3D Vision Halcon 12.0 Solution Guide Iii C 2019 04 29 09 58 30

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the power of machine vision

Solution Guide III-C
3D Vision

Machine vision in 3D world coordinates, Version 12.0.2
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by
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December 2003
July 2004
April 2005
July 2005
December 2006
June 2007
October 2007
December 2008
June 2009
March 2010
October 2010
April 2011
May 2012
November 2014
July 2015

(HALCON 7.0)
(HALCON 7.0.1)
(HALCON 7.0.2)
(HALCON 7.1)
(HALCON 7.1.2)
(HALCON 8.0)
(HALCON 8.0.1)
(HALCON 9.0)
(HALCON 9.0.1)
(HALCON 9.0.2)
(HALCON 10.0)
(HALCON 10.0.1)
(HALCON 11.0)
(HALCON 12.0)
(HALCON 12.0.1)

by MVTec Software GmbH, München, Germany

MVTec Software GmbH

Protected by the following patents: US 7,062,093, US 7,239,929, US 7,751,625, US 7,953,290, US 7,953,291, US
8,260,059, US 8,379,014, US 8,830,229. Further patents pending.
Microsoft, Windows, Windows Vista, Windows Server 2008, Windows 7, Windows 8, Windows 10, Microsoft .NET,
Visual C++, Visual Basic, and ActiveX are either trademarks or registered trademarks of Microsoft Corporation.
All other nationally and internationally recognized trademarks and tradenames are hereby recognized.

More information about HALCON can be found at: http://www.halcon.com/

About This Manual
Measurements in 3D become more and more important. HALCON provides many methods to perform
3D measurements. This Solution Guide gives you an overview over these methods, and it assists you
with the selection and the correct application of the appropriate method.
A short characterization of the various methods is given in chapter 1 on page 9. Principles of 3D transformations and poses as well as the description of the camera model can be found in chapter 2 on page
15. Afterwards, the methods to perform 3D measurements are described in detail.
The HDevelop example programs that are presented in this Solution Guide can be found in the specified
subdirectories of the directory %HALCONROOT%.

Contents
1

Introduction

Basics
2.1 3D Transformations and Poses . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 3D Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Rigid Transformations and Homogeneous Transformation Matrices
2.1.5 3D Poses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Camera Model and Parameters . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Map 3D World Points to Pixel Coordinates . . . . . . . . . . . . .
2.2.2 Area Scan Cameras . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Tilt Lenses and the Scheimpflug Principle . . . . . . . . . . . . . .
2.2.4 Line Scan Cameras . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 3D Object Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Obtaining 3D Object Models . . . . . . . . . . . . . . . . . . . . .
2.3.2 Content of 3D Object Models . . . . . . . . . . . . . . . . . . . .
2.3.3 Modifying 3D Object Models . . . . . . . . . . . . . . . . . . . .
2.3.4 Extracting Features of 3D Object Models . . . . . . . . . . . . . .
2.3.5 Matching of 3D Object Models . . . . . . . . . . . . . . . . . . .
2.3.6 Visualizing 3D Object Models . . . . . . . . . . . . . . . . . . . .

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Metric Measurements in a Specified Plane With a Single Camera
3.1 First Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 3D Camera Calibration . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Creating the Calibration Data Model . . . . . . . . . . . . . .
3.2.2 Specifying Initial Values for the Internal Camera Parameters .
3.2.3 Describing the Calibration Object . . . . . . . . . . . . . . .
3.2.4 Observing the Calibration Object in Multiple Poses (Images) .
3.2.5 Restricting the Calibration to Specific Parameters . . . . . . .
3.2.6 Performing the Calibration . . . . . . . . . . . . . . . . . . .
3.2.7 Accessing the Results of the Calibration . . . . . . . . . . . .
3.2.8 Deleting Observations from the Calibration Data Model . . .
3.2.9 Saving the Results and Destroying the Calibration Data Model
3.2.10 Troubleshooting . . . . . . . . . . . . . . . . . . . . . . . .

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3.3

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3D Position Recognition of Known Objects
4.1 Pose Estimation from Points . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Pose Estimation Using Shape-Based 3D Matching . . . . . . . . . . . . .
4.2.1 General Proceeding for Shape-Based 3D Matching . . . . . . . .
4.2.2 Enhance the Shape-Based 3D Matching . . . . . . . . . . . . . .
4.2.3 Tips and Tricks for Problem Handling . . . . . . . . . . . . . . .
4.3 Pose Estimation Using Surface-Based 3D Matching . . . . . . . . . . . .
4.3.1 General Proceeding for Surface-Based 3D Matching . . . . . . .
4.4 Pose Estimation Using Deformable Surface-Based 3D Matching . . . . .
4.4.1 General Proceeding for Deformable Surface-Based 3D Matching .
4.5 Pose Estimation Using 3D Primitives Fitting . . . . . . . . . . . . . . . .
4.6 Pose Estimation Using Calibrated Perspective Deformable Matching . . .
4.7 Pose Estimation Using Calibrated Descriptor-Based Matching . . . . . .
4.8 Pose Estimation for Circles . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Pose Estimation for Rectangles . . . . . . . . . . . . . . . . . . . . . . .

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3.4

3.5
4

Transforming Image into World Coordinates and Vice Versa . .
3.3.1 The Main Principle . . . . . . . . . . . . . . . . . . . .
3.3.2 World Coordinates for Points . . . . . . . . . . . . . . .
3.3.3 World Coordinates for Contours . . . . . . . . . . . . .
3.3.4 World Coordinates for Regions . . . . . . . . . . . . . .
3.3.5 Transforming World Coordinates into Image Coordinates
3.3.6 Compensate for Lens Distortions Only . . . . . . . . . .
Rectifying Images . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Transforming Images into the WCS . . . . . . . . . . .
3.4.2 Compensate for Lens Distortions Only . . . . . . . . . .
Inspection of Non-Planar Objects . . . . . . . . . . . . . . . . .

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3D Vision With a Stereo System
5.1 The Principle of Stereo Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 The Setup of a Stereo Camera System . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Resolution of a Stereo Camera System . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Optimizing Focus with Tilt Lenses . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Calibrating the Stereo Camera System . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Creating and Configuring the Calibration Data Model . . . . . . . . . . . . . . .
5.2.2 Acquiring Calibration Images . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Observing the Calibration Object . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4 Calibrating the Cameras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Binocular Stereo Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Comparison of the Stereo Matching Approaches Correlation-Based, Multigrid,
and Multi-Scanline Stereo . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Accessing the Calibration Results . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Acquiring Stereo Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.4 Rectifying the Stereo Images . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.5 Reconstructing 3D Information . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.6 Uncalibrated Stereo Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Multi-View Stereo Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.4.1
5.4.2

Initializing the Stereo Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Reconstructing 3D Information . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Laser Triangulation with Sheet of Light
6.1 The Principle of Sheet of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 The Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Calibrating the Sheet-of-Light Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Calibrating the Sheet-of-Light Setup using a standard HALCON calibration plate
6.3.2 Calibrating the Sheet-of-Light Setup Using a Special 3D Calibration Object . . .
6.4 Performing the Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 Calibrated Sheet-of-Light Measurement . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Uncalibrated Sheet-of-Light Measurement . . . . . . . . . . . . . . . . . . . . .
6.5 Using the Score Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 3D Cameras for Sheet of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Depth from Focus
7.1 The Principle of Depth from Focus . . . . . . . . . .
7.1.1 Speed vs. Accuracy . . . . . . . . . . . . . .
7.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Camera . . . . . . . . . . . . . . . . . . . .
7.2.2 Illumination . . . . . . . . . . . . . . . . . .
7.2.3 Object . . . . . . . . . . . . . . . . . . . . .
7.3 Working with Depth from Focus . . . . . . . . . . .
7.3.1 Rules for Taking Images . . . . . . . . . . .
7.3.2 Practical Use of Depth from Focus . . . . . .
7.3.3 Volume Measurement with Depth from Focus
7.4 Solutions for Typical Problems With DFF . . . . . .
7.4.1 Calibrating Aberration . . . . . . . . . . . .
7.5 Special Cases . . . . . . . . . . . . . . . . . . . . .
7.6 Performing Depth from Focus with a Standard Lens .

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Robot Vision
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8.1 Supported Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.1.1 Articulated Robot vs. SCARA Robot . . . . . . . . . . . . . . . . . . . . . . . 215
8.1.2 Camera and Calibration Plate vs. 3D Sensor and 3D Object . . . . . . . . . . . 216
8.1.3 Moving Camera vs. Stationary Camera . . . . . . . . . . . . . . . . . . . . . . 217
8.1.4 Calibrating the Camera in Advance vs. Calibrating It During Hand-Eye Calibration217
8.2 The Principle of Hand-Eye Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
8.3 Calibrating the Camera in Advance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.4 Preparing the Calibration Input Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.4.1 Creating the Data Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.4.2 Poses of the Calibration Object . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
8.4.3 Poses of the Robot Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
8.5 Performing the Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.6 Determine Translation in Z Direction for SCARA Robots . . . . . . . . . . . . . . . . . 227
8.7 Using the Calibration Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
8.7.1 Using the Hand-Eye Calibration for Grasping (3D Alignment) . . . . . . . . . . 228
8.7.2 How to Get the 3D Pose of the Object . . . . . . . . . . . . . . . . . . . . . . . 229

8.7.3
9

Example Application with a Stationary Camera: Grasping a Nut . . . . . . . . . 230

Calibrated Mosaicking
9.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Merging the Individual Images into One Larger Image . .
9.3.1 Definition of the Rectification of the First Image .
9.3.2 Definition of the Rectification of the Second Image
9.3.3 Rectification of the Images . . . . . . . . . . . . .

10 Uncalibrated Mosaicking
10.1 Rules for Taking Images for a Mosaic Image
10.2 Definition of Overlapping Image Pairs . . .
10.3 Detection of Characteristic Points . . . . .
10.4 Matching of Characteristic Points . . . . .
10.5 Generation of the Mosaic Image . . . . . .
10.6 Bundle Adjusted Mosaicking . . . . . . . .
10.7 Spherical Mosaicking . . . . . . . . . . . .

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11 Rectification of Arbitrary Distortions
11.1 Basic Principle . . . . . . . . . . . . . . . . . .
11.2 Rules for Taking Images of the Rectification Grid
11.3 Machine Vision on Ruled Surfaces . . . . . . . .
11.4 Using Self-Defined Rectification Grids . . . . . .

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A HDevelop Procedures Used in this Solution Guide
A.1 gen_hom_mat3d_from_three_points . . . . . .
A.2 parameters_image_to_world_plane_centered .
A.3 parameters_image_to_world_plane_entire . . .
A.4 tilt_correction . . . . . . . . . . . . . . . . . .
A.5 calc_calplate_pose_movingcam . . . . . . . .
A.6 calc_calplate_pose_stationarycam . . . . . . .
A.7 define_reference_coord_system . . . . . . . .

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Index

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283

C-9

Introduction

Chapter 1

Introduction
With HALCON you can perform 3D vision in various ways. The main applications comprise the 3D
position recognition and the 3D inspection, which both consist of several different approaches with
different characteristics, so that for a wide range of 3D vision tasks a proper solution can be provided.
This Solution Guide provides you with detailed information on the available approaches, including also
some auxiliary methods that are needed only in specific cases.
What Basic Knowledge Do You Need for 3D Vision?
Typically, you have to calibrate your camera(s) before applying a 3D vision task. Especially, if you want
to achieve accurate results, the camera calibration is essential, because it is of no use to extract edges
with an accuracy of 1/40 pixel if the lens distortion of the uncalibrated camera accounts for a couple of
pixels. This also applies if you use cameras with telecentric lenses. But don’t be afraid of the calibration
process: In HALCON, this can be done with just a few lines of code. To prepare you for the camera
calibration, chapter 2 on page 15 introduces you to the details on the camera model and parameters. The
actual camera calibration is then described in chapter 3 on page 65.
Using a camera calibration, you can transform image processing results into arbitrary 3D coordinate
systems and thus derive metrical information from images, regardless of the position and orientation of
the camera with respect to the object. In other words, you can perform inspection tasks in 3D coordinates
in specified object planes, which can be oriented arbitrarily with respect to the camera. This is, e.g.,
useful if the camera cannot be mounted such that it looks perpendicular to the object surface. Thus,
besides the pure camera calibration, chapter 3 shows how to apply a general 3D vision task with a single
camera in a specified plane. Additionally, it shows how to rectify the images such that they appear
as if they were acquired from a camera that has no lens distortions and that looks exactly perpendicular
onto the object surface. This is useful for tasks like OCR or the recognition and localization of objects,
which rely on images that are not distorted too much with respect to the training images.
Before you develop your application, we recommend to read chapter 2 and chapter 3 and then, depending
on the task at hand, to step into the section that describes the 3D vision approach you selected for your
specific application.

C-10

Introduction

How Can You Obtain an Object’s 3D Position and Orientation?
The position and orientation of 3D objects with respect to a given 3D coordinate system, which is needed,
e.g., for pick-and-place applications (3D alignment), can be determined by one of the methods described
in chapter 4 on page 105:
• The pose estimation of a known 3D object from corresponding points (section 4.1 on page 106)
is a rather general approach that includes a camera calibration and the extraction of at least three
significant points for which the 3D object coordinates are known. The approach is also known as
“mono 3D”.
• HALCON’s 3D matching locates known 3D objects based on a 3D model of the object. In particular, it automatically searches objects that correspond to a 3D model in the search data and determines their 3D poses. The model must be provided, e.g., as a Computer Aided Design (CAD)
model. Available approaches are the shape-based 3D matching (section 4.2 on page 111) that
searches the model in 2D images and the surface-based 3D matching (section 4.3 on page 123)
that searches the model in a 3D scene, i.e., in a set of 3D points that is available as 3D object
model, which can be obtained by a 3D reconstruction approach like stereo or sheet of light. Note
that the surface-based matching is also known as “volume matching”, although it only relies on
points on the object’s surface.
• HALCON’s 3D primitives fitting (section 4.5 on page 132) fits a primitive 3D shape like a cylinder, sphere, or plane into a 3D scene, i.e., into a set of 3D points that is available as a 3D object
model, which can be obtained by a 3D reconstruction approach like stereo or sheet of light followed by a 3D segmentation.
• The calibrated perspective matching locates perspectively distorted planar objects in images
based on a 2D model. In particular, it automatically searches objects that correspond to a 2D
model in the search images and determines their 3D poses. The model typically is obtained from
a representative model image. Available approaches are the calibrated perspective deformable
matching (section 4.6 on page 136) that describes the model by its contours and the calibrated
descriptor-based matching (section 4.7 on page 137) that describes the model by a set of distinctive points that are called “interest points”.
• The circle pose estimation (section 4.8 on page 138) and rectangle pose estimation (section 4.9
on page 138) use the perspective distortions of circles and rectangles to determine the pose of
planar objects that contain circles and/or rectangles in a rather convenient way.
How Can You Inspect a 3D Object?
The inspection of 3D objects can be applied by different means. If the inspection in a specified plane is
sufficient, you can use a camera calibration together with a 2D inspection as is described in chapter 3
on page 65.
If the surface of the 3D object is needed and/or the inspection can not be reduced to a single specified
plane, you can use a 3D reconstruction together with a 3D inspection. That is, you use the point,
surface, or height information returned for a 3D object by a 3D reconstruction and inspect the object,
e.g., by comparing it to a reference point, surface, or height.
Figure 1.1 provides you with an overview on the methods that are available for 3D position recognition
and 3D inspection. For an introduction to 3D object models, please refer to section 2.3 on page 41.

C-11

3D
Inspection

(chapter 7)
Camera
Calibration

(section 3.2)

3D Object Model (generated)

Stereo Vision

Sheet of Light

(chapter 5)

(chapter 6)

Pose from
Points

(section 4.1)

Photometric Stereo

Depth from Focus

TOF
etc.

Shape−Based
3D Matching

(section 4.2)
Surface−Based
3D Matching

(section 4.3)

3D Object
Model
(needed)
3D Primitives
Fitting

Calibrated
Descriptor−
Based
Matching

(section 4.7)
Calibrated
Perspective
Deformable
Matching

(section 4.6)

Circle Pose

(section 4.8)

(section 4.5)

Rectangle Pose

(section 4.9)
2D Inspection

Introduction

Planar Object Part
(Perspective View)

Arbitrary Objects

3D
Sensor

Setup

Single Camera
(Multiple Images,
additional Hardware)

Uncalibrated

(chapter 3)

Multiple
Cameras

3D Object

Calibrated

Single Camera
(Specified Plane)

Reconstruct
Surfaces

3D Inspection

Objects with Primitive Shapes

Measure Elements
and their Relations

3D Position
Recognition

Figure 1.1: Overview to the main methods used for 3D Vision.

How Can You Reconstruct 3D Objects?
To determine points on the surface of arbitrary objects, the following approaches are available:
• HALCON’s stereo vision functionality (chapter 5 on page 139) allows to determine the 3D coordinates of any point on the object surface based on two (binocular stereo) or more (multi-view
stereo) images that are acquired suitably from different points of view (typically by separate cameras). Using multi-view stereo, you can reconstruct a 3D object in full 3D, in particular, you can
reconstruct it from different sides.
• A laser triangulation with sheet of light (chapter 6 on page 177) allows to get a height profile of the
object. Note that besides a single camera, additional hardware, in particular a laser line projector
and a unit that moves the object relative to the camera and the laser, is needed.
• With depth from focus (DFF) (chapter 7 on page 199) a height profile can be obtained using
images that are acquired by a single telecentric camera but at different focus positions. In order to

C-12

Introduction

vary the focus position additional hardware like a translation stage or linear piezo stage is required.
Note that depending on the direction in which the focus position is modified, the result corresponds
either to a height image or to a distance image. A height image contains the distances between
a specific object or measure plane and the object points, whereas the distance image typically
contains the distances between the camera and the object points. Both can be called also depth
image or “Z image”.
• With photometric stereo (Reference Manual, chapter “3D Reconstruction . Photometric Stereo”)
a height image can be obtained using images that are acquired by a single telecentric camera
but with at least three different telecentric illumination sources for which the spatial relations to
the camera must be known. Note that the height image reflects only relative heights, i.e., with
photometric stereo no calibrated 3D reconstruction is possible.
• Besides the 3D reconstruction approaches provided by HALCON, you can obtain 3D information
also by specific 3D sensors like time of flight (TOF) cameras or specific setups that use structured
light. These cameras typically are calibrated and return X, Y, and Z images.
Figure 1.2 allows to compare some important features of the different 3D reconstruction approaches like
the approach-specific result types.
3D Reconstruction
Approach
Multi-View Stereo
Binocular Stereo

Sheet of Light

Depth from Focus

Photometric Stereo

3D Sensors

Hardware Requirements

Object Size

Possible Results

multiple cameras,
calibration object
two cameras,
calibration object

approx. > 10 cm

camera,
laser line projector,
unit to move the object,
and calibration object
telecentric camera,
hardware to variate
the focus position
telecentric camera,
at least three telecentric
illumination sources
special camera like
calibrated TOF

object must fit onto
the moving unit

3D object model or
X, Y, Z coordinates
X, Y, Z coordinates,
approach-specific
disparity image, or
Z image
3D object model,
X, Y, Z images, or
approach-specific
disparity image
Z image

approx. > 10 cm

approx. < 2cm

restricted by
field of view of
telecentric lens
approx. 30cm-5m

Figure 1.2: 3D reconstruction: a coarse comparison.

Z image

X, Y, Z images

C-13

A typical application area for 3D vision is robot vision, i.e., using the results of machine vision to
command a robot. In such applications you must perform an additional calibration: the so-called handeye calibration, which determines the relation between camera and robot coordinates (chapter 8 on page
215). Again, this calibration must be performed only once (offline). Its results allow you to quickly
transform machine vision results from camera into robot coordinates.
What Tasks May be Needed Additionally?
If the object that you want to inspect is too large to be covered by one image with the desired resolution,
multiple images, each covering only a part of the object, can be combined into one larger mosaic image.
This can be done either based on a calibrated camera setup with very high precision (chapter 9 on page
235) or highly automated for arbitrary and even varying image configurations (chapter 10 on page 247).
If an image shows distortions that are different to the common perspective distortions or lens distortions,
caused, e.g., by a non-flat object surface, the so-called grid rectification can be applied to rectify the
image (chapter 11 on page 263).

Introduction

How Can You Extend 3D Vision to Robot Vision?

C-14

Introduction

Basics

C-15

Basics
2.1

3D Transformations and Poses

Before we start explaining how to perform 3D vision with HALCON, we take a closer look at some basic
questions regarding the use of 3D coordinates:
• How to describe the transformation (translation and rotation) of points and coordinate systems,
• how to describe the position and orientation of one coordinate system relative to another, and
• how to determine the coordinates of a point in different coordinate systems, i.e., how to transform
coordinates between coordinate systems.
In fact, all these tasks can be solved using one and the same means: homogeneous transformation matrices and their more compact equivalent, 3D poses.

2.1.1

3D Coordinates

The position of a 3D point P is described by its three coordinates (xp , yp , zp ). The coordinates can
also be interpreted as a 3D vector (indicated by a bold-face lower-case letter). The coordinate system
in which the point coordinates are given is indicated to the upper right of a vector or coordinate. For
example, the coordinates of the point P in the camera coordinate system (denoted by the letter c) and in
the world coordinate system (denoted by the letter w ) would be written as:
 c 
 w 
xp
xp
pw =  ypw 
pc =  ypc 
zpc
zpw
Figure 2.1 depicts an example point lying in a plane where measurements are to be performed and its
coordinates in the camera and world coordinate system, respectively.

Basics

Chapter 2

C-16

Basics

2.1.2

Translation

2.1.2.1

Translation of Points

In figure 2.2, our example point has been translated along the x-axis of the camera coordinate system.
The coordinates of the resulting point P2 can be calculated by adding two vectors, the coordinate vector
p1 of the point and the translation vector t:


xp1 + xt
p2 = p1 + t =  yp1 + yt 
(2.1)
z p1 + z t
Multiple translations are described by adding the translation vectors. This operation is commutative, i.e.,
the sequence of the translations has no influence on the result.
2.1.2.2

Translation of Coordinate Systems

Coordinate systems can be translated just like points. In the example in figure 2.3, the coordinate system
c1 is translated to form a second coordinate system, c2 . Then, the position of c2 in c1 , i.e., the coordinate
vector of its origin relative to c1 (occ12 ), is identical to the translation vector:
tc1 = occ12

2.1.2.3

(2.2)

Coordinate Transformations

Let’s turn to the question how to transform point coordinates between (translated) coordinate systems.
In fact, the translation of a point can also be thought of as translating it together with its local coordinate
system. This is depicted in figure 2.3: The coordinate system c1 , together with the point Q1 , is translated
Camera coordinate system

( x c, y c , z c)

World coordinate system

xc
y

( x w, y w, z w)

zw
z

c
p = 2

P
Measurement plane

pw = 3.3
0

Figure 2.1: Coordinates of a point in two different coordinate systems.

2.1 3D Transformations and Poses

C-17

Camera coordinate system

( x c, y c , z c)

yc
zc
0

p1 = 2

p2 = 2
4

Basics

t= 0
0

Figure 2.2: Translating a point.
2

t= 0

Coordinate system 1
c1

Coordinate system 2

(x , y , z )
c1

( xc2, y c2, z c2 )

y c1
x c2

z c1
0
4

z c2

y c2

c1
q1 = 0

Q1
2
c1
q2 = 0
6

qc2
= 0
2
4

Figure 2.3: Translating a coordinate system (and point).

by the vector t, resulting in the coordinate system c2 and the point Q2 . The points Q1 and Q2 then have
the same coordinates relative to their local coordinate system, i.e., qc11 = qc22 .
If coordinate systems are only translated relative to each other, coordinates can be transformed very
easily between them by adding the translation vector:
qc21 = qc22 + tc1 = qc22 + occ12

(2.3)

In fact, figure 2.3 visualizes this equation: qc21 , i.e., the coordinate vector of Q2 in the coordinate system
c1 , is composed by adding the translation vector t and the coordinate vector of Q2 in the coordinate
system c2 (qc22 ).

C-18

Basics

The downside of this graphical notation is that, at first glance, the direction of the translation vector
appears to be contrary to the direction of the coordinate transformation: The vector points from the
coordinate system c1 to c2 , but transforms coordinates from the coordinate system c2 to c1 . According
to this, the coordinates of Q1 in the coordinate system c2 , i.e., the inverse transformation, can be obtained
by subtracting the translation vector from the coordinates of Q1 in the coordinate system c1 :
qc12 = qc11 − tc1 = qc11 − occ12
2.1.2.4

(2.4)

Summary

• Points are translated by adding the translation vector to their coordinate vector. Analogously, coordinate systems are translated by adding the translation vector to the position (coordinate vector)
of their origin.
• To transform point coordinates from a translated coordinate system c2 into the original coordinate
system c1 , you apply the same transformation to the points that was applied to the coordinate
system, i.e., you add the translation vector used to translate the coordinate system c1 into c2 .
• Multiple translations are described by adding all translation vectors; the sequence of the translations does not affect the result.

2.1.3

Rotation

2.1.3.1

Rotation of Points

In figure 2.4a, the point p1 is rotated by −90◦ around the z-axis of the camera coordinate system.
4

p4 = 0

−2

zc
0
p1 = 2
4

P1
a) first rotation

Ry (90°)

zc
2

p3 = 0
P3
Rz (−90°)

p1 = 2
4

P3
2

p3 = 0
4

b) second rotation

Figure 2.4: Rotate a point: (a) first around the zc -axis; (b) then around the yc -axis.

2.1 3D Transformations and Poses

C-19

Rotating a point is expressed by multiplying its coordinate vector with a 3 × 3 rotation matrix R. A
rotation around the z-axis looks as follows:
cos γ
p3 = Rz (γ) · p1 =  sin γ
0

− sin γ
cos γ
0

 
 

0
x p1
cos γ · xp1 − sin γ · yp1
0  ·  yp1  =  sin γ · xp1 + cos γ · yp1 
1
zp1
zp1

Rotations around the x- and y-axis correspond to the following rotation matrices:




1
0
0
cos β 0 sin β
0
1
0 
Rx (α) =  0 cos α − sin α 
Ry (β) = 
0 sin α cos α
− sin β 0 cos β
2.1.3.2

(2.5)

(2.6)

Chain of Rotations

In figure 2.4b, the rotated point is further rotated around the y-axis. Such a chain of rotations can be
expressed very elegantly by a chain of rotation matrices:
p4 = Ry (β) · p3 = Ry (β) · Rz (γ) · p1

(2.7)

Note that in contrast to a multiplication of scalars, the multiplication of matrices is not commutative, i.e.,
if you change the sequence of the rotation matrices, you get a different result.
2.1.3.3

Rotation of Coordinate Systems

In contrast to points, coordinate systems have an orientation relative to other coordinates systems. This
orientation changes when the coordinate system is rotated. For example, in figure 2.5a the coordinate
system c3 has been rotated around the y-axis of the coordinate system c1 , resulting in a different orientation of the camera. Note that in order to rotate a coordinate system in your mind’s eye, it may help to
image the points of the axis vectors being rotated.
Just like the position of a coordinate system can be expressed directly by the translation vector (see
equation 2.2 on page 16), the orientation is contained in the rotation matrix: The columns of the rotation
matrix correspond to the axis vectors of the rotated coordinate system in coordinates of the original one:

R = xcc13 ycc31 zcc13
(2.8)
For example, the axis vectors of the coordinate system c3 in figure 2.5a can be determined from the
corresponding rotation matrix Ry (90◦ ) as shown in the following equation; you can easily check the
result in the figure.

 

cos(90◦ ) 0 sin(90◦ )
0 0 1
= 0 1 0 
0
1
0
Ry (90◦ ) = 
− sin(90◦ ) 0 cos(90◦ )
−1 0 0
⇒

xcc13




0
= 0 
−1

ycc31




0
= 1 
0

zcc13




1
= 0 
0

Basics



C-20

Basics

Ry (90°)

Coordinate system 1

( xc1, y c1, z c1 )

( xc3, y c3, z c3 )
4

x c1

y c1

z c1

x c3 x c4

z c4

Q3
0
q3 = 0
4
c3

y c3

q1 = 0
4

( xc4, y c4, z c4 )

qc1
= 0
3
z c3

Coordinate system 4

y c4

x c3

Rz (−90°)

Coordinate system 3

z c3
y

q4c1= 0
0

Q4 = Q3
0

q4c4= 0

z c1

q1 = 0
Q1

a) first rotation

b) second rotation

Figure 2.5: Rotate coordinate system: (a) first around the yc1 -axis; (b) then around the zc3 -axis.

2.1.3.4

Coordinate Transformations

Like in the case of translation, to transform point coordinates from a rotated coordinate system c3 into
the original coordinate system c1 , you apply the same transformation to the points that was applied to
the coordinate system c3 , i.e., you multiply the point coordinates with the rotation matrix used to rotate
the coordinate system c1 into c3 :
qc31 = c1 Rc3 ·qc33
(2.9)
This is depicted in figure 2.5 also for a chain of rotations, which corresponds to the following equation:
qc41 = c1 Rc3 · c3 Rc4 ·qc44 = Ry (β) · Rz (γ) · qc44 = c1 Rc4 ·qc44
2.1.3.5

(2.10)

In Which Sequence and Around Which Axes are Rotations Performed?

If you compare the chains of rotations in figure 2.4 and figure 2.5 and the corresponding equations 2.7
and 2.10, you will note that two different sequences of rotations are described by the same chain of
rotation matrices: In figure 2.4, the point was rotated first around the z-axis and then around the y-axis,
whereas in figure 2.5 the coordinate system is rotated first around the y-axis and then around the z-axis.
Yet, both are described by the chain Ry (β) · Rz (γ)!
The solution to this seemingly paradox situation is that in the two examples the chain of rotation matrices
can be “read” in different directions: In figure 2.4 it is read from the right to left, and in figure 2.5 from
left to the right.
However, there still must be a difference between the two sequences because, as we already mentioned,
the multiplication of rotation matrices is not commutative. This difference lies in the second question in
the title, i.e., around which axes the rotations are performed.

2.1 3D Transformations and Poses

C-21

Let’s start with the second rotation of the coordinate system in figure 2.5b. Here, there are two possible
sets of axes to rotate around: those of the “old” coordinate system c1 and those of the already rotated,
“new” coordinate system c3 . In the example, the second rotation is performed around the “new” z-axis.
In contrast, when rotating points as in figure 2.4, there is only one set of axes around which to rotate:
those of the “old” coordinate system.
From this, we derive the following rules:

• When reading a chain from the right to left, rotations are performed around the “old” axes.
As already remarked, point rotation chains are always read from right to left. In the case of coordinate
systems, you have the choice how to read a rotation chain. In most cases, however, it is more intuitive to
read them from left to right.
Figure 2.6 shows that the two reading directions really yield the same result.
2.1.3.6

Summary

• Points are rotated by multiplying their coordinate vector with a rotation matrix.
• If you rotate a coordinate system, the rotation matrix describes its resulting orientation: The column
vectors of the matrix correspond to the axis vectors of the rotated coordinate system in coordinates
of the original one.
• To transform point coordinates from a rotated coordinate system c3 into the original coordinate
system c1 , you apply the same transformation to the points that was applied to the coordinate
system, i.e., you multiply them with the rotation matrix that was used to rotate the coordinate
system c1 into c3 .
Performing a chain of rotations:

Ry (90°) * Rz (−90°)

a) reading from left to right = rotating around "new" axes

x c3’
x c1
y c1

y c4
z c3’

Ry (90°)

x c4
z c4

c3’

Rz (−90°)

y c3’

z c1

b) reading from right to left = rotating around "old" axes

y c4

x c3
c1

x
y c1

Rz (−90°)

Ry (90°)

x c4
z c4

y c1
z c1

z c3

Figure 2.6: Performing a chain of rotations (a) from left to the right, or (b) from right to left.

Basics

• When reading a chain from the left to right, rotations are performed around the “new” axes.

C-22

Basics

p4 = 0

−2

t= 0

P5
8

p5 = 0

−2

Ry (90°)
z

p3 = 0

0
p1 = 2
4

Rz (−90°)

Figure 2.7: Combining the translation from figure 2.2 on page 17 and the rotation of figure 2.4 on page 18
to form a rigid transformation.

• Multiple rotations are described by a chain of rotation matrices, which can be read in two directions. When read from left to right, rotations are performed around the “new” axes; when read
from right to left, the rotations are performed around the “old” axes.

2.1.4

Rigid Transformations and Homogeneous Transformation Matrices

2.1.4.1

Rigid Transformation of Points

If you combine translation and rotation, you get a so-called rigid transformation. For example, in figure 2.7, the translation and rotation of the point from figures 2.2 and 2.4 are combined. Such a transformation is described as follows:
p5 = R ·p1 + t
(2.11)
For multiple transformations, such equations quickly become confusing, as the following example with
two transformations shows:
p6 = Ra ·(Rb ·p1 + tb ) + ta = Ra · Rb ·p1 + Ra ·tb + ta

(2.12)

An elegant alternative is to use so-called homogeneous transformation matrices and the corresponding
homogeneous vectors. A homogeneous transformation matrix H contains both the rotation matrix and
the translation vector. For example, the rigid transformation from equation 2.11 can be rewritten as
follows:

R
t
p5
p1
R ·p1 + t
=
·
=
= H ·p1
(2.13)
1
1
1
000
1

2.1 3D Transformations and Poses

C-23

The usefulness of this notation becomes apparent when dealing with sequences of rigid transformations,
which can be expressed as chains of homogeneous transformation matrices, similarly to the rotation
chains:

Ra
ta
Rb
tb
Ra · Rb
Ra ·tb + ta
H1 · H2 =
·
=
(2.14)
000
1
000
1
000
1

In fact, a rigid transformation is already a chain, since it consists of a translation and a rotation:

 

0
1 0 0


 0 1 0
R
t
0 
t 
 = H(t) · H(R)
· R
H=
=



0 
0 0 1
000 1
000 1
0 0 0 1

Basics

As explained for chains of rotations, chains of rigid transformation can be read in two directions. When
reading from left to right, the transformations are performed around the “new” axes, when read from
right to left around the “old” axes.

(2.15)

If the rotation is composed of multiple rotations around axes as in figure 2.7, the individual rotations can
also be written as homogeneous transformation matrices:

 
 

0
0
1 0 0



 0 1 0
Ry (β) · Rz (γ)
t
0 
0 
t 

 ·  Rz (γ)
 ·  Ry (β)
H =
=





0
0 0 1
0 
000
1
000
1
000
1
0 0 0
1
Reading this chain from right to left, you can follow the transformation of the point in figure 2.7: First,
it is rotated around the z-axis, then around the (“old”) y-axis, and finally it is translated.
2.1.4.2

Rigid Transformation of Coordinate Systems

Rigid transformations of coordinate systems work along the same lines as described for a separate translation and rotation. This means that the homogeneous transformation matrix c1 Hc5 describes the transformation of the coordinate system c1 into the coordinate system c5 . At the same time, it describes the
position and orientation of coordinate system c5 relative to coordinate system c1 : Its column vectors
contain the coordinates of the axis vectors and the origin.
c

xc15 ycc51 zcc15
occ15
c1
Hc5 =
(2.16)
0
0
0
1
As already noted for rotations, chains of rigid transformations of coordinate systems are typically read
from left to right. Thus, the chain above can be read as first translating the coordinate system, then
rotating it around its “new” y-axis, and finally rotating it around its “newest” z-axis.

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Basics

2.1.4.3

Coordinate Transformations

As described for the separate translation and the rotation, to transform point coordinates from a rigidly
transformed coordinate system c5 into the original coordinate system c1 , you apply the same transformation to the points that was applied to the coordinate system c5 , i.e., you multiply the point coordinates
with the homogeneous transformation matrix:
c1
c5
p5
p5
c1
= Hc5 ·
(2.17)
1
1
Typically, you leave out the homogeneous vectors if there is no danger of confusion and simply write:
pc51 = c1 Hc5 ·pc55
2.1.4.4

(2.18)

Summary

• Rigid transformations consist of a rotation and a translation. They are described very elegantly by
homogeneous transformation matrices, which contain both the rotation matrix and the translation
vector.
• Points are transformed by multiplying their coordinate vector with the homogeneous transformation matrix.
• If you transform a coordinate system, the homogeneous transformation matrix describes the coordinate system’s resulting position and orientation: The column vectors of the matrix correspond to
the axis vectors and the origin of the coordinate system in coordinates of the original one. Thus,
you could say that a homogeneous transformation matrix “is” the position and orientation of a
coordinate system.
• To transform point coordinates from a rigidly transformed coordinate system c5 into the original
coordinate system c1 , you apply the same transformation to the points that was applied to the
coordinate system, i.e., you multiply them with the homogeneous transformation matrix that was
used to transform the coordinate system c1 into c5 .
• Multiple rigid transformations are described by a chain of transformation matrices, which can be
read in two directions. When read from left to the right, rotations are performed around the “new”
axes; when read from the right to left, the transformations are performed around the “old” axes.
2.1.4.5

HALCON Operators

As we already anticipated at the beginning of section 2.1 on page 15, homogeneous transformation
matrices are the answer to all our questions regarding the use of 3D coordinates. Because of this, they
form the basis for HALCON’s operators for 3D transformations. Below, you find a brief overview of the
relevant operators. For more details follow the links into the Reference Manual.
• hom_mat3d_identity creates the identity transformation
• hom_mat3d_translate translates along the “old” axes: H2 = H(t) · H1

2.1 3D Transformations and Poses

C-25

• hom_mat3d_translate_local translates along the “new” axes: H2 = H1 · H(t)
• hom_mat3d_rotate rotates around the “old” axes: H2 = H(R) · H1
• hom_mat3d_rotate_local rotates around the “new” axes: H2 = H1 · H(R)
• hom_mat3d_compose multiplies two transformation matrices: H3 = H1 · H2
• hom_mat3d_invert inverts a transformation matrix: H = H -1
2

2.1.5

Basics

• affine_trans_point_3d transforms a point using a transformation matrix: p2 = H0 ·p1

3D Poses

Homogeneous transformation matrices are a very elegant means of describing transformations, but their
content, i.e., the elements of the matrix, are often difficult to read, especially the rotation part. This
problem is alleviated by using so-called 3D poses.
A 3D pose is nothing more than an easier-to-understand representation of a rigid transformation: Instead of the 12 elements of the homogeneous transformation matrix, a pose describes the rigid transformation with 6 parameters, 3 for the rotation and 3 for the translation:
(TransX, TransY, TransZ, RotX, RotY, RotZ). The main principle behind poses is that even a rotation around an arbitrary axis can always be represented by a sequence of three rotations around the axes
of a coordinate system.
In HALCON, you create 3D poses with create_pose; to transform between poses and homogeneous
matrices you can use hom_mat3d_to_pose and pose_to_hom_mat3d.
2.1.5.1

Sequence of Rotations

However, there is more than one way to represent an arbitrary rotation by three parameters. This is
reflected by the HALCON operator create_pose, which lets you choose between different pose types
with the parameter OrderOfRotation. If you pass the value ’gba’, the rotation is described by the
following chain of rotations:
Rgba = Rx (RotX) · Ry (RotY) · Rz (RotZ)

(2.19)

You may also choose the inverse order by passing the value ’abg’:
Rabg = Rz (RotZ) · Ry (RotY) · Rx (RotX)
For example, the transformation discussed in the previous sections can be represented
neous transformation matrix

cos β · cos γ − cos β · sin γ sin β


Ry (β) · Rz (γ)
t
sin γ
cos γ
0
H=
=
 − sin β · cos γ
sin β · sin γ
cos β
0 0 0
1
0
0
0

(2.20)
by the homoge
xt
yt 

zt 
1

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Basics

The corresponding pose with the rotation order ’gba’ is much easier to read:
(TransX = xt , TransY = yt , TransZ = zt , RotX = 0, RotY = 90◦ , RotZ = −90◦ )
If you look closely at figure 2.5 on page 20, you can see that the rotation can also be described by the
sequence Rz (−90◦ ) · Rx (−90◦ ). Thus, the transformation can also be described by the following pose
with the rotation order ’abg’:
(TransX = xt , TransY = yt , TransZ = zt , RotX = −90◦ , RotY = 0, RotZ = −90◦ )
2.1.5.2

HALCON Operators

Below, the relevant HALCON operators for dealing with 3D poses are briefly described. For more details
follow the links into the Reference Manual.
• create_pose creates a pose
• hom_mat3d_to_pose converts a homogeneous transformation matrix into a pose
• pose_to_hom_mat3d converts a pose into a homogeneous transformation matrix
• convert_pose_type changes the pose type
• write_pose writes a pose into a file
• read_pose reads a pose from a file
• set_origin_pose translates a pose along its “new” axes
• pose_invert inverts a pose
• pose_compose multiplies two poses, i.e., it sequentially applies two transformations (poses)
2.1.5.3

How to Determine the Pose of a Coordinate System

The previous sections showed how to describe known transformations using translation vectors, rotation
matrices, homogeneous transformation matrices, or poses. Sometimes, however, there is another task:
How to describe the position and orientation of a coordinate system with a pose.
Figure 2.8 shows how to proceed for a rather simple example. The task is to determine the pose of the
world coordinate system from figure 2.1 on page 16 relative to the camera coordinate system.
In such a case, we recommend to build up the rigid transformation from individual translations and
rotations from left to right. Thus, in figure 2.8 the camera coordinate system is first translated such that
its origin coincides with that of the world coordinate system. Now, the y-axes of the two coordinate
systems coincide; after rotating the (translated) camera coordinate system around its (new) y-axis by
180◦ , it has the correct orientation.

2.2 Camera Model and Parameters

( x c, y c , z c)

Intermediate coordinate system

t = −1.3

Ry (180°)

World coordinate system

( x w, y w, z w)

zw
zc

c’

( x c’, y c’, z c’ )

z w=c

zw
zc

x c’

zc
x w=c

y c’
z

c’

Pw = (4, −1.3, 4, 0, 180°, 0) y w=c

Figure 2.8: Determining the pose of the world coordinate system in camera coordinates.

2.2

Camera Model and Parameters

If you want to derive accurate world coordinates from your imagery, you first have to calibrate your
camera. To calibrate a camera, a model for the mapping of the 3D points of the world to the 2D image
generated by the camera, lens, and frame grabber is necessary.
HALCON supports the calibration of two different kinds of cameras: area scan cameras and line scan
cameras. While area scan cameras acquire the image in one step, line scan cameras generate the image
line by line (see Solution Guide II-A, section 6.6 on page 46). Therefore, the line scan camera must
move relative to the object during the acquisition process.
Two different types of lenses are relevant for machine vision tasks. The first type of lens effects a
perspective projection of the world coordinates into the image, just like the human eye does. With this
type of lens, objects become smaller in the image the farther they are away from the camera. This
combination of camera and lens is called a pinhole camera model because the perspective projection can
also be achieved if a small hole is drilled in a thin planar object and this plane is held parallel in front of
another plane (the image plane).
The second type of lens that is relevant for machine vision is called a telecentric lens. Its major difference
is that it effects a parallel projection of the world coordinates onto the image plane (for a certain range
of distances of the object from the camera). This means that objects have the same size in the image
independent of their distance to the camera. This combination of camera and lens is called a telecentric
camera model.
In the following, after a short overview, first the camera model for area scan cameras is described in
detail, then, the camera model for line scan cameras is explained.

2.2.1

Map 3D World Points to Pixel Coordinates

To transform a 3D point pw = (xw , yw , zw )T , which is given in world coordinates, into a 2D point
qi = (r, c)T , which is given in pixel coordinates, a chain of transformations is needed:

Basics

Camera coordinate system

C-27

C-28

Basics

pw → pc → qc → q̃c → qt → qi

(2.21)

First, pw is transformed into the camera coordinate system into pc . Then, pc is projected into the image
plane, i.e., converted to the 2D point qc , still in metric coordinates. Then, lens distortion is applied to qc ,
transforming it into the distorted point q̃c . If a tilt lens is used, q̃c only lies on a virtual image plane of a
system without tilt. This is corrected by projecting q̃c to the point qt on the tilted image plane. Finally,
the coordinates of the distorted point q̃c (or qt ) are converted to pixel coordinates, which results in the
final point qi .

2.2.2

Area Scan Cameras

Figure 2.9 displays the perspective projection effected by a pinhole camera graphically. The world point
P is projected through the optical center of the lens to the point P 0 in the image plane, which is located
at a distance of f (the focal length) behind the optical center. Actually, the term “focal length” is not
quite correct and would be appropriate only for an infinite object distance. To simplify matters, in the
following always the term “focal length” is used even if the “image distance” is meant. Note that the
focal length and thus the focus must not be changed after applying the camera calibration.
Although the image plane in reality lies behind the optical center of the lens, it is easier to pretend that
it lies at a distance of f in front of the optical center, as shown in figure 2.10. This causes the image
coordinate system to be aligned with the pixel coordinate system (row coordinates increase downward
and column coordinates to the right) and simplifies most calculations.
2.2.2.1

Transformation into Camera Coordinates (External Camera Parameters)

With this, we are now ready to describe the projection of objects in 3D world coordinates to the 2D
image plane and the corresponding camera parameters. First, we should note that the points P are given
in a world coordinate system (WCS). To make the projection into the image plane possible, they need
to be transformed into the camera coordinate system (CCS). The CCS is defined so that its x and y axes
are parallel to the column and row axes of the image, respectively, and the z axis is perpendicular to the
image plane.
The transformation from the WCS to the CCS is a rigid transformation, which can be expressed by a pose
or, equivalently, by the homogeneous transformation matrix c Hw . Therefore, the camera coordinates
pc = (xc , yc , zc )T of point P can be calculated from its world coordinates pw = (xw , yw , zw )T
simply by
pc = c Hw ·pw
(2.22)
The six parameters of this transformation (the three translations tx , ty , and tz and the three rotations α,
β, and γ) are called the external camera parameters because they determine the position of the camera
with respect to the world. In HALCON, they are stored as a pose, i.e, together with a code that describes
the order of translation and rotations.

2.2 Camera Model and Parameters

C-29

r
P’
CCD chip

Image plane coordinate system (u,v )
Image coordinate system (r , c )

u
c

Camera with
optical center

Camera coordinate system ( x c, y c , z c)

yc
zc

xw
yw

World coordinate system ( x w, y w, z w)

Figure 2.9: Perspective projection by a pinhole camera.

Basics

C-30

Basics

Camera with
optical center

Camera coordinate system ( x c, y c , z c)

yc
zc
f
Sx

Cx
Virtual image plane

c Image coordinate system (r , c )
Image plane coordinate system (u,v )

u
v

P’

xw
yw

World coordinate system ( x w, y w, z w)

Figure 2.10: Image plane and virtual image plane.

2.2.2.2

Projection

The next step is the projection of the 3D point given in the CCS into the image plane coordinate system
(IPCS). For the pinhole camera model, the projection is a perspective projection, which is given by

u
f xc
qc =
= c
(2.23)
v
z yc

2.2 Camera Model and Parameters

For the telecentric camera model, the projection is a parallel projection, which is given by
c
u
x
qc =
=
v
yc

C-31

(2.24)

As can be seen, there is no focal length f for telecentric cameras. Furthermore, the distance z of the
object to the camera has no influence on the image coordinates.
Lens Distortion

After the projection into the image plane, the lens distortion modifies the coordinates (u, v)T of qc to
q̃c = (ũ, ṽ)T . The effect is illustrated in figure 2.11: If no lens distortion were present, the projected
point P 0 would lie on a straight line from P through the optical center, indicated by the dotted line in
figure 2.11. Lens distortions cause the point P 0 to lie at a different position.
CCD chip

P’

Optical center

Figure 2.11: Schematic illustration of the effect of the lens distortion.

The lens distortion is a transformation that can be modeled in the image plane alone, i.e., 3D information
is unnecessary. In HALCON, the distortions can be modeled either by the division model or by the
polynomial model.
The division model uses one parameter (κ) to model the radial distortions. The following equations
transform the distorted image plane coordinates into undistorted image plane coordinates if the division
model is used:
u=

ũ
1 + κ(ũ2 + ṽ 2 )

and

ṽ
1 + κ(ũ2 + ṽ 2 )

(2.25)

These equations can be inverted analytically, which leads to the following equations that transform undistorted coordinates into distorted coordinates if the division model is used:

Basics

2.2.2.3

C-32

Basics

ũ =

2u
1 − 4κ(u2 + v 2 )

and

ṽ =

2v
p
1 − 4κ(u2 + v 2 )

(2.26)

Figure 2.12: Effect of radial distortions modeled with the division model with κ > 0 (left), κ = 0 (middle),
and κ < 0 (right).

The parameter κ models the magnitude of the radial distortions. If κ is negative, the distortion is barrelshaped, while for positive κ it is pincushion-shaped (see figure 2.12).
The polynomial model uses three parameters (K1 , K2 , K3 ) to model the radial distortions, and two parameters ( P1 , P2 ) to model the decentering distortions. The following equations transform the distorted
image plane coordinates into undistorted image plane coordinates if the polynomial model is used:

u = ũ + ũ(K1 r2 + K2 r4 + K3 r6 ) + 2P1 ũṽ + P2 (r2 + 2ũ2 )
v

= ṽ + ṽ(K1 r + K2 r + K3 r ) + P1 (r + 2ṽ ) + 2P2 ũṽ

(2.27)
(2.28)

√
with r = ũ2 + ṽ 2 . These equations cannot be inverted analytically. Therefore, distorted image plane
coordinates must be calculated from undistorted image plane coordinates numerically.
Some examples for the kind of distortions that can be modeled with the polynomial model are shown in
figure 2.13.
2.2.2.4

Tilt Lenses

If the lens is a tilt lens, the tilt of the lens with respect to the image plane is described by two parameters:
The rotation angle ρ (0◦ ≤ ρ < 360◦ ), which describes the direction of the tilt axis, and the tilt angle
τ (0◦ ≤ τ < 90◦ ), by which the sensor plane is tilted with respect to the optical axis (see figure 2.14).
For projective tilt lenses, the projection of q̃c = (ũ, ṽ)T into the point qt = (û, v̂)T , which lies in the
tilted image plane, is described by a projective 2D transformation, i.e., by the homogeneous matrix H:
qt = H ·q̃c
where

(2.29)

(a)

(b)

(c)

(d)

(e)

(f)

(a)
(b)
(c)
(d)
(e)
(f)

K1
<0
<0
<0
<0
<0
<0

K2
0
>0
>0
0
0
>0

K3
0
0
<0
0
0
<0

P1
0
0
0
<0
<0
<0

C-33

Basics

2.2 Camera Model and Parameters

P2
0
0
0
0
>0
>0

Figure 2.13: Effect of distortions modeled with the polynomial model with different values for the parameters K1 , K2 , K3 , P1 , and P2 .



h11
H =  h21
h31

with


q11
Q =  q21
q31

q12
q22
q32

h12
h22
h32

 
h13
q11 q33 − q13 q31
h23  =  q12 q33 − q13 q32
h33
q13 /f

 
q13
(cos ρ)2 (1 − cos τ ) + cos τ


q23
cos ρ sin ρ(1 − cos τ )
=
q33
− sin ρ sin τ

q21 q33 − q23 q31
q22 q33 − q23 q32
q23 /f


0
0 
q33

cos ρ sin ρ(1 − cos τ )
(sin ρ)2 (1 − cos τ ) + cos τ
cos ρ sin τ

(2.30)


sin ρ sin τ
− cos ρ sin τ 
cos τ
(2.31)

C-34

Basics

Figure 2.14: The tilt of the lens is described by the two parameters Tilt and Rot. Rot describes the
orientation of the tilt axis. Tilt describes the actual tilt of the lens.

For telecentric tilt lenses, the projection onto the tilted image plane is described by a linear 2D transformation, i.e., by a 2 × 2 matrix:
H=

h11
h21

h12
h22

1
=
q11 q22 − q12 q21

q22
−q21

−q12
q11

(2.32)

where Q is defined as above for projective lenses.
2.2.2.5

Transformation into Pixel Coordinates

Finally, the point q̃c = (ũ, ṽ)T (or qt if a tilt lens is present) is transformed from the image plane
coordinate system into the image coordinate system (the pixel coordinate system):


v̂ + C

y
r
S
y

qi =
=
û + C
c
x
Sx

(2.33)

Here, Sx and Sy are scaling factors. For pinhole cameras, they represent the horizontal and vertical
distance of the sensors elements on the CCD chip of the camera. For cameras with telecentric lenses,
they represent the size of a pixel in world coordinates (not taking into account the lens distortions).
The point (Cx , Cy )T is the principal point of the image. For pinhole cameras, this is the perpendicular
projection of the optical center onto the image plane, i.e., the point in the image from which a ray through
the optical center is perpendicular to the image plane. It also defines the center of the radial distortions.
For telecentric cameras, no optical center exists. Therefore, the principal point is solely defined by the
radial distortions.

2.2 Camera Model and Parameters

C-35

The parameters f, κ, K1 , K2 , K3 , P1 , P2 , τ, ρ, Sx , Sy , Cx , Cy are called the internal camera parameters
because they determine the projection from 3D to 2D performed by the camera. Depending on the
camera type, lens type, and lens distortion model, only a subset of the parameters is actually used, as the
following table shows:

’area_scan_telecentric_division’
’area_scan_tilt_division’
’area_scan_telecentric_tilt_division’
’area_scan_polynomial’
’area_scan_telecentric_polynomial’
’area_scan_tilt_polynomial’
’area_scan_telecentric_tilt_polynomial’
’line_scan’

CameraParam
[Focus, Kappa, Sx, Sy, Cx, Cy, ImageWidth,
ImageHeight]
[0, Kappa, Sx, Sy, Cx, Cy, ImageWidth,
ImageHeight]
[Focus, Kappa, Tilt, Rot, Sx, Sy, Cx, Cy,
ImageWidth, ImageHeight]
[0, Kappa, Tilt, Rot, Sx, Sy, Cx, Cy, ImageWidth,
ImageHeight]
[Focus, K1, K2, K3, P1, P2, Sx, Sy, Cx, Cy,
ImageWidth, ImageHeight]
[0, K1, K2, K3, P1, P2, Sx, Sy, Cx, Cy,
ImageWidth, ImageHeight]
[Focus, K1, K2, K3, P1, P2, Tilt, Rot, Sx, Sy, Cx,
Cy, ImageWidth, ImageHeight]
[0, K1, K2, K3, P1, P2, Tilt, Rot, Sx, Sy, Cx, Cy,
ImageWidth, ImageHeight]
[Focus, Kappa, Sx, Sy, Cx, Cy, ImageWidth,
ImageHeight, Vx, Vy, Vz]

#
8
8
10
10
12
12
14
14
11

Note that, in HALCON, the internal camera parameters always contain the focal length and, in addition
to the above described parameters, the width (ImageWidth) and the height (ImageHeight) of the image.
The differentiation among the two camera models (pinhole and telecentric) is done based on the value of
the focal length. If it has a positive value, a pinhole camera with the given focal length is assumed. If
the focal length is set to zero, the telecentric camera model is used. The differentiation among the two
models for the lens distortion (division model and polynomial model) is done based on the number of
values of the internal camera parameters. If eight or ten values are passed, the division model is used. If
twelve or fourteen values are passed, the polynomial model is used.
With this, we can see that camera calibration is the process of determining the internal camera parameter
and the external camera parameters (tx , ty , tz , α, β, γ).

2.2.3

Tilt Lenses and the Scheimpflug Principle

In a normal setup, the image plane is orthogonal to the optical axis of the lens (see figure 2.15). In case
of strong magnification, that leads to the sometimes undesired effect, that objects that are viewed from
an angle are partly out of focus, because of the limited depth of field (see figure 2.16).
To avoid this situation the lens or the image plane may be tilted to meet the Scheimpflug condition: The
image plane, the lens plane, and the focus plane intersect in a single line, the Scheimpflug line. As a
consequence, it is now possible to take images of objects that are completely in focus, even if they are
not parallel to the camera.

Basics

CameraType
’area_scan_division’

C-36

Basics

F
d

I
L

L
S
F

Figure 2.15: The Scheimpflug principle: Left: Without tilt lens the focus plane F is parallel to the image
plane I. Therefore, the object is only partly in focus if it exceeds the depth of field d. Right:
When using a tilt lens, the focus plane, the image plane, and the lens plane L are intersecting
in the Scheimpflug line S. Using this principle, a planar object can be completely in focus
even if it is viewed from an angle.

Figure 2.16: Image of a caliper taken from an angle in two different setups: Left: Without tilt lens, parts of
the object are out of focus. Right: With tilt lens, the whole object is in focus.

The use of tilt lenses is especially useful if the depth-of-field is small, e.g., due to a small field of view,
a large magnification, or a low f-number. In these cases, it might be necessary to use a tilt lens to adjust
the focus plane to better fit the observed scene.
This is typically the case in one of the following conditions:
Physical Obstacles If the space directly above the object is blocked by obstacles, e.g., some machine
parts above the conveyor belt, the camera has to be mounted in a way that the objects can only be
seen from an angle. Tilt lenses can be used to align the focus plane with the object plane.
Stereo Vision In most stereo setups, the used cameras observe the scene at different angles. That means,
when using normal lenses, their focal planes are also not parallel, i.e., each camera covers a dif-

2.2 Camera Model and Parameters

C-37

ferent volume that is in focus. With smaller depth-of-field or larger angle between the cameras,
the volume that is in focus for all cameras gets smaller, what might lead to serious problems for
the reconstruction. Tilt lenses can be used to align the focus planes of the different cameras (see
section 5.1.3).

Note that, if no tilt lens is available, in some applications it may be sufficient to increase the depth of
field by using a higher f-number, i.e., a smaller aperture. Of course that implies that either the exposure
times have to be increased (which may lead to a longer overall cycle time) or a stronger illumination has
to be used.

2.2.4

Line Scan Cameras

A line scan camera has only a one-dimensional line of sensor elements, i.e., to acquire an image, the
camera must move relative to the object (see figure 2.17). This means that the camera moves over a fixed
object, the object travels in front of a fixed camera, or camera and object are both moving.
CCD sensor line





Vx 

Motion vector Vy 
Vz 








Optical center

Figure 2.17: Principle of line scan image acquisition.

The relative motion between the camera and the object is modeled in HALCON as part of the internal
camera parameters. In HALCON, the following assumptions for this motion are made:
1. the camera moves — relative to the object — with constant velocity along a straight line

Basics

Sheet of Light In a sheet-of-light setup, a laser line is projected onto the scene, while a camera is observing the line’s reflection of the object. To get the most accurate results, it is desireable, that the
focus plane of the camera is aligned with the sheet of light emitted by the laser. This is not always
possible with normal lenses (see section 6.2).

C-38

Basics

2. the orientation of the camera is constant with respect to the object
3. the motion is equal for all images
The motion is described by the motion vector V = (Vx , Vy , Vz )T , which must be given in [meters/scanline] in the camera coordinate system. The motion vector describes the motion of the camera,
i.e., it assumes a fixed object. In fact, this is equivalent to the assumption of a fixed camera with the
object traveling along −V .
Camera with
optical center

xc
y

Camera coordinate system (xc , y c , z c )

zc
f
Virtual image plane

Cx
P′

Sensor line coordinate system (r s , cs )
Image plane coordinate system (u, v)





xw
y



−Vx
−Vy 

−Vz
World coordinate system (xw , y w , z w )

Figure 2.18: Coordinate systems in regard to a line scan camera.

The camera coordinate system of line scan cameras is defined as follows (see figure 2.18): The origin
of the coordinate system is the center of projection. The z-axis is identical to the optical axis and it is
directed so that the visible points have positive z coordinates. The y-axis is perpendicular to the sensor
line and to the z-axis. It is directed so that the motion vector has a positive y-component, i.e., if a
fixed object is assumed, the y-axis points in the direction in which the camera is moving. The x-axis is
perpendicular to the y- and z-axis, so that the x-, y-, and z-axis form a right-handed coordinate system.
Similarly to area scan cameras, the projection of a point given in world coordinates into the image is

2.2 Camera Model and Parameters

C-39

modeled in two steps: First, the point is transformed into the camera coordinate system. Then, it is
projected into the image.

The transformation from the WCS to the CCS of the first image line is a rigid transformation, which can
be expressed by a pose or, equivalently, by the homogeneous transformation matrix c Hw . Therefore,
the camera coordinates pc = (xc , yc , zc )T of point P can be calculated from its world coordinates
pw = (xw , yw , zw )T simply by
pc = c Hw ·pw
(2.34)
The six parameters of this transformation (the three translations tx , ty , and tz and the three rotations α,
β, and γ) are called the external camera parameters because they determine the position of the camera
with respect to the world. In HALCON, they are stored as a pose, i.e, together with a code that describes
the order of translation and rotations.
For line scan cameras, the projection of the point pc that is given in the camera coordinate system of the
first image line into a (sub-)pixel [r,c] in the image is defined as follows:
Assuming


x
pc =  y  ,
z


the following set of equations must be solved for m, ũ, and t:
m · D · ũ =

−m · D · pv
m·f

x − t · Vx

= y − t · Vy
= z − t · Vz

with
D
pv

1
1 + κ(ũ2 + pv 2 )
= Sy · Cy
=

This already includes the compensation for radial distortions.
Finally, the point is transformed into the image coordinate system, i.e., the pixel coordinate system:
c=

ũ
+ Cx
Sx

and r = t

Sx and Sy are scaling factors. Sx represents the distance of the sensor elements on the CCD line, Sy is
the extent of the sensor elements in y-direction. The point (Cx , Cy )T is the principal point. Note that

Basics

As the camera moves over the object during the image acquisition, also the camera coordinate system
moves relative to the object, i.e., each image line has been imaged from a different position. This means
that there would be an individual pose for each image line. To make things easier, in HALCON all
transformations from world coordinates into camera coordinates and vice versa are based on the pose of
the first image line only. The motion V is taken into account during the projection of the point pc into
the image.

C-40

Basics

in contrast to area scan images, (Cx , Cy )T does not define the position of the principal point in image
coordinates. It rather describes the relative position of the principal point with respect to the sensor line.
The nine parameters (f, κ, Sx , Sy , Cx , Cy , Vx , Vy , Vz ) of the pinhole line scan camera are called the
internal camera parameters because they determine the projection from 3D to 2D performed by the
camera.
As for area scan cameras, the calibration of a line scan camera is the process of determining
the internal camera parameters (f, κ, Sx , Sy , Cx , Cy , Vx , Vy , Vz ) and the external camera parameters
(tx , ty , tz , α, β, γ) of the first image line.

2.3 3D Object Models

2.3

C-41

3D Object Models

A 3D object model is a data structure that describes 3D objects. 3D object models can be obtained by
several means and they may contain different types of data. Additionally, the different operations that use
3D object models have different requirements concerning the model’s content. Thus, not every operation
can be applied to every 3D object model. To provide you with the basic knowledge needed to work with
3D object models, the following sections show

Basics

• how to obtain 3D object models (section 2.3.1),
• which information typically is stored in 3D object models (section 2.3.2 on page 46),
• how to modify 3D object models (section 2.3.3 on page 48),
• how to access specific features of 3D object models (section 2.3.4 on page 53),
• how to register 3D object models, i.e., how to match different models of the same object or of
overlapping object parts (section 2.3.5 on page 55), and
• how to visualize 3D object models (section 2.3.6 on page 62).

2.3.1

Obtaining 3D Object Models

The following sections give an impression on the various ways that can be used to obtain a 3D object
model. Generally, 3D object models can be
• created from scratch by explicitly setting the coordinates of points lying on the object’s surface or
by explicitly setting the parameters of a simple 3D shape (see section 2.3.1.1),
• obtained from Computer Aided Design (CAD) data (see section 2.3.1.2 on page 43), or
• derived by one of the available 3D reconstruction approaches (see section 2.3.1.3 on page 43).
2.3.1.1

Creating 3D Object Models from Scratch

3D object models can be created from scratch either by using given points that approximate the surfaces
of the objects or by using the parameters of simple 3D shapes like boxes, spheres, cylinders, or planes,
which are called “3D primitives”. In particular, the following operators are available to create a 3D object
model from scratch:
• gen_empty_object_model_3d creates an empty 3D object model that can
filled with content, e.g., with the operator set_object_model_3d_attrib
set_object_model_3d_attrib_mod as is described in section 2.3.3.2 on page 48,
• gen_object_model_3d_from_points creates a 3D object model consisting of points,
• gen_box_object_model_3d creates a box-shaped 3D primitive,

be
or

C-42

Basics

• gen_sphere_object_model_3d or gen_sphere_object_model_3d_center creates a sphereshaped 3D primitive,
• gen_cylinder_object_model_3d creates a cylinder-shaped 3D primitive, and
• gen_plane_object_model_3d creates a plane-shaped 3D primitive.
For example, in the HDevelop example program hdevelop\3D-Object-Model\Creation\
set_object_model_3d_attrib.hdev an empty 3D object model is created with
gen_empty_object_model_3d and filled with the point coordinates of a cube using
set_object_model_3d_attrib_mod (see also section 2.3.3.2 on page 48). The resulting 3D
object model, which is a simple point set, is shown in figure 2.19 on the left side.

Figure 2.19: 3D object models created from scratch: (left) 3D object model defined by point coordinates,
(right) 3D primitives defined by shape parameters.
gen_empty_object_model_3d (ObjectModel3D)
PointCoordX := [0.5,0,1,1,0,0,1,1,0] - 0.5
PointCoordY := [0,1,1,1,1,0,0,0,0] - 0.5
PointCoordZ := [0.5,0,0,1,1,0,0,1,1] - 0.5
set_object_model_3d_attrib_mod (ObjectModel3D, ['point_coord_x', \
'point_coord_y','point_coord_z'], [], \
[PointCoordX,PointCoordY,PointCoordZ])

In contrast, the HDevelop example program hdevelop\3D-Object-Model\Creation\
gen_primitives_object_model_3d.hdev creates different 3D primitives by specifying their
parameters. The resulting 3D object models are displayed in figure 2.19 on the right side.
gen_plane_object_model_3d ([0,0,0,0,0,0,0], [], [], ObjectModel3DPlane1)
gen_sphere_object_model_3d ([0,0,3,0,0,0,0], 0.5, ObjectModel3DSphere1)
gen_sphere_object_model_3d_center (-1, 0, 1, 1, ObjectModel3DSphere2)
gen_cylinder_object_model_3d ([1,-1,2,0,0,60,0], 0.5, -1, 1, \
ObjectModel3DCylinder)
gen_box_object_model_3d ([-1,2,1,0,0,90,0], 1, 2, 1, ObjectModel3DBox)

2.3 3D Object Models

2.3.1.2

C-43

Obtaining 3D Object Models from Computer Aided Design (CAD) Data

read_object_model_3d ('pipe_joint', 'm', [], [], ObjectModel3D, Status)

Figure 2.20: 3D object model obtained from a CAD model that is available as a PLY file.

2.3.1.3

Deriving 3D Object Models by 3D Reconstruction

All (calibrated) 3D reconstruction approaches are suitable to explicitly or implicitly derive a 3D object
model. For example, with multi-view stereo you can explicitly obtain a 3D object model, whereas with
a common 3D sensor X, Y, and Z images and with depth from focus depth images are obtained.
X, Y, and Z images implicitly contain the information needed for a 3D object model. Thus, you can
derive a 3D object model from X, Y, and Z images using the operator xyz_to_object_model_3d,
as is shown, e.g., in the HDevelop example program hdevelop\3D-Object-Model\Features\
select_object_model_3d.hdev (see figure 2.21).

Basics

If a 3D model of an object is already available as a Computer Aided Design (CAD) model, e.g., as a DXF
or PLY file, the model simply can be read as 3D object model with the operator read_object_model_3d
as is shown, e.g., in the HDevelop example program hdevelop\3D-Object-Model\Features\
smallest_bounding_box_object_model_3d.hdev (see figure 2.20). Please refer to the description
of the operator read_object_model_3d in the Reference Manual for the complete list of supported file
formats and their descriptions.

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Basics

ImagePath := 'time_of_flight/'
read_image (Image, ImagePath + 'engine_cover_xyz_01')
scale_image (Image, Image, .001, .0)
zoom_image_factor (Image, Image, 2, 2, 'constant')
decompose3 (Image, X, Y, Z)
xyz_to_object_model_3d (X, Y, Z, ObjectModel3DID)

Figure 2.21: 3D object model obtained from X,Y, and Z images.

If only a (calibrated) depth image, i.e., a “Z image” is available, you can create the corresponding X and
Y images as follows: The X and Y images must have the same size as the Z image. The X image is
created by assigning the column numbers of the Z image to each row of the X image and the Y image
is created by assigning the row numbers of the Z image to each column of the Y image as is illustrated
in a simple (pixel-precise) version in figure 2.22. The thus created X, Y, and Z images can then be
transformed again into a 3D object model.

X image

Y image

0
0

1
1

2
2

3
3

4
4

Figure 2.22: Partial content of (pixel-precise) X and Y images created to complete a Z image to an X, Y,
and Z image.

2.3 3D Object Models

C-45

Sheet of Light

3D Object Model (3D Points)

3D Sensor

DFF

Binocular Stereo

Multi−View Stereo

Z Image
Disparity Image
Add Artificial
X and Y Images
X, Y, Z Image

disparity_image_to_xyz

xyz_to_object_model_3d

3D Object Model (3D Points and 2D Mapping)

Surface−Based
3D Matching

prepare_object_model_3d

3D Object Model
(3D Points and
Point Normals)

3D Object Model
(3D Points and
Meshing)

3D Primitives
Fitting

Shape−Based
3D Matching
3D Pose

3D Object Model
with the Primitive’s
Parameters

Figure 2.23: Overview on the 3D object model.

3D Position Recognition

read_object_model_3d

3D Object Model (Extended Information)

Basics

3D Object Model
(the Primitive’s
Parameters)

Modifications, e.g., by triangulate_object_model_3d

gen_object_model_3d_from_points
or
gen_empty_object_model_3d and
set_object_model_3d_attrib_mod

3D Reconstruction

gen_box_object_model_3d,
gen_sphere_object_model_3d,
gen_sphere_object_model_3d_center,
gen_cylinder_object_model_3d,
or gen_plane_object_model_3d

Creation of 3D Object Models
from Scratch

Figure 2.23 guides you through the different ways how to derive a 3D object model that can be used,
e.g., for a following 3D position recognition approach.

C-46

Basics

2.3.2

Content of 3D Object Models

The following sections give an impression on
• what kind of information generally may be stored in 3D object models (see section 2.3.2.1) and
• how to query the information contained in a specific 3D object model (see section 2.3.2.2).
2.3.2.1

Kind of Information Contained in 3D Object Models

Because 3D object models can be obtained by several means, the information that is contained in 3D
object models differs from model to model. For example, if a 3D object model is created explicitly from
given points using the operator gen_object_model_3d_from_points, the basic information in this
model is a set of point coordinates. In contrast, a model that is created explicitly by the parameters of a
3D primitive contains no points but the parameters of the corresponding simple 3D shape.
Generally, the content of a specific 3D object model depends on the specific process that was used to
create or derive it. For example, if a 3D object model is derived from a (calibrated) 3D reconstruction
method like stereo vision, sheet of light, or depth from focus, it contains points. If within such a 3D
object model a 3D primitives fitting is applied, the model of the resulting 3D primitive contains points
as well as the primitive’s parameters. That is, the 3D object model of such a 3D primitive contains more
information than that of a 3D primitive that was created explicitly by its parameters.
A 3D object model that is obtained from X, Y, and Z images typically contains the coordinates of the
3D points and the corresponding 2D mapping, i.e., a mapping of the 3D points to 2D image coordinates, whereas a 3D object model that is obtained by multi-view stereo can contain a lot of further
information. For example, the surface may be approximated by triangles or polygons. A triangulation can be applied also explicitly to a 3D object model that contains points using the operator triangulate_object_model_3d. Additional operations to modify 3D object models are introduced in
section 2.3.3 on page 48.
The following lists the different kind of data that may be contained in a 3D object model:
• Points: Coordinates of the 3D points
If contained in the 3D object model, further parameters can be:
– Triangles: Indices of the 3D points that represent triangles
– Lines: Indices of the 3D points that represent polylines
– Faces: Indices of the 3D points that represent faces
– Normals: Normal vectors
– xyz Mapping: Mapping of a 3D point to image coordinates
• Primitive
If contained in the 3D object model, further parameters are:
– Primitive Type: Type of the primitive (plane, sphere, box, cylinder)
– Primitive Pose: Pose that describes the position and orientation of the primitive

2.3 3D Object Models

C-47

– Primitive Rms: Accuracy of the primitive parameters (only available if they were determined
by fitting the primitive into a point cloud)
Note that a 3D object model may contain at most one primitive.
• Extended Attributes
– Attribute Names: Names of extended attributes defined for the 3D object model

• Additional Attributes
– Shape Based Data: Flag that indicates if the 3D object model has been prepared for shapebased 3D matching
– Distance Computation: Flag that indicates if the 3D object model has been prepared for
distance compuation
The content is represented in form of attributes and can be accessed with the operator
get_object_model_3d_params. For the complete list and a detailed description of the available attributes, see the reference manual entry of get_object_model_3d_params.
2.3.2.2

Querying Information from Specific 3D Object Models

Because many approaches exist to obtain and modify a 3D object model, also many different combinations of information can be contained in the models. To query the actual content of a specific 3D object
model you can apply the operator get_object_model_3d_params. With this operator, you can query
if a specific information, a so-called ’attribute’, is contained in the model. For example, you can query if
the 3D object model contains primitive data, points, point normals, triangles, faces, or a 2D mapping. If
an attribute is contained in the model, you can query its explicit values with the same operator.
Note that here only the most common attributes were listed. The complete list of attributes that
may be contained in a 3D object model and their descriptions are provided with the operator
get_object_model_3d_params in the Reference Manual.
Another possiblity to check the actual content of a 3D object model is given with a special inspection
window in HDevelop. Here, the parametric properties of the 3D object model that are listed in the
previous section are displayed as described in the HDevelop User’s Guide, section 6.7.7 on page 193.

Basics

– Attribute Types: Types of extended attributes defined for the 3D object model

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Basics

2.3.3

Modifying 3D Object Models

The following sections show how to modify 3D object models by
• preparing them for a following 3D pose recognition (see section 2.3.3.1),
• adding attributes to their content (see section 2.3.3.2),
• reducing their content to selected attributes (see section 2.3.3.3 on page 50),
• changing their contained point sets, e.g., by a reduction of the number of points or by a selection
of points with specific characteristics (see section 2.3.3.4 on page 51),
• transforming them (see section 2.3.3.5 on page 52), or
• combining several 3D object models to a single 3D object model (see section 2.3.3.6 on page 53).
2.3.3.1

Preparing 3D Object Models for a Following 3D Pose Recognition

Some 3D position recognition approaches need 3D object models as input. Especially the shape-based
3D matching (see section 4.2 on page 111) and the segmentation that is applied in the context of a 3D
primitives fitting (see section 4.5 on page 132) need specific information that may be contained only
implicitly in the 3D object model. To prepare a 3D object model for one of these 3D position recognition approaches, i.e., to enable a faster internal access to this information, you can use the operator
prepare_object_model_3d. There, for the shape-based 3D matching the parameter Purpose must
be set to ’shape_based_matching_3d’, whereas for the 3D segmentation it must be set to ’segmentation’. Note that for a surface-based 3D matching (see section 4.3 on page 123) such a preparation
is not needed for the actual matching, but may also be required if a visualization with the operator
project_object_model_3d using one of the generic parameters ’hidden_surface_removal’ or
’min_face_angle’ is applied. Then, similar to the shape-based 3D matching the parameter Purpose
of prepare_object_model_3d must be set to ’shape_based_matching_3d’.
2.3.3.2

Adding Attributes to 3D Object Models

Sometimes, specific attributes are needed, e.g., for a following operation, that are not contained explicitly
but only implicitly in a 3D object model. Then, different operators are available that can be used to make
the implicit information explicit or to manually add or modify specific attributes, e.g.,
• set_object_model_3d_attrib and set_object_model_3d_attrib_mod can be used to
manually add attributes to a 3D object model or to modify the already contained attributes. The
main difference between both operators is that set_object_model_3d_attrib returns a new 3D
object model, whereas set_object_model_3d_attrib_mod changes the input 3D object model.
Besides standard attributes that can be obtained during the creation or modification of a 3D object
model and that are expected by various operators, so-called “extended” attributes can be set, i.e.,
new types of attributes can be defined by the user and attached to a model. These attributes must
be indicated in the parameter Name by a preceding “&”.

2.3 3D Object Models

C-49

• surface_normals_object_model_3d can be used to add the normals attribute to 3D object
models.

• fit_primitives_object_model_3d can be used to obtain the attributes that are related to 3D
primitives. In particular, it can be used to add the parameters of a 3D primitive that fits best into
the set of points given in the original 3D object model. Note that such a 3D primitives fitting is
often preceded by a segmentation of the 3D object model into different 3D object models having
similar characteristics as is described in section 4.5 on page 132.
How to add attributes using set_object_model_3d_attrib or set_object_model_3d_attrib_mod
is shown, e.g., in the HDevelop example program hdevelop\3D-Object-Model\Creation\
set_object_model_3d_attrib.hdev. There, the coordinates of the corner points of a cube were
added to an empty 3D object model (see figure 2.24, left) as was already introduced in section 2.3.1.1
on page 41. Additionally, the attribute for triangles and the corresponding triangle information (see
figure 2.24, right) is added.
T1 := [0,8,5]
T2 := [0,7,8]
T3 := [0,6,7]
T4 := [0,5,6]
T5 := [5,6,2]
T6 := [5,2,1]
T7 := [6,7,3]
T8 := [6,3,2]
T9 := [7,3,4]
T10 := [7,4,8]
T11 := [8,4,1]
T12 := [8,1,5]
T13 := [2,3,4]
T14 := [2,4,1]
set_object_model_3d_attrib (ObjectModel3D, 'triangles', [], [T1,T2,T3,T4,T5, \
T6,T7,T8,T9,T10,T11,T12,T13,T14], \
ObjectModel3D2)

Note that one of the sides of the cube contains an additional point that is needed to demonstrate how to
modify a point coordinate with the operator set_object_model_3d_attrib_mod in a following step
(see figure 2.25). Thus, this side of the cube is not represented by two but by four triangles.
PointCoordY[0,3,6,7] := [-1,0.3,-.8,-.3]
set_object_model_3d_attrib_mod (ObjectModel3D2, 'point_coord_y', [], \
PointCoordY)

Basics

• triangulate_object_model_3d can be used to add the triangles attribute to a 3D object model
that consists of points and their normals. In particular, the returned 3D object model contains a
mesh of triangles that either fits perfectly to the set of points contained in the 3D object model
(“greedy” algorithm) or that approximates the set of points (“implicit” algorithm).

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Basics

Figure 2.24: 3D object model of a cube: (left) model with point coordinates and (right) with triangles
added.

Figure 2.25: 3D object model after modifying a point coordinate.

2.3.3.3

Removing Attributes from 3D Object Models

To reduce the amount of data stored in a 3D object model, selected attributes of a 3D object
model can be removed from the model if they are not needed anymore. In particular, the operator
copy_object_model_3d can be used to copy a 3D object model such that only selected attributes are
copied to the new 3D object model. For example, in the HDevelop example program hdevelop\3DObject-Model\Segmentation\segment_object_model_3d.hdev a simultaneous segmentation and
3D primitives fitting has been applied to a 3D object model that was obtained from X, Y, and Z images.
To save memory, information like the the point coordinates and the 2D mapping, which were obtained
automatically during the creation of the 3D object model from the X, Y, and Z images, are removed from
the model by copying only the data related to the obtained 3D primitives.

2.3 3D Object Models

C-51

segment_object_model_3d (ObjectModel3DID, [ParSegmentation,ParFitting], \
[ValSegmentation,ValFitting], ObjectModel3DOutID)
for Index := 0 to |ObjectModel3DOutID| - 1 by 1
copy_object_model_3d (ObjectModel3DOutID[Index], 'primitives_all', \
CopiedObjectModel3DID)
endfor

Modifying the Point Sets of 3D Object Models

Various operators are provided that enable the modification of the 3D object model’s point set or the
segmentation of a 3D object model into different parts. For example,
• set_object_model_3d_attrib_mod can be used to directly modify point coordinates as was
shown already in section 2.3.3.2 on page 48.
• select_points_object_model_3d applies thresholds to selected attributes to reduce the set of
points to those points that lie within the specified attribute value ranges.
• reduce_object_model_3d_by_view can be used to reduce the point set to a set of points lying
within a specified 2D region that is defined for a specified 2D projection of the 3D object model.
• sample_object_model_3d can be used to achieve 3D object models with a uniform point density
with a specified distance between the points. This is suitable to generally reduce the point density,
e.g., to enable a faster following operation like a triangulation, or if the 3D object model contains
parts for which the point density is higher than for other parts of the object. The latter case
may result, e.g., from a multi-view stereo reconstruction (see section 5.4 on page 167) or from
the registration and fusion of multiple 3D object models (see section 2.3.5 on page 55). Note that,
depending on the specified distance between the points, the point density of the resulting 3D object
model may even be higher than that of the original one.
• simplify_object_model_3d can be used to reduce the number of points of triangulated 3D
object models, especially for smooth parts of the 3D object model, i.e., for parts where a high point
density is often not necessary. This may be used, for example, to speed up subsequent operator
calls by using the resulting 3D object model with reduced complexity. Typically, the point density
of the simplified 3D object model is nonuniform. The point density is higher for parts where more
points are required to represent the object’s geometry and it is lower for smoother parts. This is in
contrast to the results of sample_object_model_3d, where a uniform point density is achieved.
• smooth_object_model_3d smoothes the surface of the 3D object model. Typically, it is used to
prepare a 3D object model for a surface triangulation or to smooth noisy point data within a 3D
object model.
• segment_object_model_3d segments a 3D object model into parts with similar characteristics,
e.g., the same orientation of the normals or a similar curvature. Such a segmentation can be
applied, e.g., in the context of a 3D primitives fitting (see section 4.5 on page 132).
• connection_object_model_3d segments a 3D object model into parts that consist of connected components. Whether two components are considered as being connected depends on

Basics

2.3.3.4

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Basics

user-specified criteria, in particular attributes or distance functions and their corresponding threshold values.
For example, in the HDevelop example program hdevelop\3D-Object-Model\Features\
select_object_model_3d.hdev select_points_object_model_3d is used to select points of a
3D object model using a threshold on the z coordinates. Thus the individual parts of the engine cover
shown in figure 2.26 on the left side are seperated from their background as is shown in figure 2.26 on
the right side.
select_points_object_model_3d (ObjectModel3DID, 'point_coord_z', MinValue, \
MaxValue, ObjectModel3DIDReduced)

After excluding the background from the model, the remaining parts are further segmented into connected parts with connection_object_model_3d as is shown in figure 2.27.
connection_object_model_3d (ObjectModel3DIDReduced, 'distance_3d', 0.010, \
ObjectModel3DIDConnections)

Figure 2.26: 3D object model (left) in its initial state and (right) after removing points of the background
using a threshold.

2.3.3.5

Transforming 3D Object Models

3D object models can be spatially transformed by several means. In particular, you can transform 3D
object models by
• a rigid 3D transformation using the operator rigid_trans_object_model_3d,
• an arbitrary affine 3D transformation using the operator affine_trans_object _model_3d, or
• an
arbitrary
projective
3D
tive_trans_object_model_3d.

transformation

using

the

operator

projec-

C-53

Basics

2.3 3D Object Models

Figure 2.27: 3D object model after the segmentation into its connected components.

Note that the transformed 3D object model contains only data that can be represented by a 3D object
model and that could be transformed. For example, after an affine 3D transformation no 3D primitives
will be contained in the transformed 3D object model. For the transformation of 3D primitives only the
rigid transformation is suitable.
2.3.3.6

Combining 3D Object Models

To combine several 3D object models to a single 3D object model, you can apply the operator
union_object_model_3d. Note that the resulting model contains only those attributes that are contained in all of the input 3D object models.

2.3.4

Extracting Features of 3D Object Models

The following sections show how to
• calculate or access specific features of 3D object models and
• how to select 3D object models by their specific features.
2.3.4.1

Calculating or Accessing Features of 3D Object Models

Several kinds of features are contained explicitly or implicitly in 3D object models. On the one hand,
the 3D object models explicitly contain different attributes as was introduced in section 2.3.2 on page
46. These attributes can be related to features like point coordinates, normals, triangles, faces, or the
parameters of 3D primitives and can be accessed with the operator get_object_model_3d_params.
On the other hand, the 3D object models contain implicit information about specific geometric features,
i.e., the contained information can be used to explicitly calculate these features using one of the many
available operators that are provided by HALCON:

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Basics

• area_object_model_3d calculates the area of the surface of the 3D object model,
• distance_object_model_3d calculates the distances of the points in one 3D object model to
the points, triangles, or primitive in another 3D object model,
• max_diameter_object_model_3d calculates the maximum diameter of the 3D object model,
• moments_object_model_3d calculates the mean or the central moment of second order of the
3D object model,
• smallest_bounding_box_object_model_3d and smallest_sphere_object_model_3d
calculate the smallest bounding box or the smallest sphere surrounding the 3D object model,
• volume_object_model_3d_relative_to_plane calculates the volume of the 3D object model
relative to a plane if triangles or a list of polygons is contained in the 3D object model, and
• intersect_plane_object_model_3d calculates an intersetion between the 3D object model
and a plane and returns the cross section as a set of 3D points that are connected by lines (see, e.g.,
figure 2.28, which shows the result of the HDevelop example program hdevelop\3D-ObjectModel\Transformations\intersect_plane_object_model_3d.hdev).

Figure 2.28: (left) 3D object model of a mug and (right) its intersection with a plane.

In
the
HDevelop
example
program
hdevelop\3D-Object-Model\Features\
select_object_model_3d.hdev the maximum diameters and the volumes for the connected
components of the 3D object model that were introduced in section 2.3.3.4 on page 51 are calculated
(see figure 2.29).
volume_object_model_3d_relative_to_plane (ObjectModel3DIDConnections, [0,0, \
MaxValue,0,0,0,0], 'signed', \
'true', Volume)
max_diameter_object_model_3d (ObjectModel3DIDConnections, Diameter)

C-55

Basics

2.3 3D Object Models

Figure 2.29: Features calculated for the connected components of a 3D object model.

2.3.4.2

Selecting 3D Object Models by Their Features

If many 3D object models are available but only those 3D object models are needed that have specific
features, the operator select_object_model_3d can be applied. It selects 3D object models from an
array of 3D object models according to global features like the existance of specific attributes or specific
value ranges for features like the object’s mean diameter or the object’s volume.
For example, in the HDevelop example program hdevelop\3D-Object-Model\Features\
select_object_model_3d.hdev the maximum diameter and the volume of the connected components
of the 3D object model are used to select only those components that have a certain size. The selected
models are visualized in figure 2.30.
select_object_model_3d (ObjectModel3DTranslated, ['volume', \
'diameter_object'], 'and', [MinVolume,MinDiameter], \
[MaxVolume,MaxDiameter], ObjectModel3DSelected)

2.3.5

Matching of 3D Object Models

HALCON provides functionality for matching 3D object models that represent the same object or overlapping parts of the same object. This matching is also called “registration” of 3D object models.
The following sections
• introduce pair-wise and global registration of 3D object models (see section 2.3.5.1) and
• show how registration and further modifications of 3D object models can be used to derive a
surface model for a surface-based 3D matching (see section 2.3.5.2).

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Basics

Figure 2.30: Connected components of a 3D object model, selected by their maximum diameter and
volume.

2.3.5.1

Registering 3D Object Models

With the operator register_object_model_3d_pair you can apply a matching between two 3D object models to get the pose that describes the spatial relation between them. This process is also called
“pair-wise registration” of 3D object models. The matching determines an initial pose that is then consecutively refined so that the mutual difference between the overlapping parts of both 3D object models
becomes minimal. The final pose can be used to transform the first 3D object model into the coordinate
system of the second 3D object model. Note that if both models are already available in the same coordinate system, the initial pose between them is aready implicitly known. Then, the operator can also be
applied in a mode that performs no initial matching but only the pose refinement (Method=’icp’).
With the operator register_object_model_3d_global you can improve the relative positions between many 3D object models, which is also called “global registration” of 3D object models. In particular, if a 3D object consists of several overlapping 3D object models, the specific overlaps are used to
determine an array of homogeneous transformation matrices with which the models can be transformed
such that the relation between all models is refined, i.e., a minimal mutual difference between all 3D
object models is realized, which leads to a better representation of the complete object.
For both operators, the result of the registration is typically used to transform the initial 3D object models
with the operator affine_trans_object_model_3d.
2.3.5.2

Example Application: Deriving a Surface Model from a Set of 3D Images

The
HDevelop
example
program
hdevelop\Applications\Robot-Vision\
reconstruct_3d_object_model_for_matching.hdev shows how registration is used to derive a
unique surface model for a surface-based 3D matching (see section 4.3 on page 123) from a set of views
on the same object that are obtained by a 3D sensor. In particular, for each view on the object a 3D
object model and a corresponding gray value image are available (see figure 2.31).

2.3 3D Object Models

C-57

Step 1:

With a pair-wise registration, a following global registration, and the corresponding affine transformations, the 3D object models of the initial views on the object are aligned. In particular, a pair-wise
registration is applied for all 3D object models of successive views.
read_object_model_3d ('universal_joint_part/universal_joint_part_xyz_00.om3', \
'm', [], [], ObjectModel3D, Status)
PreviousOM3 := ObjectModel3D
RegisteredOM3s := ObjectModel3D
for Index := 1 to NumTrainingImages - 1 by 1
read_object_model_3d ('universal_joint_part/universal_joint_part_xyz_' + Index$'02d', \
'm', [], [], ObjectModel3D, Status)
register_object_model_3d_pair (ObjectModel3D, PreviousOM3, 'matching', \
'default_parameters', 'accurate', Pose, \
Score)
pose_to_hom_mat3d (Pose, HomMat3D)
RegisteredOM3s := [RegisteredOM3s,ObjectModel3D]
Offsets := [Offsets,HomMat3D]
PreviousOM3 := ObjectModel3D
endfor

For example, in figure 2.32 the corresponding images for the 3D object models of view 12 and view
13 are shown and the result of transforming both 3D object models into the same coordinate system is
displayed.
After all pair-wise registrations have been applied, all 3D object models can be transformed into the coordinate system of the first model (see figure 2.33, left). The relations between the thus transformed models
are then refined by a global registration and the corresponding affine transformations (see figure 2.33,
right).
register_object_model_3d_global (RegisteredOM3s, Offsets, 'previous', [], \
'max_num_iterations', 1, HomMat3DRefined, \
Score)
affine_trans_object_model_3d (RegisteredOM3s, HomMat3DRefined, \
GloballyRegisteredOM3s)

Basics

Figure 2.31: Some of the images obtained together with the corresponding 3D object models by a 3D
sensor.

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Basics

Figure 2.32: Pair-wise registration of two 3D object models: (top) images corresponding to the registered
3D object models, (bottom) visualization of the pair-wise registered and transformed 3D
object models.

Figure 2.33: The 3D object models (left) after the successively applied pair-wise registration and (right)
after the global registration.

Step 2:

Extract surface model from the registered 3D object models

The registered 3D object models consist of different structures, in particular of a universal joint, i.e., the
actual object that should be used as surface model, and parts of the background. The extraction of the
surface model from the set of aligned 3D object models is done in several steps. First, the models are
combined into a single 3D object model using union_object_model_3d (see figure 2.34, left). To get
an object with a smaller and homogeneous point density, sample_object_model_3d is applied (see
figure 2.34, right).

2.3 3D Object Models

C-59

After that, the object is further smoothed and the surface normals are calculated. In this example, the
object is moved temporarily, so that the origin of the coordinate system lies below the object. This way,
in combination with the ’mls_force_inwards’ option of smooth_object_model_3d, the normals of the
smoothed object model will point downwards making the surface-based 3D matching more robust.
get_object_model_3d_params (SampleExact, 'center', Center)
get_object_model_3d_params (SampleExact, 'bounding_box1', BoundingBox)
hom_mat3d_identity (HomMat3DTrans)
hom_mat3d_translate_local (HomMat3DTrans, -Center[0], -Center[1], \
-BoundingBox[2], HomMat3DTranslate)
affine_trans_object_model_3d (SampleExact, HomMat3DTranslate, \
SampleExactTrans)
smooth_object_model_3d (SampleExactTrans, 'mls', 'mls_force_inwards', \
'true', SmoothObject3DTrans)
hom_mat3d_invert (HomMat3DTranslate, HomMat3DInvert)
affine_trans_object_model_3d (SmoothObject3DTrans, HomMat3DInvert, \
SmoothObject3D)

Figure 2.34: The 3D object models (left) united into a single 3D object model and (right) subsampled and
smoothed.

Then, the resulting model is triangulated with triangulate_object_model_3d and the connected
components are determined with connection_object_model_3d (see figure 2.35).
triangulate_object_model_3d (SmoothObject3D, 'greedy', [], [], Surface3D, \
Information)
connection_object_model_3d (Surface3D, 'mesh', 1, ObjectModel3DConnected)

Basics

union_object_model_3d (GloballyRegisteredOM3s, 'points_surface', \
UnionOptimized)
MinNumPoints := 5
SampleDistance := 0.5
sample_object_model_3d (UnionOptimized, 'accurate', SampleDistance, \
'min_num_points', MinNumPoints, SampleExact)

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Basics

Figure 2.35: 3D object model (left) after the triangulation and (right) after the segmentation into connected
components.

To separate the component that should be used as model for the surface-based 3D matching from the
background, select_object_model_3d is applied with common shape features (see figure 2.36).
select_object_model_3d (ObjectModel3DConnected, ['has_triangles', \
'num_triangles'], 'and', [1,2000], [1,100000], \
ObjectModel3DSelected)
select_object_model_3d (ObjectModel3DSelected, ['central_moment_2_x', \
'central_moment_2_y'], 'and', [150,200], [400,230], \
ObjectModel3DCross)

Figure 2.36: 3D object model selected to be used as surface model for a surface-based 3D matching.

2.3 3D Object Models

Step 3:

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Use the surface model for a surface-based 3D matching

Basics

The obtained 3D object model is now used for a surface-based 3D matching. The matching is demonstrated first on the data that was used to create the surface model, i.e., on the initial views on the model
object that were obtained by a 3D sensor (see figure 2.37). Then, a matching is applied in multi-view
stereo images representing different views on a set of universal joints (see figure 2.38). For detailed information about surface-based 3D matching and multi-view stereo reconstruction, please refer to section 4.3
on page 123 and section 5.4 on page 167.

Figure 2.37: Result of surface-based 3D matching in one of the views of the training data.

Figure 2.38: Result of surface-based 3D matching in a scene obtained from multi-view stereo images.

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Basics

2.3.6

Visualizing 3D Object Models

For the visualization of 3D object models, different operators and procedures are available.
• Visualization of individual 3D object models with disp_object_model_3d.
– Visualization of individual 3D object models without the need to set camera parameters and
pose.
– disp_object_model_3d provides an easy to use interface for the visualization of one or a
small number of 3D object models.
• Visualization of multiple 3D object models, a so called ’3D scene’, with display_scene_3d.
– Best choice if multiple 3D object models should be displayed.
– Multiple cameras can be defined to view the 3D scene from different positions.
– Labels can be attached to specific 3D points.
– Smooth display of dynamic scenes since the handle can manage the caching of the objects
on the graphics card.
• Interactive visualization of 3D object models with the procedure visualize_object_model_3d.
– Visualization of 3D object models with the possibility to interactively change their position
and orientation.
Note that the visualization of 3D object models requires OpenGL 2.1, GLSL 1.2, and the OpenGL
extensions GL_EXT_framebuffer_object and GL_EXT_framebuffer_blit. If these requirements are not
fulfilled, e.g., because of the use of Windows Remote Desktop or SSH forwarding, a compatibility mode
with less requirements but lower quality is automatically enabled.
2.3.6.1

Visualization of individual 3D object models

To simply display 3D object models within a window on the screen, the operator
disp_object_model_3d can be used. Here, camera parameters and pose can be specified to
define the view on the objects. In addition, with the operator get_disp_object_model_3d_info you
can then query specific information for each position in the displaying window. In particular, for each
row and column pair, you can query the index of the 3D object model that is displayed in this pixel or
the depth of the 3D object model in this pixel (measured from the camera to the projected point).
The HDevelop example program disp_object_model_3d.hdev shows how
disp_object_model_3d, e.g., to display three 3D object models with different colors:

use

disp_object_model_3d (WindowHandle, ObjectModel3DIDs, CameraParam, ObjPoses, \
'colored', 12)

Note that the simplest way to call disp_object_model_3d just requires the handles of the 3D object
model and the window, in which it should be displayed:

2.3 3D Object Models

C-63

disp_object_model_3d (WindowHandle, ObjectModel3D, [], [], [], [])

In this case, the pose and the camera parameters are determined internally such that the complete object
is visible.

If the contours of a 3D object model for a specific view on the object are needed,
project_object_model_3d can be used.
2.3.6.2

Visualization of multiple 3D object models using handles

Another way of displaying 3D data are the 3D scene operators.
The HDevelop example program display_scene_3d.hdev shows how to create and display a 3D scene.
First a scene is created via create_scene_3d, which enables the visualization of 3D objects.
create_scene_3d (Scene3D)

A scene needs at least one camera. A camera can be added to the scene with the operator
add_scene_3d_camera. Optionally, the pose of the camera can be set.
add_scene_3d_camera (Scene3D, CameraParam, CameraIndex)
set_scene_3d_camera_pose (Scene3D, CameraIndex, [0,0,-0.4,0,0,0,0])

Optionally, a light source can be selected for the scene. The light source can be added to the scene with
the operator add_scene_3d_light and its properties, e.g., the RGB color of its diffuse part, can be set
with set_scene_3d_light_param.
add_scene_3d_light (Scene3D, [1.0,1.0,1.0], 'point_light', LightIndex)
set_scene_3d_light_param (Scene3D, LightIndex, 'diffuse', [0.8,0.8,0.8])
set_scene_3d_light_param (Scene3D, LightIndex, 'ambient', [0.2,0.2,0.2])

Now objects can be added to the scene using add_scene_3d_instance. This operator requires a 3D
object model and its pose relative to the coordinate system of the 3D scene. Note that the pose of the 3D
scene in the world coordinate system can be set with set_scene_3d_to_world_pose.
add_scene_3d_instance (Scene3D, ObjectModel3D, Pose1,
set_scene_3d_instance_param (Scene3D, Instance1Index,
add_scene_3d_instance (Scene3D, ObjectModel3D, Pose2,
set_scene_3d_instance_param (Scene3D, Instance2Index,

Optionally, labels can be attached to specific 3D points.

Instance1Index)
'alpha', 0.7)
Instance2Index)
'color', '#f28f26')

Basics

If a view like that generated by disp_object_model_3d should not only be displayed but stored in an
image, you can use render_object_model_3d instead.

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Basics

add_scene_3d_label (Scene3D, 'Instance 1', Pose1[0:2], 'bottom_left', \
'point', Instance1LabelID)
add_scene_3d_label (Scene3D, 'Instance 2', Pose2[0:2], 'top', 'point', \
Instance2LabelID)

The 3D objects can then be displayed using the operator display_scene_3d.
display_scene_3d (WindowHandle, Scene3D, CameraIndex)

An image of the scene can also be rendered with render_scene_3d.
render_scene_3d (Image, Scene3D, CameraIndex)

Finally, to free the allocated memory, the scene needs to be deleted.
clear_scene_3d (Scene3D)

2.3.6.3

Interactive visualization of 3D object models

A convenient way to visualize 3D object models such that the object can be moved and rotated interactively is provided by the external procedure visualize_object_model_3d.

Metric Measurements in a Specified Plane With a Single Camera

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Metric Measurements in a
Specified Plane With a Single
Camera
In HALCON it is easy to obtain undistorted measurements in world coordinates from images. In general,
this can only be done if two or more images of the same object are taken at the same time with cameras
at different spatial positions. This is the so-called stereo approach (see chapter 5 on page 139).
In industrial inspection, we often have only one camera available and time constraints do not allow us
to use the expensive process of finding corresponding points in the stereo images (the so-called stereo
matching process).
Nevertheless, it is possible to obtain measurements in world coordinates for objects acquired through
telecentric lenses and objects that lie in a known plane, e.g., on an assembly line, for pinhole cameras.
Both of these tasks can be solved by intersecting an optical ray (also called line of sight) with a plane.
With this, it is possible to measure objects that lie in a plane, even when the plane is tilted with respect
to the optical axis. The only prerequisite is that the camera has been calibrated. In HALCON, the
calibration process is very easy as can be seen in the example described in section 3.1, which introduces
the operators that are necessary for the calibration process.
The easiest way to perform the calibration is to use the HALCON standard calibration plates. You just
need to take a few images of a calibration plate (see figure 3.1 for examples), where in one image the
calibration plate has been placed directly on the measurement plane.
The sections that follow the example show how to
• calibrate a camera (section 3.2 on page 68),
• transform image into world coordinates (section 3.3 on page 86), and
• rectify images, i.e., remove perspective and/or lens distortions from an image (section 3.4 on page
91).
Finally, we briefly look at the case of inspecting a non-planar object (section 3.5 on page 99).

Single Camera

Chapter 3

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Metric Measurements in a Specified Plane With a Single Camera

Figure 3.1: The HALCON calibration plates.

3.1

First Example

The HDevelop example solution_guide\3d_vision\camera_calibration_multi_image.hdev
shows how easy it is to calibrate a camera and use the calibration results to transform measurements into
3D world coordinates.
First, we specify general parameters for the calibration.
create_calib_data ('calibration_object', 1, 1, CalibDataID)
set_calib_data_cam_param (CalibDataID, 0, 'area_scan_division', StartCamPar)
set_calib_data_calib_object (CalibDataID, 0, 'calplate_80mm.cpd')

Then, images of the calibration plate are read. With the operator find_calib_object, the calibration
plate is searched, the contours and centers of the marks are extracted, and the pose of the calibration
plate is estimated. The obtained information is stored in the calibration data model.
for I := 1 to NumImages by 1
read_image (Image, ImgPath + 'calib_image_' + I$'02d')
find_calib_object (Image, CalibDataID, 0, 0, I, [], [])
endfor

Now, we perform the actual calibration with the operator calibrate_cameras.
calibrate_cameras (CalibDataID, Errors)

Afterwards, we can access the calibration results, i.e., the internal camera parameters and the pose of the
calibration plate in a reference image.
get_calib_data (CalibDataID, 'camera', 0, 'params', CamParam)
get_calib_data (CalibDataID, 'calib_obj_pose', [0,1], 'pose', Pose)

3.1 First Example

C-67

This pose is used as the external camera parameters, i.e., the pose of the 3D world coordinate system in
camera coordinates. In the example, the world coordinate system is located on the ruler (see figure 3.2).
To compensate for the thickness of the calibration plate, the pose is moved by the corresponding value.
set_origin_pose (Pose, 0, 0, 0.002, Pose)

read_image (Image, ImgPath + 'ruler')
gen_measure_rectangle2 (690, 680, rad(-0.25), 480, 8, 1280, 960, 'bilinear', \
MeasureHandle)
measure_pairs (Image, MeasureHandle, 0.5, 5, 'all', 'all', RowEdgeFirst, \
ColumnEdgeFirst, AmplitudeFirst, RowEdgeSecond, \
ColumnEdgeSecond, AmplitudeSecond, IntraDistance, \
InterDistance)
Row := (RowEdgeFirst + RowEdgeSecond) / 2.0
Col := (ColumnEdgeFirst + ColumnEdgeSecond) / 2.0

With the internal and external camera parameters, the measurement results are transformed into 3D world
coordinates with the operator image_points_to_world_plane.
image_points_to_world_plane (CamParam, Pose, Row, Col, 'mm', X1, Y1)

Figure 3.2: After the calibration, marks on the ruler are measured and the results are transformed into 3D
world coordinates with the calibration results.

Single Camera

Now, we perform the measurement. For that, we have acquired an additional image of the ruler without
occlusions by the calibration plate.

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Metric Measurements in a Specified Plane With a Single Camera

3.2

3D Camera Calibration

This section describes the process of 3D camera calibration in detail. Note that you can easily calibrate your camera with the help of HDevelop’s Calibration Assistant (see the HDevelop User’s Guide,
section 7.2 on page 246 for more details).
The calibration process consists of three steps:
• the preparation,
• the calibration, and
• the access to the results.
After accessing the results, you can store them and then release the memory of the calibration data model
(see section 3.2.9 on page 85).
Preparations Before Performing the Calibration
All information for the calibration is passed in the so-called calibration data model. In particular, you
• create the model and specify basic information like the number of cameras to calibrate (section 3.2.1),
• specify initial values for the internal camera parameters (section 3.2.2),
• describe the calibration object (section 3.2.3 on page 74),
• observe the calibration object in multiple poses (images) and store the extracted information (section 3.2.4 on page 78), and
• optionally restrict the calibration to certain parameters, keeping the others fixed (section 3.2.5 on
page 81).
Performing the Actual Calibration
In HALCON, you calibrate single or multiple cameras with the operator calibrate_cameras, which
needs the calibration data model as input (see section 3.2.6 on page 81 for more information).
Accessing the Results of the Calibration
calibrate_cameras again stores its results, in particular the calibrated camera parameters and poses
of the calibration objects, in the calibration data model. You can access them (and all other calibration
parameters) with the operator get_calib_data (see section 3.2.7 on page 81).

3.2 3D Camera Calibration

3.2.1

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Creating the Calibration Data Model

You create a calibration data model with the operator create_calib_data, specifying the number of
cameras and the numbers of calibration objects. When using a single camera, you typically use a single
calibration object as well.
create_calib_data ('calibration_object', 1, 1, CalibDataID)

Then, you proceed to
• specify initial values for the internal camera parameters (section 3.2.2) and

3.2.2

Specifying Initial Values for the Internal Camera Parameters

You set internal camera parameters with the operator set_calib_data_cam_param.
StartCamPar := [0.012,0,0.00000375,0.00000375,640,480,1280,960]
set_calib_data_cam_param (CalibDataID, 0, 'area_scan_division', StartCamPar)

Besides the calibration data model, the operator needs the following parameters as input:
• CameraIdx: the index of the camera (0 for a single camera),
• CameraType: the camera type, and
• CameraParam: a tuple with values for the internal camera parameters.
Below, you find
• a list of the supported camera types and their parameters (section 3.2.2.1),
• tips how to choose the suitable distortion model (section 3.2.2.2),
• special tips for area scan cameras (section 3.2.2.3 on page 71), and
• special tips for line scan cameras (section 3.2.2.4 on page 72).
3.2.2.1

Camera Type and Camera Parameters

The values of CameraType and the sequence of the values of CameraParam depend on the used camera,
lens type, and the selected model for the lens distortion. Please, refer to section 2.2 on page 27 for the
description of the underlying camera models and to section 3.2.2.2 for tips that help you to decide which
distortion model to use. The values of CameraType and CameraParam are set as follows:

Single Camera

• describe the calibration object (section 3.2.3 on page 74).

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Metric Measurements in a Specified Plane With a Single Camera

CameraType
’area_scan_division’
’area_scan_telecentric_division’
’area_scan_tilt_division’

CameraParam
[Focus, Kappa, Sx, Sy, Cx, Cy, ImageWidth, ImageHeight]
[0, Kappa, Sx, Sy, Cx, Cy, ImageWidth, ImageHeight]
[Focus, Kappa, Tilt, Rot, Sx, Sy, Cx, Cy, ImageWidth,
ImageHeight]
’area_scan_telecentric_tilt_division’
[0, Kappa, Tilt, Rot, Sx, Sy, Cx, Cy, ImageWidth,
ImageHeight]
’area_scan_polynomial’
[Focus, K1, K2, K3, P1, P2, Sx, Sy, Cx, Cy, ImageWidth,
ImageHeight]
’area_scan_telecentric_polynomial’
[0, K1, K2, K3, P1, P2, Sx, Sy, Cx, Cy, ImageWidth,
ImageHeight]
’area_scan_tilt_polynomial’
[Focus, K1, K2, K3, P1, P2, Tilt, Rot, Sx, Sy, Cx, Cy,
ImageWidth, ImageHeight]
’area_scan_telecentric_tilt_polynomial’ [0, K1, K2, K3, P1, P2, Tilt, Rot, Sx, Sy, Cx, Cy,
ImageWidth, ImageHeight]
’line_scan’
[Focus, Kappa, Sx, Sy, Cx, Cy, ImageWidth, ImageHeight,
Vx, Vy, Vz]
Note that in addition to the internal camera parameters, the width (ImageWidth) and height
(ImageHeight) of the image must be given.
Also note that for all operators that use internal camera parameters as input, the parameter values are
checked as to whether they fullfill certain restrictions. However, these restrictions may differ slightly for
some operators. Please see the reference manual entry of the operator calibrate_cameras for details
about the respective restrictions.
For most of the internal camera parameters, initial values can be determined from the specifications of
the CCD sensor and the lens. The following sections contain additional tips on how to find suitable initial
values for
• area scan cameras (section 3.2.2.3) and
• line scan cameras (section 3.2.2.4 on page 72).
3.2.2.2

Which Distortion Model to Use

For area scan cameras, two distortion models can be used: The division model and the polynomial model.
The division model uses one parameter to model the radial distortions while the polynomial model uses
five parameters to model radial and decentering distortions (see section 2.2.2 on page 28).
The advantages of the division model are that the distortions can be applied faster, especially the inverse
distortions, i.e., if world coordinates are projected into the image plane. Furthermore, if only few calibration images are used or if the field of view is not covered sufficiently, the division model typically
yields more stable results than the polynomial model. The main advantage of the polynomial model is
that it can model the distortions more accurately because it uses higher order terms to model the radial
distortions and because it also models the decentering distortions. Note that the polynomial model cannot be inverted analytically. Therefore, the inverse distortions must be calculated iteratively, which is
slower than the calculation of the inverse distortions with the (analytically invertible) division model.

3.2 3D Camera Calibration

C-71

Typically, the division model should be used for the calibration. If the accuracy of the calibration is not
high enough, the polynomial model can be used. But note that the calibration sequence used for the
polynomial model must provide a complete coverage of the area in which measurements will later be
performed. The distortions may be modeled inaccurately outside of the area that was covered by the
calibration plate. This holds for the image border as well as for areas inside the field of view that were
not covered by the calibration plate.
3.2.2.3

Tips for Area Scan Cameras

Focus f:

The initial value is the nominal focal length of the used lens, e.g., 0.016 m. For
telecentric cameras you must pass 0.

κ:

Use 0.0 as initial value (omitted if the polynomial model is used).
Or:

K1, K2, K3,
P1, P2:

Use the initial value 0.0 for each of the five coefficients (omitted if the division
model is used).

Tilt and Rot:

The tilt angle τ (0◦ ≤ τ ≤ 90◦ ) describes the angle by which the optical axis is
tilted with respect to the normal of the sensor plane. The rotation angle ρ (0◦ ≤ ρ <
360◦ ) describes the direction in which the optical axis is tilted. These parameters
are only used if a tilt lens is part of the camera setup. These angles are typically
roughly known based on the considerations that led to the use of the tilt lens or can
be read off from the mechanism by which the lens is tilted.

Sx and Sy:

For pinhole cameras, this corresponds to the horizontal and vertical distance between two neighboring cells on the sensor. For telecentric cameras, this represents
the horizontal and vertical size of a pixel in world coordinates (i.e., the distance of
the cells on the sensor divided by the magnification of the lens). Since in most
cases the image signal is sampled line-synchronously, Sy is determined by the dimension of the sensor and does not need to be estimated for pinhole cameras by the
calibration process.
The initial values depend on the dimensions of the used chip of the camera. See
the technical specification of your camera for the actual values. Attention: These
values increase if the image is subsampled!

Cx and Cy:

Initial values for the coordinates of the principal point are the coordinates of the
image center, i.e., half the image width and half the image height. Notice: The
values of Cx and Cy decrease if the image is subsampled! Appropriate initial values
are, for example:
Cx
Cy

Full image (640*480)
320.0
240.0

Subsampling (320*240)
160.0
120.0

Single Camera

In the following, some hints for the determination of the initial values for the internal camera parameters
of an area scan camera are given:

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Metric Measurements in a Specified Plane With a Single Camera

ImageWidth
and
ImageHeight:

These two parameters are set by the used frame grabber and therefore are not calibrated. Appropriate initial values are, for example:
ImageWidth
ImageHeight

3.2.2.4

Full image (640*480)
640
480

Subsampling (320*240)
320
240

Tips for Line Scan Cameras

In the following, some hints for the determination of the initial values for the internal camera parameters
of a line scan camera are given:
Focus f:

The initial value is the nominal focal length of the used lens, e.g., 0.035 m.

κ:

Use 0.0 as initial value.

Sx:

The horizontal distance between two neighboring sensor elements can be taken
from the technical specifications of the camera. Typical initial values are 7·10−6
m, 10·10−6 m, and 14·10−6 m. Notice: The value of Sx increases if the image is
subsampled. Note also that Sx will not be calibrated for line scan cameras because
Sx and Focus cannot be determined simultaneously for line scan cameras.

Sy:

The size of a cell in the direction perpendicular to the sensor line can also be taken
from the technical specifications of the camera. Typical initial values are 7·10−6
m, 10·10−6 m, and 14·10−6 m. Notice: The value of Sy increases if the image is
subsampled. Note also that Sy will not be calibrated for line scan cameras because
it cannot be determined separately from the parameter Cy.

Cx:

The initial value for the x-coordinate of the principal point is half the image width.
Notice: The values of Cx decreases if the image is subsampled! Appropriate initial
values are:
Image width:
Cx:

1024
512

2048
1024

4096
2048

8192
4096

Cy:

Normally, the initial value for the y-coordinate of the principal point can be set to
0.

ImageWidth
and
ImageHeight:

These two parameters are determined by the used frame grabber and therefore are
not calibrated.

3.2 3D Camera Calibration

The initial values for the x-, y-, and z-component of the motion vector depend on
the image acquisition setup. Assuming a fixed camera that looks perpendicularly
onto a conveyor belt, such that the y-axis of the camera coordinate system is antiparallel to the moving direction of the conveyor belt (see figure 3.3 on page 74),
the initial values are Vx = Vz = 0. The initial value for Vy can then be determined,
e.g., from a line scan image of an object with known size (e.g., calibration plate or
ruler):
Vy
with:

= L[m]/L[row]

L[m]

= Length of the object in object coordinates [meter]

L[row]

= Length of the object in image coordinates [rows]

If, compared to the above setup, the camera is rotated 30 degrees around its optical
axis, i.e., around the z-axis of the camera coordinate system (figure 3.4 on page
75), the above determined initial values must be changed as follows:
Vz x

Vz y

Vz z

sin(30◦ ) · Vy

cos(30◦ ) · Vy
Vz = 0

If, compared to the first setup, the camera is rotated -20 degrees around the x-axis
of the camera coordinate system (figure 3.5 on page 76), the following initial values
result:
Vxx

= Vx = 0

Vxy

= cos(−20◦ ) · Vy

Vxz

= sin(−20◦ ) · Vy

The quality of the initial values for Vx, Vy, and Vz are crucial for the success of
the whole calibration. If they are not accurate enough, the calibration may fail.
Section 3.2.10.1 on page 86 provides you with tips what to do in this case.

Single Camera

Vx, Vy, Vz:

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Metric Measurements in a Specified Plane With a Single Camera

Camera with
optical center

xc
y

Camera coordinate system (xc , y c , z c )

zc
f
Virtual image plane

Cx
P′

Sensor line coordinate system (r s , cs )
Image plane coordinate system (u, v)







−Vx
−Vy 

−Vz

xw
yw

World coordinate system (xw , y w , z w )

Figure 3.3: Line scan camera looking perpendicularly onto a conveyor belt.

3.2.3

Describing the Calibration Object

With the operator set_calib_data_calib_object you specify the needed information about the calibration object.
If you are using a HALCON calibration plate, the name of the corresponding description file is sufficient.
set_calib_data_calib_object (CalibDataID, 0, 'calplate_80mm.cpd')

How to obtain a HALCON calibration plate is explained in section 3.2.3.1.
However, you can also use your own calibration object. Then, you pass the coordinates of the markers
instead of the file name to the operator (see section 3.2.3.2 on page 77 for more information).
3.2.3.1

How to Obtain a Suitable Calibration Plate

The simplest method to determine the camera parameters of a CCD camera is to use a HALCON calibration plate (see figure 3.6 on page 76 for examples). In this case, the whole process of finding the
calibration plate, extracting the calibration marks, and determining the correspondences between the
extracted calibration marks and the respective 3D world coordinates can be carried out automatically.
Even more important, these calibration plates are highly accurate, down to 1 µm or below, which is a
prerequisite for high accuracy applications. Therefore, we recommend to obtain such a calibration plate
from the local distributor from which you purchased HALCON.

3.2 3D Camera Calibration

Camera with
optical center

xc
y

C-75

Camera coordinate system (xc , y c , z c )

zc
f

Cx
P′

Virtual image plane
Cy

Sensor line coordinate system (r s , cs )
cs
Image plane coordinate system (u, v)





xw
yw



−Vx
−Vy 

−Vz
World coordinate system (xw , y w , z w )

Figure 3.4: Line scan camera rotated around the optical axis.

Two types of HALCON calibration plates are supported. In particular, calibration plates with hexagonally arranged marks and calibration marks with rectangularly arranged marks are available. The calibration plates with hexagonally arranged marks are introduced with HALCON 12 and are recommended
for most applications, as they provide the following advantages compared to the calibration plates with
rectangularly arranged marks:
• The calibration plates with rectangularly arranged marks should fill a fourth of the image area. To
cover the whole field of view including different tilts, many images (at least 10 to 20) are needed.
As calibration plates with hexagonally arranged marks contain a significantly larger number of
calibration marks and therefore can and should cover the whole image area with a single image,
less images (6 to 7) are needed to get a comparable calibration result.
• The calibration plates with rectangularly arranged marks must be completely visible in the images whereas plates with hexagonally arranged marks may protude beyond the rim of the image.
Thus, with the latter less care must be taken when placing the calibration plate in the images.
The acquisition of the calibration images becomes faster and more convenient without a loss of
robustness.
The calibration plates are available in different materials (ceramics for front light, glass for back light
applications, aluminum for very large calibration plates) and sizes. Thus, you can choose the one that is
optimal for your application. A calibration plate with hexagonally arranged marks should fill the whole
image area in a single image. A calibration plate with rectangularly arranged marks should cover at very

Single Camera

C-76

Metric Measurements in a Specified Plane With a Single Camera

Camera with
optical center

Camera coordinate system (xc , y c , z c )

yc
zc
f
Virtual image plane
Sy
Cy

Cx
v





xw
yw

P′
u



Sensor line coordinate system (r s , cs )
Image plane coordinate system (u, v)

−Vx
−Vy 

−Vz
World coordinate system (xw , y w , z w )

Figure 3.5: Line scan camera rotated around the x-axis.

Figure 3.6: HALCON calibration plates of different materials and sizes: (a) with hexagonally arranged
marks and (b) with rectangularly arranged marks.

least the ninth part of the image area. To improve the calibration quality, we recommend, however, a
plate that covers at least a quarter of the image area. Detailed information about the available materials,
sizes, and the accuracy can be obtained from your distributor.
The main differences between the calibration plates with hexagonally arranged marks and the calibration
plates with rectangularly arranged marks are:

3.2 3D Camera Calibration

C-77

• Different arrangement of the marks: For calibration plates with hexagonally arranged marks the
marks are arranged in a hexagonal lattice such that each mark has six equidistant neighbors
whereas calibration plates with rectangularly arranged marks are arranged in a rectangular grid
with equidistant rows and columns.
• Different number of contained marks: Calibration plates with hexagonally arranged marks have a
significantly larger number of marks than calibration plates with rectangularly arranged marks.

• Different origin: The coordinate system of the calibration plate with hexagonally arranged marks
is located at the center of the central mark of the first finder pattern whereas the coordinate system
of the calibration plate with rectangularly arranged marks is located in the barycenter of all marks.
• Slightly different handling: Due to the differences described above, the rules for the acquisition
of calibration images differ in some points. First of all, for calibration plates with hexagonally
arranged marks less calibration images are needed and the calibration plates do not have to be
completly visible in the image. Please refer to section 3.2.4.1 for further details.
Each calibration plate comes with a description file. Place this file in the subdirectory calib of the folder
where you installed HALCON. Then, you can use its file name directly in the operator caltab_points
(see section 3.2.4). Note that the description files for calibration plates with hexagonally arranged marks
have the file extension ’.cpd’ and those of the calibration plates with rectangularly arranged marks have
the file extension ’.descr’. Calibration plates with rectangularly arranged marks always have dark marks
on a light background, whereas calibration plates with hexagonally arranged marks usually have light
marks on a dark background. Nevertheless, also calibration plates with dark hexagonally arranged marks
on a light background are available, e.g., for back light applications. The corresponding description files
are indicated by the suffix ’dark_on_light’.
For test purposes, you can create a calibration plate yourself with the operator create_caltab for
calibration plates with hexagonally arranged marks and with the operator gen_caltab for calibration
plates with rectangularly arranged marks. Print the resulting PostScript file and mount it on a planar and
rigid surface, e.g., an aluminum plate or a solid cardboard. If you do not mount the printout on a planar
and rigid surface, you will not get meaningful results by HALCON’s camera calibration as the operators
create_caltab and gen_caltab assume that the calibration marks lie within a plane. Such self-made
calibration plates should only be used for test purposes as you will not achieve the high accuracy that can
be obtained with an original HALCON calibration plate. Note that the printing process is typically not
accurate enough to create calibration plates smaller than 16 cm for calibration plates with hexagonally
arranged marks and 3 cm for calibration plates with rectangularly arranged marks.
3.2.3.2

Using Your Own Calibration Object

With HALCON, you are not restricted to using a planar calibration object like the HALCON calibration
plate. You can use a 3D calibration object or even arbitrary characteristic points (natural landmarks).

Single Camera

• Different special patterns: The calibration plates with hexagonally arranged marks contain one
to five unique finder patterns, i.e., special mark hexagons (each consisting of a mark and its six
neighbors) where either four or six marks contain a hole. To find the calibration plate in the
image and estimate the pose of it relative to the observing camera using find_calib_object,
at least one finder pattern must be completely visible in the image. The calibration plates with
rectangularly arranged marks contain a triangular orientation mark and a surrounding frame. Here,
the complete calibration plate must be visible to find the plate and estimate its pose.

C-78

Metric Measurements in a Specified Plane With a Single Camera

The only requirement is that the 3D world position of the model points is known with high accuracy.
Then, you simply pass the 3D coordinates of all points (markers) of the calibration object as a tuple in
the parameter CalibObjDescr of set_calib_data_calib_object. All x, y, and z coordinates of all
points must be packed sequentially in the tuple in the form [X, Y, Z].
Note however, that if you use your own calibration object, you cannot use the operator
find_calib_object anymore. Instead, you must determine the 2D locations of the model points and
the correspondence to the 3D points by yourself.

3.2.4

Observing the Calibration Object in Multiple Poses (Images)

The main input data for the calibration are the so-called observations. For this, the calibration object is
placed in different poses. For each pose, the camera acquires an image. In this image, the markers of the
calibration object are extracted, and their (pixel) coordinates, together with the indices of camera, calibration object, and calibration object pose and with a tuple containing the indices of the corresponding
markers, are stored in the calibration data model.
If you are using a standard HALCON calibration plate, you can apply the extraction of the coordinates
with the operator find_calib_object, which automatically stores the obtained information, including
the coordinates of the markers and the list of marker correspondences, in the calibration data model.
for I := 1 to NumImages by 1
read_image (Image, ImgPath + 'calib_image_' + I$'02d')
find_calib_object (Image, CalibDataID, 0, 0, I, [], [])
endfor

If you are using your own calibration object, you must extract its markers and determine the
correspondences by yourself and then store the information into the calibration data model with
set_calib_data_observ_points.
The following sections contain
• rules for acquiring suitable calibration images (section 3.2.4.1) and
• additional information about the extraction of the calibration marks of a standard HALCON calibration plate (section 3.2.4.2 on page 80).
3.2.4.1

Rules for Acquiring Calibration Images

If you want to achieve accurate results, please follow the rules given in this section:
• Use a clean calibration plate.
• Cover the whole field of view with multiple images, i.e, place the calibration plate in all areas of
the field of view at least once.

3.2 3D Camera Calibration

C-79

• Vary the orientations of the calibration plate. This includes rotations around the x- and y-axis
of the calibration plate, such that the perspective distortions of the calibration pattern are clearly
visible. Without some tilted calibration plates the focal length can not be calculated properly (a tilt
angle of approximately 45 degrees is recommended).
• Use at least 6 images for calibration plates with hexagonally arranged marks and 10 – 15 images
for calibration plates with rectangularly arranged marks.
• For calibration plates with rectangularly arranged marks use an illumination where the background
is darker than the calibration plate.
• The light parts of the calibration plate should have a gray value of at least 100.
• The contrast between the light and the dark parts of the calibration plate should be more than 100
gray values.

• The images should not be overexposed (the gray values of the light parts of the image should be
strictly below 255).
• The diameter of a circle should be at least 20 pixels.
• The size of the calibration plate should be selected such that it covers the whole image for calibration plates with hexagonally arranged marks and at least a quarter of the whole image for
calibration plates with rectangularly arranged marks.
• For calibration plates with hexagonally arranged marks at least one finder pattern must be completely visible in the image. If at least two finder patterns are visible, it is possible to detect whether
the calibration plate is mirrored or not. For calibration plates with rectangularly arranged marks,
the calibration plate must be completely visible inside the image.
• The images should contain as little noise as possible.
• The images should be sharply focused, i.e., transitions between objects should be clearly delimited.
Note that a good calibration result can be obtained only for a homogeneous distribution of calibration
marks within the field of view of the camera. You can imagine the part of the 3D space that corresponds
to the field of view as a calibration volume like shown in figure 3.7. There, two poses of calibration
plates and the positions of their calibration marks, when seen from different views, are illustrated. You
can see, e.g., in the view from side 1, that large parts are not covered by marks. To get a homogeneous
distribution of the marks and thus enable a good calibration result, you have to place the calibration
plates in your other images so that the empty parts of the calibration volume are minimized for all views.
Be aware that when having very small calibration plates (compared to the field of view), this means that
it may be necessary to use significantly more than the recommended number of calibration images.
If only one image is used for the calibration process or if the orientations of the calibration plate do not
vary over the different calibration images it is not possible to determine both the focal length and the pose
of the camera correctly; only the ratio between the focal length and the distance between calibration plate
and camera can be determined in this case. Nevertheless, it is possible to measure world coordinates in
the plane of the calibration plate but it is not possible to adapt the camera parameters in order to measure
in another plane, e.g., the plane onto which the calibration plate was placed.

Single Camera

• Use an illumination where the calibration plate is homogeneous.

C-80

Metric Measurements in a Specified Plane With a Single Camera

Side 1
Top

Side 2

Side 1

Side 2

Top

Figure 3.7: Investigation of the calibration volume: (left) calibration volume with two calibration plate poses
and (right) the corresponding distribution of calibration marks when seen from different views.
For a good calibration result, the areas without calibration marks (which are especially large
in the view from side 1) have to be minimized by a cautious selection of the further calibration
plate poses.

The accuracy of the resulting world coordinates depends — apart from the measurement accuracy in the
image — very much on the number of images used for the calibration process. The more images (with
significantly different calibration plate poses) are used, the more accurate results will be achieved.
3.2.4.2

Extracting the Marks from a HALCON Calibration Plate

The operator find_calib_object searches for the calibration plate, determines the image coordinates
of the calibration marks with high precision, and stores the results in a calibration data model.
Before HALCON 11, this task was solved by the sequence of the operators find_caltab,
which searches for the calibration plate, find_marks_and_pose, which extracts the calibration
marks from the calibration plate region and precisely determines their image coordinates, and
set_calib_data_observ_points, which stores the image coordinates in a previously created calbration data model.

Note that the operators find_caltab and find_marks_and_pose can be used only for calibration
plates with rectangularly arranged marks! Additionally, they need complex parameter adjustments.
In contrast, the operator find_calib_object can be used for all standard HALCON calibration plates,
selects suitable parameters automatically and therefore is very easy to use. Additionally, the search for
the calibration plate is more robust and the resulting calibration data model leads to a more accurate
camera calibration, because it contains also the calibration mark contours, which can be accessed with

3.2 3D Camera Calibration

C-81

the operator get_calib_data_observ_contours. Because of these advantages, we recommend to
always use find_calib_object when applying a camera calibration.
Nevertheless, set_calib_data_observ_points is still needed if the calibration object is no standard
HALCON calibration plate (see section 3.2.3.2 on page 77).

3.2.5

Restricting the Calibration to Specific Parameters

If certain camera parameters are already known, you can exclude them from the calibration with the
operator set_calib_data. Analogously, you can restrict the calibration to certain parameters.
Please refer to the Reference Manual for more information and a short example.

Performing the Calibration

After preparing the calibration data model as described in the previous sections, you perform the calibration by calling the operator calibrate_cameras, using the calibration data model as input.
calibrate_cameras (CalibDataID, Errors)

As a direct result, only the calibration error is returned. It corresponds to the average distance (in pixels)
between the backprojected calibration points and their extracted image coordinates. An error of up to 0.1
pixels indicates that the calibration was successful. You can further analyse the quality of the calibration
results with the operator get_calib_data (see the Reference Manual for details).
The main results of the calibration, e.g., the internal camera parameters, are stored in the calibration data
model. How to access them is described in the following section.
If the calibration fails, please refer to section 3.2.10 on page 85 for additional information.

3.2.7

Accessing the Results of the Calibration

The main results of the operator calibrate_cameras comprise the internal camera parameters and
the pose of the calibration plate in each of the images from which the corresponding points were determined. The operator stores them in the calibration data model. You can access them with the operator
get_calib_data.
The example program solution_guide\3d_vision\camera_calibration_internal.hdev shows
how to access the internal camera parameters and write them into a file.
get_calib_data (CalibDataID, 'camera', 0, 'params', CamParam)
write_cam_par (CamParam, 'camera_parameters.dat')

The external camera parameters cannot be queried directly, because the needed information about the
world coordinate system is not stored in the calibration data model. However, if the calibration plate
was placed directly on the measurement plane, its pose can be used to easily derive the external camera
parameters, which are the pose of the measurement plane. This is described in the following section.

Single Camera

3.2.6

C-82

Metric Measurements in a Specified Plane With a Single Camera

3.2.7.1

Determining the External Camera Parameters

The external camera parameters describe the relation between the measurement plane and the camera,
i.e., only if the external parameters are known it is possible to transform coordinates from the camera coordinate system (CCS) into the coordinate system of the measurement plane and vice versa. In
HALCON, the measurement plane is defined as the plane z = 0 of the world coordinate system (WCS).
The external camera parameters can be determined in different ways:
1. Use the pose obtained from one of the calibration images in which the calibration plate is placed
directly on the measurement plane. In this case, you just need to access this pose with the operator
get_calib_data.
2. Separate the determination of the internal camera parameters from the determination of the external camera parameters by using an additional image in which the calibration plate is placed
directly on the measurement plane. Apply find_calib_object to extract the calibration marks
and the pose.
3. Determine the correspondences between 3D world points and their projections in the image by
yourself and then call vector_to_pose.
If you only need to accurately measure the dimensions of an object, regardless of the absolute position
of the object in a given coordinate system, one of the first two cases can be used.
The latter two cases have the advantage that the external camera parameters can be determined independently from the internal camera parameters. This is more flexible and might be useful if the measurements should be done in several planes from a single camera or if it is not possible to calibrate the camera
in situ.
In the following, the different cases are described in more detail.
Placing the Calibration Plate on the Measurement Plane in One of the Calibration Images
The first case is the easiest way of determining the external parameters. The calibration plate must be
placed directly on the measurement plane, e.g., the assembly line, in one of the (many) images used for
the determination of the internal parameters.
Since the pose of the calibration plate is determined by the operator calibrate_cameras, you can
simply access its pose with the operator get_calib_data. This way, internal and external parameters are determined in one single calibration step as is shown in the HDevelop example program
solution_guide\3d_vision\camera_calibration_multi_image.hdev. Here, the pose of the
calibration plate (calibration object index 0) in the first calibration image is determined. Please note
that each pose consists of seven values.
get_calib_data (CalibDataID, 'calib_obj_pose', [0,1], 'pose', Pose)

The resulting pose would be the true pose of the measurement plane if the calibration plate were infinitely
thin. Because real calibration plates have a thickness d > 0, the pose of the calibration plate is shifted
by an amount −d perpendicular to the measurement plane, i.e., along the z axis of the WCS. To correct

3.2 3D Camera Calibration

C-83

this, we need to shift the pose by d along the z axis of the WCS. To perform this shift, the operator
set_origin_pose can be used.
set_origin_pose (Pose, 0, 0, 0.002, Pose)

In general, the calibration plate can be oriented arbitrarily within the WCS as long as the spatial relation
between the calibration plate and the measurement plane is known (see figure 3.8). Then, to derive the
pose of the measurement plane from the pose of the calibration plate, a rigid transformation is necessary.
In the following example, the pose of the calibration plate is adapted by a translation along the y axis
followed by a rotation around the x axis.

Placing the Calibration Plate on the Measurement Plane in a Separate Image
If the advantages of using the HALCON calibration plate should be combined with the flexibility given
by the separation of the internal and external camera parameters the second method for the determination
of the external camera parameters can be used.
First, the camera is calibrated as described in the previous sections. This can be done, e.g., prior to the
mounting of the camera at its final usage site.
Then, after setting up the camera at its final usage site, the external parameters can be determined. The
only thing to be done is to take an additional image in which the calibration plate is placed directly on
the measurement plane. From this image the external parameters can be determined as is shown in the
HDevelop example program solution_guide\3d_vision\camera_calibration_external.hdev.
There, the internal camera parameters, the image in which the calibration plate was placed directly on
the measurement plane, and the world coordinates of the calibration marks are read from file.
read_cam_par ('camera_parameters.dat', CamParam)
read_image (Image, ImgPath + 'calib_11')

Then, the calibration marks and the pose of the calibration plate are extracted.
find_calib_object (Image, CalibDataID, 0, 0, 1, [], [])

Finally, to take the thickness of the calibration plate into account, the z value of the origin given by the
camera pose is translated by the thickness of the calibration plate.
set_origin_pose (PoseForCalibrationPlate, 0, 0, 0.00075, \
PoseForCalibrationPlate)

Note that it is very important to fix the focus of your camera if you want to separate the calibration
process into two steps as described in this section, because changing the focus is equivalent to changing
the focal length, which is part of the internal parameters.

Single Camera

pose_to_hom_mat3d (FinalPose, HomMat3D)
hom_mat3d_translate_local (HomMat3D, 0, 3.2, 0, HomMat3DTranslate)
hom_mat3d_rotate_local (HomMat3DTranslate, rad(-14), 'x', HomMat3DAdapted)
hom_mat3d_to_pose (HomMat3DAdapted, PoseAdapted)

C-84

Metric Measurements in a Specified Plane With a Single Camera

Camera with
optical center

Camera coordinate system ( x c, y c , z c)

yc
zc
f
Sx

Cx
Virtual image plane

c Image coordinate system (r , c )
Image plane coordinate system (u,v )

c
c

Hcp

Calibration plate
y cp
x cp
z cp

xw
w

Calibration plate coordinate system (x cp, y cp, z cp)

y
zw
Measurement plane (z w= 0)

World coordinate system ( x w, y w, z w)

Figure 3.8: Relation between calibration plate and measurement plane.

Using Known 3D Points and Their Corresponding Image Points
If it is necessary to perform the measurements within a given world coordinate system, the third case
for the determination of the external camera parameters can be used. Here, you need to know the 3D
world coordinates of at least three points that do not lie on a straight line. Then, you must determine the
corresponding image coordinates of the projections of these points. Now, the operator vector_to_pose
can be used for the determination of the external camera parameters.
An example for this possibility of determining the external parameters is given in the following program.
First, the world coordinates of three points are set.

3.2 3D Camera Calibration

C-85

X := [0,50,100,80]
Y := [5,0,5,0]
Z := [0,0,0,0]

Then, the image coordinates of the projections of these points in the image are determined. In this
example, they are simply set to some approximate values. In reality, they should be determined with
subpixel accuracy since they define the external camera parameters.
RCoord := [414,227,85,128]
CCoord := [119,318,550,448]

vector_to_pose (X, Y, Z, RCoord, CCoord, CamParam, 'iterative', 'error', \
FinalPose, Errors)

Again, it is very important to fix the focus of your camera because changing the focus is equivalent to
changing the focal length, which is part of the internal parameters.

3.2.8

Deleting Observations from the Calibration Data Model

To determine the effect of an observation on the calibration or to remove observations of
bad quality from the calibration data model, an observation can be deleted using the operator
remove_calib_data_observ. For acquiring calibration images of suitable quality please refer to the
rules listed in section 3.2.4.1 on page 78.
RemoveObservationIdx := 3
remove_calib_data_observ (CalibDataID, 0, 0, RemoveObservationIdx)

When performing the camera calibration again as described in section 3.2.6 on page 81, it can be noted
that the calibration error changes.

3.2.9

Saving the Results and Destroying the Calibration Data Model

After accessing the results (and perhaps storing them using the operators write_cam_par and
write_pose), you can destroy the calibration data model with the operator clear_calib_data.

3.2.10

Troubleshooting

Below, you find information for the case that the calibration of a line scan camera fails.

Single Camera

Finally, the operator vector_to_pose is called with the correspondences and the internal camera parameters.

C-86

Metric Measurements in a Specified Plane With a Single Camera

3.2.10.1

Problems With Calibrating Line Scan Cameras

In general, the procedure for the calibration of line scan cameras is identical to the one for the calibration
of area scan cameras.
However, line scan imaging suffers from a high degree of parameter correlation. For example, any small
rotation of the linear array around the x-axis of the camera coordinate system can be compensated by
changing the y-component of the translation vector of the respective pose. Even the focal length is
correlated with the scale factor Sx and with the z-component of the translation vector of the pose, i.e.,
with the distance of the object from the camera.
The consequences of these correlations for the calibration of line scan cameras are that some parameters
cannot be determined with high absolute accuracy. Nevertheless, the set of parameters is determined
consistently, what means that the world coordinates can be measured with high accuracy.
Another consequence of the parameter correlations is that the calibration may fail in some cases where
the start values for the internal camera parameters are not accurate enough. If this happens, try the
following approach: In many cases, the start values for the motion vector are the most difficult to set. To
achieve better start values for the parameters V x, V y, and V z, reduce the number of parameters to be
estimated such that the camera calibration succeeds. Try first to estimate the parameters V x, V y, V z, α,
β, γ, tx, ty, and tz by calling set_calib_data with ItemType = ’camera’, ItemIdx = ’general’,
DataName = ’calib_settings’, and DataValue = [’vx’, ’vy’, ’vz’, ’alpha’, ’beta’, ’gamma’, ’transx’,
’transy’, ’transz’] and if this does not work, try DataValue = [’vx’, ’vy’, ’vz’, ’transx’, ’transy’, ’transz’].
Then, determine the whole set of parameters using the above determined values for V x, V y, and V z as
start values.
If none of the above proposed tips works, try to determine better start values directly from the camera
setup. If possible, change the setup such that it is easier to determine appropriate start values, e.g., mount
the camera such that it looks approximately perpendicularly onto the conveyor belt (see figure 3.3 on
page 74).
If the calibration plates lie in a plane, you may get the error 8440 (“Camera calibration did not
converge”), because in this case the parameters are even more correlated. A possible solution may
be to exclude the rotation around the x axis, i.e., ’alpha’ from the calibration with the operator
set_calib_data.

3.3

Transforming Image into World Coordinates and Vice
Versa

In this section, you learn how to obtain world coordinates from images based on the calibration data. On
the one hand, it is possible to process the images as usual and then to transform the extraction results
into the world coordinate system. In many cases, this will be the most efficient way of obtaining world
coordinates. On the other hand, some applications may require that the segmentation itself must be
carried out in images that are already transformed into the world coordinate system (see section 3.4 on
page 91).
In general, the segmentation process reduces the amount of data that needs to be processed. Therefore,
rectifying the segmentation results is faster than rectifying the underlying image. What is more, it is

3.3 Transforming Image into World Coordinates and Vice Versa

C-87

often better to perform the segmentation process directly on the original images because smoothing or
aliasing effects may occur in the rectified image, which could disturb the segmentation and may lead to
inaccurate results. These arguments suggest to rectify the segmentation results instead of the images.
In the following, first some general remarks on the underlying principle of the transformation of image
coordinates into world coordinates are given. Then, it is described how to transform points, contours,
and regions into the world coordinate system. Finally, we show that it is possible to transform world
coordinates into image coordinates as well, e.g., in order to visualize information given in the world
coordinate system.

The Main Principle

Given the image coordinates of one point, the goal is to determine the world coordinates of the corresponding point in the measurement plane. For this, the line of sight, i.e., a straight line from the optical
center of the camera through the given point in the image plane, must be intersected with the measurement plane (see figure 3.9).
The calibration data is necessary to transform the image coordinates into camera coordinates and finally
into world coordinates.
All these calculations are performed by the operators of the family ..._to_world_plane.
Again, please remember that in HALCON the measurement plane is defined as the plane z = 0 with
respect to the world coordinate system. This means that all points returned by the operators of the family
..._to_world_plane have a z-coordinated equal to zero, i.e., they lie in the plane z = 0 of the world
coordinate system.

3.3.2

World Coordinates for Points

The world coordinates of an image point (r, c) can be determined using the operator
image_points_to_world_plane. In the following code example, the row and column coordinates
of pitch lines are transformed into world coordinates.
image_points_to_world_plane (CamParam, FinalPose, RowPitchLine, \
ColPitchLine, 1, X1, Y1)

As input, the operator requires the internal and external camera parameters as well as the row and column
coordinates of the point(s) to be transformed.
Additionally, the unit in which the resulting world coordinates are to be given is specified by the parameter Scale (see also the description of the operator image_to_world_plane in section 3.4.1 on page
91). This parameter is the ratio between the unit in which the resulting world coordinates are to be given
and the unit in which the world coordinates of the calibration target are given (equation 3.1).
Scale =

unit of resulting world coordinates
unit of world coordinates of calibration target

(3.1)

In many cases the coordinates of the calibration target are given in meters. In this case, it is possible to
set the unit of the resulting coordinates directly by setting the parameter Scale to ’m’ (corresponding

Single Camera

3.3.1

C-88

Metric Measurements in a Specified Plane With a Single Camera

Camera with
optical center

Camera coordinate system ( x c, y c , z c)

yc
zc
f
Sx

Cx
Virtual image plane

c Image coordinate system (r , c )
Image plane coordinate system (u,v )

P’

s
Line of

ight

xw
y

World coordinate system ( x w, y w, z w)

z ww
Measurement plane (z = 0)

Figure 3.9: Intersecting the line of sight with the measurement plane.

to the value 1.0, which could be set alternatively for the parameter Scale), ’cm’ (0.01), ’mm’ (0.001),
’microns’ (1e-6), or ’µm’ (again, 1e-6). Then, if the parameter Scale is set to, e.g., ’m’, the resulting
coordinates are given in meters. If, e.g., the coordinates of the calibration target are given in µm and the
resulting coordinates have to be given in millimeters, the parameter Scale must be set to:
Scale =

mm
1 · 10−3 m
=
= 1000
µm
1 · 10−6 m

(3.2)

3.3 Transforming Image into World Coordinates and Vice Versa

3.3.3

C-89

World Coordinates for Contours

If you want to convert an XLD object containing pixel coordinates into world coordinates, the operator contour_to_world_plane_xld can be used. Its parameters are similar to those of the operator
image_points_to_world_plane, as can be seen from the following example program.
lines_gauss (ImageReduced, Lines, 1, 3, 8, 'dark', 'true', 'bar-shaped', \
'true')
contour_to_world_plane_xld (Lines, ContoursTrans, CamParam, PoseAdapted, 1)

World Coordinates for Regions

In HALCON, regions cannot be transformed directly into the world coordinate system. Instead, you
must first convert them into XLD contours using the operator gen_contour_region_xld, then apply
the transformation to these XLD contours as described in the previous section.
If the regions have holes and if these holes would influence your further calculations, set the parameter
Mode of the operator gen_contour_region_xld to ’border_holes’. Then, in addition to the outer
border of the input region the operator gen_contour_region_xld returns the contours of all holes.

3.3.5

Transforming World Coordinates into Image Coordinates

In this section, the transformation between image coordinates and world coordinates is performed in
the opposite direction, i.e., from world coordinates to image coordinates. This is useful if you want to
visualize information given in world coordinates or it may be helpful for the definition of meaningful
regions of interest (ROI).
First, the world coordinates must be transformed into the camera coordinate system. For this, the homogeneous transformation matrix CCS HW CS is needed, which can easily be derived from the pose of the
measurement plane with respect to the camera by the operator pose_to_hom_mat3d. The transformation itself can be carried out using the operator affine_trans_point_3d. Then, the 3D coordinates,
now given in the camera coordinate system, can be projected into the image plane with the operator
project_3d_point. An example program is given in the following:
There, the world coordinates of four points defining a rectangle in the WCS are defined.
ROI_X_WCS := [-2,-2,112,112]
ROI_Y_WCS := [0,0.5,0.5,0]
ROI_Z_WCS := [0,0,0,0]

Then, the transformation matrix CCS HW CS is derived from the respective pose.
pose_to_hom_mat3d (FinalPose, CCS_HomMat_WCS)

With this transformation matrix, the world points are transformed into the camera coordinate system.

Single Camera

3.3.4

C-90

Metric Measurements in a Specified Plane With a Single Camera

affine_trans_point_3d (CCS_HomMat_WCS, ROI_X_WCS, ROI_Y_WCS, ROI_Z_WCS, \
CCS_RectangleX, CCS_RectangleY, CCS_RectangleZ)

Finally, the points are projected into the image coordinate system.
project_3d_point (CCS_RectangleX, CCS_RectangleY, CCS_RectangleZ, CamParam, \
RectangleRow, RectangleCol)

3.3.6

Compensate for Lens Distortions Only

All operators discussed above automatically compensate for lens distortions. In some cases, you might
want to compensate for lens distortions only without transforming results or images into world coordinates.
Note that in the following, only the compensation for radial distortions using the division model is
described. The compensation for radial and decentering distortions using the polynomial model is done
analogously by replacing κ = 0 with K1 = K2 = K3 = P1 = P2 = 0.
The procedure is to specify the original internal camera parameters and those of a virtual camera that
does not produce lens distortions, i.e., with κ = 0.
The easiest way to obtain the internal camera parameters of the virtual camera would be to simply set κ
to zero. This can be done directly by changing the respective value of the internal camera parameters.
CamParVirtualFixed := CamParOriginal
CamParVirtualFixed[1] := 0

Alternatively, the operator change_radial_distortion_cam_par can be used with the parameter
Mode set to ’fixed’ and the parameter Kappa set to 0.
change_radial_distortion_cam_par ('fixed', CamParOriginal, 0, \
CamParVirtualFixed)

Then,
for the rectification of the segmentation results,
the HALCON operator
change_radial_distortion_contours_xld can be used, which requires as input parameters
the original and the virtual internal camera parameters. If you want to change the lens distortion of
image coordinates (Row, Col), you can alternatively use change_radial_distortion_points.
change_radial_distortion_contours_xld (Edges, EdgesRectifiedFixed, \
CamParOriginal, CamParVirtualFixed)

The rectification of the segmentation results changes the visible part of the scene (see figure 3.10b).
To obtain virtual camera parameters such that the whole image content lies within the visible part of
the scene, the parameter Mode of the operator change_radial_distortion_cam_par must be set to
’fullsize’ (see figure 3.10c). Again, to eliminate the lens distortions, the parameter Kappa must be set to
0, or all coefficients of the polynomial model must be set to zero, respectively.

3.4 Rectifying Images

C-91

change_radial_distortion_cam_par ('fullsize', CamParOriginal, 0, \
CamParVirtualFullsize)

If the lens distortions are eliminated in the image itself using the rectification procedure described in
section 3.4.2 on page 98, the mode ’fullsize’ may lead to undefined pixels in the rectified image. The
mode ’adaptive’ (see figure 3.10d) slightly reduces the visible part of the scene to prevent such undefined
pixels.
change_radial_distortion_cam_par ('adaptive', CamParOriginal, 0, \
CamParVirtualAdaptive)

change_radial_distortion_cam_par ('preserve_resolution', CamParOriginal, 0, \
CamParVirtualPreservedResolution)

Note that this compensation for lens distortions is not possible for line scan images because of the
acquisition geometry of line scan cameras. To eliminate radial distortions from segmentation results of
line scan images, the segmentation results must be transformed into the WCS (see section 3.3.2 on page
87, section 3.3.3 on page 89, and section 3.3.4 on page 89).

3.4

Rectifying Images

For applications like blob analysis or OCR, it may be necessary to have undistorted images. Imagine
that an OCR has been trained based on undistorted image data. Then, it will not be able to recognize
characters in heavily distorted images. In such a case, the image data must be rectified, i.e., the lens and
perspective distortions must be eliminated before the OCR can be applied.

3.4.1

Transforming Images into the WCS

The operator image_to_world_plane rectifies an image by transforming it into the measurement plane,
i.e., the plane z = 0 of the WCS. The rectified image shows no lens and no perspective distortions. It
corresponds to an image captured by a camera that produces no lens distortions and that looks perpendicularly to the measurement plane.
image_to_world_plane (Image, ImageMapped, CamParam, PoseForCenteredImage, \
WidthMappedImage, HeightMappedImage, \
ScaleForCenteredImage, 'bilinear')

If more than one image must be rectified, a projection map can be determined with
the operator gen_image_to_world_plane_map, which is used analogously to the operator

Single Camera

The mode ’preserve_resolution’ (see figure 3.10e) works similar to the mode ’fullsize’ but prevents
undefined pixels by additionally increasing the size of the modified image so that the image resolution
does not decrease in any part of the image.

C-92

Metric Measurements in a Specified Plane With a Single Camera

(a)

(b)

(c)

(d)

(e)
Figure 3.10: Eliminating radial distortions: The original image overlaid with (a) edges extracted from
the original image; (b) edges rectified by setting κ to zero; (c) edges rectified with mode
’fullsize’; (d) edges rectified with mode ’adaptive’; (e) edges rectified with mode ’preserved_resolution’.

image_to_world_plane, followed by the actual transformation of the images, which is carried out
by the operator map_image.

3.4 Rectifying Images

C-93

gen_image_to_world_plane_map (Map, CamParam, PoseForCenteredImage, \
WidthOriginalImage, HeightOriginalImage, \
WidthMappedImage, HeightMappedImage, \
ScaleForCenteredImage, 'bilinear')
map_image (Image, Map, ImageMapped)

The size of the rectified image can be chosen with the parameters Width and Height for the operator
image_to_world_plane and with the parameters WidthMapped and HeightMapped for the operator
gen_image_to_world_plane_map. The size of the rectified image must be given in pixels.
The pixel size of the rectified image is specified by the parameter Scale (see also the description of
the operator image_points_to_world_plane in section 3.3.2 on page 87). This parameter is the ratio
between the pixel size of the rectified image and the unit in which the world coordinates of the calibration
target are given (equation 3.3).
pixel size of rectified image
unit of world coordinates of calibration target

(3.3)

In many cases the coordinates of the calibration targets are given in meters. In this case, it is possible to
set the pixel size directly by setting the parameter Scale to ’m’ (corresponding to the value 1.0, which
could be set alternatively for the parameter Scale), ’cm’ (0.01), ’mm’ (0.001), ’microns’ (1e-6), or ’µm’
(again, 1e-6). Then, if the parameter Scale is set to, e.g., ’µm’, one pixel of the rectified image has a
size that corresponds to an area of 1 µm × 1 µm in the world. The parameter Scale should be chosen
such that in the center of the area of interest the pixel size of the input image and of the rectified image
is similar. Large scale differences would lead to aliasing or smoothing effects. See below for examples
of how the scale can be determined.
The parameter Interpolation specifies whether bilinear interpolation (’bilinear’) should be applied
between the pixels in the input image or whether the gray value of the nearest neighboring pixel (’none’)
should be used.
The rectified image ImageWorld is positioned such that its upper left corner is located exactly at the
origin of the WCS and that its column axis is parallel to the x-axis of the WCS. Since the WCS is
defined by the external camera parameters CamPose the position of the rectified image ImageWorld can
be translated by applying the operator set_origin_pose to the external camera parameters. Arbitrary
transformations can be applied to the external camera parameters based on homogeneous transformation
matrices. See below for examples of how the external camera parameters can be set.
In figure 3.11, the WCS has been defined such that the upper left corner of the rectified image corresponds
to the upper left corner of the input image. To illustrate this, in figure 3.11, the full domain of the rectified
image, transformed into the virtual image plane of the input image, is displayed. As can be seen, the
upper left corner of the input image and of the projection of the rectified image are identical.
Note that it is also possible to define the WCS such that the rectified image does not lie or lies only partly
within the imaged area. The domain of the rectified image is set such that it contains only those pixels
that lie within the imaged area, i.e., for which gray value information is available. In figure 3.12, the
WCS has been defined such that the upper part of the rectified image lies outside the imaged area. To
illustrate this, the part of the rectified image for which no gray value information is available is displayed
dark gray. Also in figure 3.12, the full domain of the rectified image, transformed into the virtual image
plane of the input image, is displayed. It can be seen that for the upper part of the rectified image no
image information is available.

Single Camera

Scale =

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Metric Measurements in a Specified Plane With a Single Camera

If several images must be rectified using the same camera parameters the operator
gen_image_to_world_plane_map in combination with map_image is much more efficient than
the operator image_to_world_plane because the transformation must be determined only once. In
this case, a projection map that describes the transformation between the image plane and the world
plane is generated first by the operator gen_image_to_world_plane_map. Then, this map is used by
the operator map_image to transform the image very efficiently.
The following example from solution_guide\3d_vision\transform_image_into_wcs.hdev
shows how to perform the transformation of images into the world coordinate system using the
operators gen_image_to_world_plane_map together with map_image as well as the operator
image_to_world_plane.
In the first part of the example program the parameters Scale and CamPose are set such that a given
point appears in the center of the rectified image and that in the surroundings of this point the scale of
the rectified image is similar to the scale of the original image.

3.4 Rectifying Images

Camera with
optical center

C-95

Camera coordinate system ( x c, y c , z c)

yc
zc
f
c Image coordinate system (r , c )

Virtual image plane

Single Camera

World coordinate system ( x w, y w, z w)

z w Rectified image

Measurement plane ( z w= 0)

Sc
al

ni
t

gh
t

Width
xw

Figure 3.11: Projection of the image into the measurement plane.

First, the size of the rectified image is defined.
WidthMappedImage := 652
HeightMappedImage := 494

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Metric Measurements in a Specified Plane With a Single Camera

Camera with
optical center

Camera coordinate system ( x c, y c , z c)

yc
zc
f
c Image coordinate system (r , c )

Virtual image plane

xw
yw

z w Rectified image

World coordinate system ( x w, y w, z w)

Measurement plane ( z w= 0)
Figure 3.12: Projection of the image into the measurement plane with part of the rectified image lying
outside the image area.

Then, the scale is determined based on the ratio of the distance between points in the WCS and of the
respective distance in the ICS.

3.4 Rectifying Images

C-97

Now, the pose of the measurement plane is modified such that a given point will be displayed in the
center of the rectified image.
DX := CenterX - ScaleForCenteredImage * WidthMappedImage / 2.0
DY := CenterY - ScaleForCenteredImage * HeightMappedImage / 2.0
DZ := 0
set_origin_pose (Pose, DX, DY, DZ, PoseForCenteredImage)

These
calculations
are
implemented
parameters_image_to_world_plane_centered.

the

HDevelop

procedure

procedure parameters_image_to_world_plane_centered (: : CamParam, Pose,
CenterRow, CenterCol,
WidthMappedImage,
HeightMappedImage:
ScaleForCenteredImage,
PoseForCenteredImage)

which is part of the HDevelop example program
solution_guide\3d_vision\transform_image_into_wcs.hdev (see appendix A.2 on page 278).
Finally, the image can be transformed.
gen_image_to_world_plane_map (Map, CamParam, PoseForCenteredImage, \
WidthOriginalImage, HeightOriginalImage, \
WidthMappedImage, HeightMappedImage, \
ScaleForCenteredImage, 'bilinear')
map_image (Image, Map, ImageMapped)

The
second
part
of
the
example
program
solution_guide\3d_vision\
transform_image_into_wcs.hdev shows how to set the parameters Scale and CamPose such
that the entire image is visible in the rectified image.
First, the image coordinates of the border of the original image are transformed into world coordinates.

Single Camera

Dist_ICS := 1
image_points_to_world_plane (CamParam, Pose, CenterRow, CenterCol, 1, \
CenterX, CenterY)
image_points_to_world_plane (CamParam, Pose, CenterRow + Dist_ICS, \
CenterCol, 1, BelowCenterX, BelowCenterY)
image_points_to_world_plane (CamParam, Pose, CenterRow, \
CenterCol + Dist_ICS, 1, RightOfCenterX, \
RightOfCenterY)
distance_pp (CenterY, CenterX, BelowCenterY, BelowCenterX, \
Dist_WCS_Vertical)
distance_pp (CenterY, CenterX, RightOfCenterY, RightOfCenterX, \
Dist_WCS_Horizontal)
ScaleVertical := Dist_WCS_Vertical / Dist_ICS
ScaleHorizontal := Dist_WCS_Horizontal / Dist_ICS
ScaleForCenteredImage := (ScaleVertical + ScaleHorizontal) / 2.0

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Metric Measurements in a Specified Plane With a Single Camera

full_domain (Image, ImageFull)
get_domain (ImageFull, Domain)
gen_contour_region_xld (Domain, ImageBorder, 'border')
contour_to_world_plane_xld (ImageBorder, ImageBorderWCS, CamParam, Pose, 1)

Then, the extent of the image in world coordinates is determined.
smallest_rectangle1_xld (ImageBorderWCS, MinY, MinX, MaxY, MaxX)
ExtentX := MaxX - MinX
ExtentY := MaxY - MinY

The scale is the ratio of the extent of the image in world coordinates and of the size of the rectified image.
ScaleX := ExtentX / WidthMappedImage
ScaleY := ExtentY / HeightMappedImage

Now, the maximum value must be selected as the final scale.
ScaleForEntireImage := max([ScaleX,ScaleY])

Finally, the origin of the pose must be translated appropriately.
set_origin_pose (Pose, MinX, MinY, 0, PoseForEntireImage)

These calculations are implemented in the HDevelop procedure
procedure parameters_image_to_world_plane_entire (Image: : CamParam, Pose,
WidthMappedImage,
HeightMappedImage:
ScaleForEntireImage,
PoseForEntireImage)

which
is
part
of
the
example
program
solution_guide\3d_vision\
transform_image_into_wcs.hdev (see appendix A.3 on page 279).
If the object is not planar the projection map that is needed by the operator map_image may be determined by the operator gen_grid_rectification_map, which is described in section 11.3 on page
268.
If only the lens distortions should be eliminated the projection map can be determined by the operator
gen_radial_distortion_map, which is described in the following section.

3.4.2

Compensate for Lens Distortions Only

The principle of the compensation for lens distortions has already be described in section 3.3.6 on page
90.

3.5 Inspection of Non-Planar Objects

C-99

If only one image must be rectified the operator change_radial_distortion_image can be used.
It is used analogously to the operator change_radial_distortion_contours_xld described in section 3.3.6, with the only exception that a region of interest (ROI) can be defined with the parameter
Region.
change_radial_distortion_image (GrayImage, ROI, ImageRectifiedAdaptive, \
CamParOriginal, CamParVirtualAdaptive)

If more than one image must be rectified, a projection map can be determined with
the operator gen_radial_distortion_map, which is used analogously to the operator
change_radial_distortion_image, followed by the actual transformation of the images, which is
carried out by the operator map_image, described in section 3.4.1 on page 91. If a ROI is to be specified,
it must be rectified separately (see section 3.3.4 on page 89).
gen_radial_distortion_map (MapFixed, CamParOriginal, CamParVirtualFixed, \
'bilinear')
map_image (GrayImage, MapFixed, ImageRectifiedFixed)

Note that this compensation for lens distortions is not possible for line scan images because of the
acquisition geometry of line scan cameras. To eliminate radial distortions from line scan images, the
images must be transformed into the WCS (see section 3.4.1 on page 91).

3.5

Inspection of Non-Planar Objects

Note that the measurements described so far will only be accurate if the object to be measured is planar,
i.e., if it has a flat surface. If this is not the case the perspective projection of the pinhole camera (see
equation 2.23 on page 30) will make the parts of the object that lie closer to the camera appear bigger than
the parts that lie farther away. In addition, the respective world coordinates are displaced systematically.
If you want to measure the top side of objects with a flat surface that have a significant thickness that is
equal for all objects it is best to place the calibration plate onto one of these objects during calibration.
With this, you can make sure that the optical rays are intersected with the correct plane.
The displacement that results from deviations of the object surface from the measurement plane can be
estimated very easily. Figure 3.14 shows a vertical section of a typical measurement configuration. The
measurement plane is drawn as a thick line, the object surface as a dotted line. Note that the object surface
does not correspond to the measurement plane in this case. The deviation of the object surface from the
measurement plane is indicated by ∆z, the distance of the projection center from the measurement plane
by z, and the displacement by ∆r. The point N indicates the perpendicular projection of the projection
center (P C) onto the measurement plane.

Single Camera

Again, the internal parameters of the virtual camera that does not show lens distortions can be determined
by setting κ to zero for the division model or K1 , K2 , K3 , P1 , and P2 to zero for the polynomial
model (see figure 3.13b). Alternatively, the internal parameters of the virtual camera can be obtained
by using the operator change_radial_distortion_cam_par with the parameter Mode set to ’fixed’
(equivalent to setting κ or the coefficients of the polynomial model to zero; see figure 3.13b), ’adaptive’
(see figure 3.13c), ’fullsize’ (see figure 3.13d), or ’preserve_resolution’ (see figure 3.13e).

C-100

Metric Measurements in a Specified Plane With a Single Camera

(a)

(b)

(c)

(d)

(e)
Figure 3.13: Eliminating radial distortions: (a) The original image; (b) the image rectified by setting κ to
zero; (c) the image rectified with mode ’fullsize’; (d) the image rectified with mode ’adaptive’;
(e) the image rectified with mode ’preserved_resolution’.

For the determination of the world coordinates of point Q, which lies on the object surface, the optical ray from the projection center of the camera through Q0 , which is the projection of Q into the
image plane, is intersected with the measurement plane. For this reason, the operators of the family

3.5 Inspection of Non-Planar Objects

C-101

Camera
Projection center (PC)

Virtual image plane
P’ = Q’
z

∆r

Object surface
Measurement plane

Single Camera

∆z

Figure 3.14: Displacement ∆r caused by a deviation of the object surface from the measurement plane.

..._to_world_plane do not return the world coordinates of Q, but the world coordinates of point P ,
which is the perspective projection of point Q0 onto the measurement plane.
If we know the distance r from P to N , the distance z, which is the shortest distance from the projection
center to the measurement plane, and the deviation ∆z of the object’s surface from the measurement
plane, the displacement ∆r can be calculated by:
∆r = ∆z ·

r
z

(3.4)

Often, it will be sufficient to have just a rough estimate for the value of ∆r. Then, the values r, z, and
∆z can be approximately determined directly from the measurement setup.
If you need to determine ∆r more precisely, you first have to calibrate the camera. Then you have to select a point Q0 in the image for which you want to know the displacement ∆r. The transformation of Q0
into the WCS using the operator image_points_to_world_plane yields the world coordinates of point
P . Now, you need to derive the world coordinates of the point N . An easy way to do this is to transform
the camera coordinates of the projection center P C, which are (0, 0, 0)T , into the world coordinate system, using the operator affine_trans_point_3d. To derive the homogeneous transformation matrix
W CS
HCCS needed for the above mentioned transformation, first, generate the homogeneous transformation matrix CCS HW CS from the pose of the measurement plane via the operator pose_to_hom_mat3d
and then, invert the resulting homogeneous transformation matrix (hom_mat3d_invert). Because N is
the perpendicular projection of P C onto the measurement plane, its x and y world coordinates are equal
to the respective world coordinates of P C and its z coordinate is equal to zero. Now, r and z can be derived as follows: r is the distance from P to N , which can be calculated by the operator distance_pp;
z is simply the z coordinate of P C, given in the WCS.

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Metric Measurements in a Specified Plane With a Single Camera

The following HALCON program (solution_guide\3d_vision\height_displacement.hdev)
shows how to implement this approach. First, the camera parameters are read from file.
read_cam_par ('camera_parameters.dat', CamParam)
read_pose ('pose_from_three_points.dat', Pose)

Then, the deviation of the object surface from the measurement plane is set.
DeltaZ := 2

Finally, the displacement is calculated, according to the method described above.
get_mbutton (WindowHandle, RowQ, ColumnQ, Button)
image_points_to_world_plane (CamParam, Pose, RowQ, ColumnQ, 1, WCS_PX, \
WCS_PY)
pose_to_hom_mat3d (Pose, CCS_HomMat_WCS)
hom_mat3d_invert (CCS_HomMat_WCS, WCS_HomMat_CCS)
affine_trans_point_3d (WCS_HomMat_CCS, 0, 0, 0, WCS_PCX, WCS_PCY, WCS_PCZ)
distance_pp (WCS_PX, WCS_PY, WCS_PCX, WCS_PCY, r)
z := fabs(WCS_PCZ)
DeltaR := DeltaZ * r / z

Assuming a constant ∆z, the following conclusions can be drawn for ∆r:
• ∆r increases with increasing r.
• If the measurement plane is more or less perpendicular to the optical axis, ∆r increases towards
the image borders.
• At the point N , ∆r is always equal to zero.
• ∆r increases the more the measurement plane is tilted with respect to the optical axis.
The maximum acceptable deviation of the object’s surface from the measurement plane, given a maximum value for the resulting displacement, can be derived by the following formula:
∆z = ∆r ·

z
r

(3.5)

The values for r and z can be determined as described above.
If you want to inspect an object that has a surface that consists of several parallel planes you can first
use equation 3.5 to evaluate if the measurement errors stemming from the displacements are acceptable within your project or not. If the displacements are too large, you can calibrate the camera such
that the measurement plane corresponds to, e.g., the uppermost plane of the object. Now, you can derive a pose for each plane, which is parallel to the uppermost plane simply by applying the operator
set_origin_pose. This approach is also useful if objects of different thickness may appear on the
assembly line. If it is possible to classify these objects into classes corresponding to their thickness, you
can select the appropriate pose for each object. Thus, it is possible to derive accurate world coordinates
for each object.

3.5 Inspection of Non-Planar Objects

C-103

Note that if the plane in which the object lies is severely tilted with respect to the optical axis, and if the
object has a significant thickness, the camera will likely see some parts of the object that you do not want
to measure. For example, if you want to measure the top side of a cube and the plane is tilted, you will
see the side walls of the cube as well, and therefore might measure the wrong dimensions. Therefore,
it is usually best to align the camera so that its optical axis is perpendicular to the plane in which the
objects are measured. If the objects do not have significant thickness, you can measure them accurately
even if the plane is tilted.
What is more, it is even possible to derive world coordinates for an object’s surface that consists of
several non-parallel planes if the relation between the individual planes is known. In this case, you may
define the relative pose of the tilted plane with respect to an already known measurement plane.
RelPose := [0,3.2,0,-14,0,0,0]

pose_to_hom_mat3d
pose_to_hom_mat3d
hom_mat3d_compose
hom_mat3d_to_pose

(FinalPose, HomMat3D)
(RelPose, HomMat3DRel)
(HomMat3D, HomMat3DRel, HomMat3DAdapted)
(HomMat3DAdapted, PoseAdapted)

Alternatively, you can use the operators of the family hom_mat3d_..._local to adapt the pose.
hom_mat3d_translate_local (HomMat3D, 0, 3.2, 0, HomMat3DTranslate)
hom_mat3d_rotate_local (HomMat3DTranslate, rad(-14), 'x', HomMat3DAdapted)
hom_mat3d_to_pose (HomMat3DAdapted, PoseAdapted)

Now, you can obtain world coordinates for points lying on the tilted plane, as well.
contour_to_world_plane_xld (Lines, ContoursTrans, CamParam, PoseAdapted, 1)

If the object is to complex to be approximated by planes, or if the relations between the planes are
not known, it is not possible to perform precise measurements in world coordinates using the methods
described in this section. In this case, it is necessary to use two cameras and to apply the HALCON
stereo operators described in chapter 5 on page 139.

Single Camera

Then, you can transform the known pose of the measurement plane into the pose of the tilted plane.

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Metric Measurements in a Specified Plane With a Single Camera

3D Position Recognition of Known Objects

C-105

Chapter 4

3D Position Recognition of Known
Objects

The most general approach determines the pose of a known 3D object using at least three corresponding
points, i.e., points with known 3D object coordinates for which the corresponding image coordinates are
extracted. The approach is also known as “mono 3D” (section 4.1).
If a model of a known 3D object is available, 3D matching can be applied to locate the object. The
available 3D matching approaches perform a full 3D object recognition, i.e., they not only estimate a
pose but first locate the object in the respective search data. The following approaches are available:
• Shape-based 3D matching (section 4.2 on page 111) can be used to locate a complex 3D object in a single 2D image. The model of the 3D object must be available as a Computer Aided
Design (CAD) model in, e.g., DXF, OFF, or PLY format (see the Reference Manual entry of
read_object_model_3d for details about the supported formats) and the object needs “hard geometric edges” to be recognized.
• Surface-based 3D matching (section 4.3 on page 123) can be used to quickly locate a complex 3D
object in a 3D scene, i.e., in a set of 3D points that is available as a so-called 3D object model
(see also section 2.3 on page 41). The model of the 3D object must be available also as a 3D
object model and can be obtained either from a CAD model (see the Reference Manual entry of
read_object_model_3d for details about the supported formats) or from a reference 3D scene
that is obtained by a 3D reconstruction approach, e.g., stereo or sheet of light. Here, the object may
also consist of a “smooth surface”. Note that this approach is also known as “volume matching”.
If the poses of simple 3D shapes like boxes, cylinders, spheres, or planes, which are called “3D primitives”, are searched in a 3D scene that is available as a 3D object model, the 3D primitives fitting
(section 4.5 on page 132) can be used. There, the 3D scene is segmented into sub-parts so that into each

Pose Estimation

Estimating the 3D pose of an object is an important task in many application areas, e.g., during completeness checks or for 3D alignment in robot vision applications (see section 8.7.1 on page 228). HALCON
provides multiple methods to determine the position or pose of known 3D objects.

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3D Position Recognition of Known Objects

sub-part a primitive of a selected type can be fitted. For each sub-part the fitting returns the parameters
of the best fitting primitive, e.g., the pose for a fitted plane.
Sometimes, a full 3D object recognition or 3D matching is not necessary because you can estimate
the pose of the object with simpler means. For example, if the object contains a characteristic planar
part, you can estimate its 3D pose from a single image using perspective matching. Similar to the 3D
matching approaches, they locate the object before they estimate its pose. Two approaches are available:
• The calibrated perspective deformable matching determines the 3D pose of a planar object that
was defined by a template object by using automatically derived contours of the object (section 4.6
on page 136).
• The calibrated descriptor-based matching determines the 3D pose of a planar object that was
defined by a template object by using automatically derived distinctive points of the object, which
are called “interest points” (section 4.7 on page 137).
If a circle or rectangle is contained in the plane for which the 3D pose is needed, the pose estimation can
be applied also by a simple circle pose (section 4.8 on page 138) or rectangle pose (section 4.9 on page
138) estimation. There, the circle or rectangle must be extracted from the image and the internal camera
parameters as well as the dimensions of the circle or rectangle must be known.
An example application for pose estimation in a robot vision system is described in section 8.7.3 on page
230. Note that an introduction to the different 3D matching approaches can be found also in the Solution
Guide I, chapter 10 on page 145. The approaches for the perspective matching are described in more
detail in the Solution Guide II-B.

4.1

Pose Estimation from Points

If the internal camera parameters are known, the pose of an object can be determined by a call of the
operator vector_to_pose.
The individual steps are illustrated based on the example program solution_guide\3d_vision\
pose_of_known_3d_object.hdev, which determines the pose of a metal part with respect to a given
world coordinate system.
First, the camera must be calibrated, i.e., the internal camera parameters and, if the pose of the object
is to be determined relative to a given world coordinate system, the external camera parameters must
be determined. See section 3.2 on page 68 for a detailed description of the calibration process. The
world coordinate system can either be identical to the calibration plate coordinate system belonging
to the calibration plate from one of the calibration images, or it can be modified such that it fits to
some given reference coordinate system (figure 4.1). This can be achieved, e.g., by using the operator
set_origin_pose
set_origin_pose (PoseOfWCS, -0.0568, 0.0372, 0, PoseOfWCS)

or if other transformations than translations are necessary, via homogeneous transformation matrices
(section 2.1 on page 15).

4.1 Pose Estimation from Points

C-107

pose_to_hom_mat3d (PoseOfWCS, camHwcs)
hom_mat3d_rotate_local (camHwcs, rad(180), 'x', camHwcs)
hom_mat3d_to_pose (camHwcs, PoseOfWCS)

With the homogeneous transformation matrix c Hw , which corresponds to the pose of the world coordinate system, world coordinates can be transformed into camera coordinates.
Camera with
optical center

Camera coordinate system ( x c, y c , z c)

c
c

Pose Estimation

Hcp

Calibration plate
y cp
x cp
z cp

Calibration plate coordinate system (x cp, y cp, z cp)

World coordinate system ( x w, y w, z w)

Figure 4.1: Determination of the pose of the world coordinate system.

Then, the pose of the object can be determined from at least three points (control points) for which both
the 3D object coordinates and the 2D image coordinates are known.
The 3D coordinates of the control points need to be determined only once. They must be given in a
coordinate system that is attached to the object. You should choose points that can be extracted easily
and accurately from the images. The 3D coordinates of the control points are then stored in three tuples,
one for the x coordinates, one for the y coordinates, and the last one for the z coordinates.

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In each image from which the pose of the object should be determined, the control points must be
extracted. This task depends heavily on the object and on the possible poses of the object. If it is known
that the object will not be tilted with respect to the camera the detection can, e.g., be carried out by
shape-based matching (for a detailed description of shape-based matching, please refer to the Solution
Guide II-B, section 3.3 on page 64).
Once the image coordinates of the control points are determined, they must be stored in two tuples that
contain the row and the column coordinates, respectively. Note that the 2D image coordinates of the
control points must be stored in the same order as the 3D coordinates.
In the example program, the centers of the three holes of the metal part are used as control points. Their
image coordinates are determined with the HDevelop procedure determine_control_points,
procedure determine_control_points (Image: Intersections: : RowCenter,
ColCenter)

which
is
part
of
the
pose_of_known_3d_object.hdev.

example

program

solution_guide\3d_vision\

Now, we simply call the operator vector_to_pose, passing the 3D object coordinates and the 2D image
coordinates of the control points together with the internal camera parameters.
vector_to_pose (ControlX, ControlY, ControlZ, RowCenter, ColCenter, \
CamParam, 'iterative', 'error', PoseOfObject, Errors)

If both the pose of the world coordinate system and the pose of the object coordinate system are known
with respect to the camera coordinate system (see figure 4.2), it is easy to determine the transformation
matrices for the transformation of object coordinates into world coordinates and vice versa:

=
=

Hc · c Ho

−1

( Hw )

(4.1)
c

· Ho

(4.2)

where w Ho is the homogeneous transformation matrix for the transformation of object coordinates into
world coordinates and c Hw and c Ho are the homogeneous transformation matrices corresponding to
the pose of the world coordinate system and the pose of the object coordinate system, respectively, each
with respect to the camera coordinate system.
The transformation matrix for the transformation of world coordinates into object coordinates can be
derived by:

(w Ho )−1

(4.3)

The calculations described above can be implemented in HDevelop as follows. First, the homogeneous
transformation matrices are derived from the respective poses.

4.1 Pose Estimation from Points

Camera with
optical center

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Camera coordinate system ( x c, y c , z c)

yc
zc

Hw
z

Object

Object coordinate system ( x o , y o , z o )
Pose Estimation

xo
w
o

zw
yw
xw

World coordinate system ( x w, y w, z w)

Figure 4.2: Pose of the object coordinate system and transformation between object coordinates and
world coordinates.

pose_to_hom_mat3d (PoseOfWCS, camHwcs)
pose_to_hom_mat3d (PoseOfObject, camHobj)

Then, the transformation matrix for the transformation of object coordinates into world coordinates is
derived.
hom_mat3d_invert (camHwcs, wcsHcam)
hom_mat3d_compose (wcsHcam, camHobj, wcsHobj)

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3D Position Recognition of Known Objects

Now, known object coordinates
affine_trans_point_3d.

can

transformed

into

world

coordinates

with

affine_trans_point_3d (wcsHobj, CornersXObj, CornersYObj, CornersZObj, \
CornersXWCS, CornersYWCS, CornersZWCS)

In the example program solution_guide\3d_vision\pose_of_known_3d_object.hdev, the world
coordinates of the four corners of the rectangular hole of the metal part are determined from their respective object coordinates. The object coordinate system and the world coordinate system are visualized as
well as the respective coordinates for the four points (see figure 4.3).

Object coordinates:

World coordinates:

1: (6.39,26.78,0.00) [mm]
2: (6.27,13.62,0.00) [mm]
3: (17.62,13.60,0.00) [mm]
4: (17.68,26.66,0.00) [mm]

1: (44.98,31.58,2.25)
2: (49.55,19.62,5.28)
3: (60.12,23.73,5.20)
4: (55.54,35.58,2.20)

[mm]
[mm]
[mm]
[mm]

4
1
3

2
x
z

WCS

OBJ

Figure 4.3: Object coordinates and world coordinates for the four corners of the rectangular hole of the
metal part.

4.2 Pose Estimation Using Shape-Based 3D Matching

4.2

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Pose Estimation Using Shape-Based 3D Matching

Figure 4.4: Shape-based 3D matching: (left) original image containing two tile spacers, (right) 3D shape
model of the tile spacer projected in the image with the poses of the found model instances.

If you need the 3D pose of a planar object or a planar object part, we recommend to use the calibrated
perspective deformable matching (see section 4.6 on page 136) or the calibrated descriptor-based matching (see section 4.7 on page 137). Both are significantly faster, because no 2D projections of the model
must be computed. Instead, a single 2D model is derived from images.
In the following, the general proceeding for shape-based 3D Matching is introduced (see section 4.2.1),
it is shown how to generally enhance the matching robustness and speed (see section 4.2.2 on page 116),
and tips and tricks for the handling of specific problems are provided (see section 4.2.3 on page 119).

4.2.1

General Proceeding for Shape-Based 3D Matching

Shape-based 3D matching consists of the following basic steps:
• the 3D object model is accessed from file,
• the 3D shape model is created from it,
• the 3D object model is destroyed,
• the 3D shape model is used to search the object in search images, and

Pose Estimation

For the 3D pose estimation with shape-based 3D matching, a 3D shape model is generated from a 3D
computer aided design (CAD) model. The 3D shape model consists of 2D projections of the 3D object
seen from different views. To restrain the needed memory and runtime for the shape-based 3D matching,
you should restrict the allowed pose range of the shape model and thus minimize the number of 2D
projections that have to be computed and stored in the 3D shape model. Analogously to the shape-based
matching of 2D structures described in the Solution Guide II-B, section 3.3 on page 64, the 3D shape
model is used to recognize instances of the object in the image. But here, instead of a 2D position,
orientation, and scaling, the 3D pose of each instance is returned.

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• the 3D shape model is destroyed.
An example for the shape-based 3D matching of tile spacers is the HDevelop example program
hdevelop\3D-Matching\Shape-Based\create_shape_model_3d_lowest_model_level.hdev
(see figure 4.4).
Step 1:

Read the 3D object model

The 3D object model describing the search object is loaded in HALCON with the operator
read_object_model_3d. It must be available as a CAD model in one of the supported formats, e.g.,
DXF, STL, or PLY. The list of supported CAD formats and tips on how to obtain a suitable model, including the specification of the requirements the CAD models must fulfill, are provided with the description
of the operator in the Reference Manual. For additional tips on handling selected CAD formats, please
contact your distributor.
read_object_model_3d ('tile_spacer.dxf', 0.0001, [], [], ObjectModel3DID, \
DXFStatus)

Step 2:

Create the 3D shape model

The 3D shape model is created with the operator create_shape_model_3d. It needs the 3D object
model and camera parameters as input. Additionally, a set of parameters has to be adjusted. The camera
parameters can be obtained by a camera calibration as is described in detail in section 3.2 on page 68.
In the example, the camera parameters are known and just assigned to the variable CamParam. Before
creating the 3D shape model, it is recommended to prepare the 3D object model for the shape-based 3D
matching using prepare_object_model_3d. Otherwise, the preparation is applied internally within
create_shape_model_3d, which may slow down the application if the same 3D object model is used
several times.
CamParam := [0.0269462,-354.842,1.27964e-005,1.28e-005,254.24,201.977,512, \
384]
prepare_object_model_3d (ObjectModel3DID, 'shape_based_matching_3d', 'true', \
[], [])
create_shape_model_3d (ObjectModel3DID, CamParam, 0, 0, 0, 'gba', -rad(60), \
rad(60), -rad(60), rad(60), 0, rad(360), 0.26, 0.27, \
10, 'lowest_model_level', 3, ShapeModel3DID)

The 3D shape model is generated by computing different views of the 3D object model within a userspecified pose range. The views are obtained by placing virtual cameras around the object model and
projecting the 3D object model into the image plane of each camera position. The resulting 2D shape
representations of all views are stored in the 3D shape model.
An important task is to specify the pose range. To ease this task, imagine a sphere that surrounds the
object. On the surface of the sphere, a camera is placed that looks at the object. Now, the pose range can
be defined by restricting the position of the camera to a part of the sphere’s surface. Additionally, the
minimum and maximum distance of the camera to the object, i.e., the radii of different spheres, must be
specified.
In the following, the position of the sphere relative to the object and the definition of the surface part
are described. The position of the sphere is defined by placing its center at the center of the object’s

4.2 Pose Estimation Using Shape-Based 3D Matching

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bounding box. The radius of the sphere corresponds to the distance of the camera to the center of the
object. To define a specific part of the sphere’s surface, the geographical coordinates longitude (λ) and
latitude (ϕ) are used (see figure 4.5a). For these, minimum and maximum values are specified so that a
quadrilateral on the sphere is obtained (see figure 4.5b).
N
ϕ = 90°

N
ϕ = 90°

λ min

λ max
ϕmax

z
λ = −90°

x
λ = 0°
ϕ = 0°

selected pose
range

zero meridian

λ = 90°

λ = −90°

x
λ = 0°
ϕ = 0°

ϕmin

λ = 90°

equator

ϕ = −90°
S

Figure 4.5: Geographic coordinate system: (a) the geographical coordinates longitude (λ) and latitude (ϕ)
describe positions on the sphere’s surface, (b) the minimum and maximum values for λ and
ϕ describe a quadrilateral on the sphere, which defines the pose range.

To describe the orientation of the geographical coordinate system we introduce an object-centered coordinate system. This is obtained by moving the origin of the object coordinate system (CAD model) to
the center of the sphere. The xz-plane of the object-centered coordinate system defines the equatorial
plane of the geographical coordinate system. The north pole lies on the negative y-axis. The origin of the
geographical coordinate system (λ = ϕ = 0°), i.e., the intersection of the equator with the zero meridian,
lies in the negative z-axis.
To illustrate the above description, let us assume that we have the object model shown in figure 4.6a.
For illustrative purpose additionally the object coordinate system is visualized. In figure 4.6b, the corresponding geographical coordinate system is shown together with a camera placed at its origin (λ = ϕ =
0°). Consequently, this camera view corresponds to a bottom view of the object.
Note that the coordinate system introduced here is only used to specify the pose range. The pose resulting
from the shape-based 3D matching always refers to the original object coordinate system used in the
CAD file and not to the center of the object’s bounding box.
In most cases, the specification of the pose range can be simplified by changing the origin of the geographical coordinate system (i.e., the orientation of the sphere) such that it coincides with a mean viewing
direction of the real camera to the object. This can be achieved by rotating the object-centered coordinate
system, which defines the geographical coordinate system as described above. The rotation can be specified by passing the rotation angles to the parameters RefRotX, RefRotY, and RefRotZ of the operator
create_shape_model_3d. That is, you can specify the pose range either by adjusting the longitude
and latitude, leaving the origin of the sphere at its initial position, or by rotating the object-centered

Pose Estimation

zero meridian

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3D Position Recognition of Known Objects

equator
z
x

z
x

z
x
y

zero meridian

Figure 4.6: Object coordinate systems: (a) original object coordinate system of the CAD model, (b) objectcentered geographical coordinate system obtained by moving the object to the center of the
sphere: the camera is placed on the sphere’s surface at the position λ = ϕ = 0° and the
object is placed at the center of the sphere. Note that in contrast to the previous image, the
sphere is tilted, i.e., it is visualized in a way that the equator is vertical and the zero meridian
is completely visible.

coordinate system to get a mean reference view and then specify the pose range in a more intuitive way
(see figure 4.7 on page 115), which is recommended in most cases.
Figure 4.8 shows a rotation-symmetric object, for which a reference view is specified. Rotationsymmetric objects have the specific advantage that they have the same appearance from all directions
that are perpendicular to their rotation axis. Thus, if the mean reference view is selected such that the
camera view is perpendicular to the rotation axis (and the rotation axis crosses the poles of the sphere),
the longitude range can collapse to a single value so that the number of calculated 2D projections is
significantly reduced. That is, less memory is needed and the matching becomes faster. Here, the initial
z-axis of the object-centered coordinate system corresponds to the rotation axis of the object. To obtain
the intended reference view, the orientation of the sphere is changed by rotating the object-centered coordinate system by -90° around its x-axis. Now, the pose range for ϕ is specified like described before
and the pose range for λ is restricted to 0°.
Besides the definition of the pose range, also the camera roll angle, i.e., the allowed range for the rotation
of the virtual camera around its z-axis, must be set. In most cases, it is recommended to allow a full
circle for the camera roll angle. For details, we recommend to read the description of the operator
create_shape_model_3d in the Reference Manual.
Step 3:

Destroy the 3D object model

After creating the 3D shape model, the 3D object model often is not needed anymore and can be destroyed for memory reasons using the operator clear_object_model_3d. If the 3d object model is
still needed, e.g., for visualization purposes, this step must be moved, e.g., to the end of the application.

4.2 Pose Estimation Using Shape-Based 3D Matching

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Pose Range

z
z
x

z
equator

equator

z
x

zero meridian

a)
λ min/max ϕ min/max

RefRotX RefRotY RefRotZ

a) 180° +/− α

0° +/− α

0°

0° +/− α

180°

0°

zero meridian

pose range with
λ min/max = 0° and
ϕ min/max = 0°+/− α

equator

x
x

x
S

N
equator

y
y
z
x
y

RefRotX = −90°
RefRotY = 0°
RefRotZ = 0°

Figure 4.8: Change the origin of the geographical coordinate system for (a) a rotation-symmetric object
with the z-axis of the object-centered coordinate system corresponding to the rotation axis
such that (b) the z-axis becomes perpendicular to the rotation axis.

clear_object_model_3d (ObjectModel3DID)

Step 4:

Find the 3D shape model in search images

With the 3D shape model that was created by create_shape_model_3d or read from file by
read_shape_model_3d, the object can be searched for in images. For the search, the operator

Pose Estimation

Figure 4.7: Specify pose range: (a) only by longitude and latitude, or (b) by additionally rotating the objectcentered coordinate system to a reference view. Note that because of the rotation around the
x-axis, the positions of the poles have changed.

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find_shape_model_3d is applied.
for I := 1 to NumImages by 1
read_image (Image, 'tile_spacers/tile_spacers_color_' + I$'02')
find_shape_model_3d (Image, ShapeModel3DID, 0.7, 0.85, 0, \
['num_matches','max_overlap','border_model'], [3, \
0.75,'true'], Pose, CovPose, Score)
endfor

Several parameters can be set to control the search process. For detailed information, we recommend
to read the description of the operator in the Reference Manual. The operator returns the pose of the
matching model, the standard deviation of the pose, and the score of the found instances of the 3D shape
model that describes how much of the model is visible in the image.
Step 5:

Destroy the 3D shape model

When the 3D shape model is not needed anymore, it is destroyed with the operator
clear_shape_model_3d.
clear_shape_model_3d (ShapeModel3DID)

Besides the basic steps, it is often required to inspect the 3D object model or the 3D shape model, to
re-use the 3D shape model, or to visualize the result of the matching. These steps are described in the
Solution Guide I, chapter 10 on page 145.

4.2.2

Enhance the Shape-Based 3D Matching

The following sections generally show how to enhance the robustness (section 4.2.2.1) and speed (section 4.2.2.2) of shape-based 3D matching.
4.2.2.1

Enhance the Robustness

For a robust shape-based 3D matching it is important that the edges of the object are cleary visible in
the image. Thus, the following general tips may help you to enhance your application already when
acquiring the images of the object:
• If possible, use a background with a good contrast to the object, so that the background can be
cleary separated from the object.
• Carefully adjust the lighting for the image acquisition.

To get clearly visible edges of the object in your images, take special care of the lighting conditions. In general, the edges that are included in the 3D shape model should also be visible
in the image. The edges that are included in the model can be adjusted with the generic parameter min_face_angle of create_shape_model_3d. The effect of this parameter can be inspected by visualizing the resulting edges with the procedure inspect_object_model_3d, which
can be found in the example program hdevelop\Applications\Position-Recognition-3D\
3d_matching_clamps.hdev.

4.2 Pose Estimation Using Shape-Based 3D Matching

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• If possible, use multi-channel images.

Figure 4.9: A fuse illuminated from three different directions by three differently colored light sources.

4.2.2.2

Enhance the Speed

There are several means to speed up the shape-based 3D matching:
• Use a homogeneous background during the image acquisition.

Generally, you should try to adjust the lighting for the image acquisition so that the edges of the
object but no surface texture are visible in the images.

• Reduce the resolution of the image.

A reduced resolution of the image can speed up the online as well as the offline phase of the
shape-based 3D matching, because less 2D projections have to be generated. Note that if you
reduce the resolution of the images (e.g., from 1 Megapixel to 640x480 pixels), you must also
change the camera parameters used for the creation of the 3D shape model by adapting Sx, Sy,
Cx, Cy, ImageWidth, and ImageHeight (see the description of write_cam_par in the Reference
Manual). For example, if the image is scaled down by a factor of 0.5, Sx and Sy must be multiplied
by 2, whereas Cx, Cy, ImageWidth, and ImageHeight must be multiplied by 0.5.

• Use a region of Interest.

Using a region of interest, you can speed up the search. The more the region in which the objects are
searched can be restricted, the faster and more robust the search will be. For detailed information
see the Solution Guide I, chapter 3 on page 33 or the Solution Guide II-B, section 2.1.2 on page
21.

Pose Estimation

Multi-channel, e.g., color images contain more information and thus typically lead to a more robust
edge extraction. An especially robust edge extraction can be obtained if color images are used and
the object is illuminated from different directions by three differently colored, typically red, green
and blue, light sources (see figure 4.9).

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3D Position Recognition of Known Objects

• Restrict the pose range.

The more the pose range is restricted while creating a 3D shape model with
create_shape_model_3d, the faster is the search process. But note that only those object
instances are found that correspond to the selected pose range.

• Eliminate unnecessary edges from the 3D shape model.

If edges are contained in the 3D shape model that are not visible in the search image, the
performance, i.e., the robustness and speed, of the shape-based 3D matching decreases. Especially for objects that contain curved surfaces, which are approximated by multiple planar
faces in the 3D object model, the generic parameter min_face_angle should be adjusted within
create_shape_model_3d to eliminate unnecessary edges from the 3D shape model. To check
which edges are visible with a specific minimum face angle, you can display the corresponding
contours with the operator project_object_model_3d.

• Select the value for the number of used pyramid levels as large as possible.

In create_shape_model_3d as well as in find_shape_model_3d, the number of used pyramid
levels can be set by the parameter num_levels. To speed up the search process, the value should
be as large as possible, but the object should be still recognizable in the model. To check a view
on the object in a specific pyramid level, you can query the corresponding contours by the operator
get_shape_model_3d_contours.

• Disable the pregeneration of the model views on lower pyramid levels.

When specifying a large pose range, the number of model views on lower pyramid levels may become very large, which leads to a slow model generation and high memory consumption. To speed
up the model generation, you can disable the pregeneration of the model views on lower pyramid
levels using the generic parameter ’lowest_model_level’ within create_shape_model_3d.
Note that you nevertheless obtain a high accuracy during the search with find_shape_model_3d,
because the pose is still refined (but now on the fly) on the original pyramid level.

• Select the value for MinScore as large as possible.

In find_shape_model_3d you can adjust the parameter MinScore. A large MinScore speeds up
the search process, but allows less invisible edges for the object, so that possibly some objects are
not recognized.

• Select the value for Greediness as large as possible.

In find_shape_model_3d you can adjust the parameter Greediness. A large Greediness
speeds up the search process. But because the search becomes less robust, possibly some objects
are not recognized.

• Adjust the pose refinement.

In find_shape_model_3d you can adjust the generic parameter pose_refinement. If it is set
to none, the search is fast but the pose is determined with limited accuracy. A tradeoff between
runtime and accuracy is to set the pose refinement to least_squares_high. For complex models
with a large number of faces it is reasonable to speed up the pose refinement by splitting it such
that some of the needed calculations are already performed during the creation of the model. Thus,
the generic parameter ’fast_pose_refinement’ of create_shape_model_3d is by default set
to ’true’. This leads to a faster matching but also to a higher memory consumption. If the storage
is more critical than the speed of the matching, you can set the parameter to ’false’.

4.2 Pose Estimation Using Shape-Based 3D Matching

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• Check the setting of border_model.

In find_shape_model_3d you can adjust the generic parameter border_model. If it is set to
true, also objects that extend beyond the image borders can be found. Because this is rather
time-consuming if you only search for objects that are completely contained in the image, we
recommend to leave the default value false unchanged.

• Speed up the visualization.

When visualizing the 3D object model or the 3D shape model by projecting them into the image
using project_object_model_3d or project_shape_model_3d, respectively, you can speed
up the visualization by setting the parameter HiddenLineRemoval to false. Then, also those
edges of the model are visualized that would be hidden by faces.

4.2.3

Tips and Tricks for Problem Handling

In section 4.2.2 on page 116, general tips to enhance the robustness and speed of shape-based 3D matching were listed. In the following, the focus is on the handling of specific problems, i.e., possible reasons
for an unsuccessful, erroneous, or very slow recognition or model generation are introduced. In particular, possible reasons for problems and tips to solve them are provided for the cases that

• the recognition is not successful, i.e., the object is not found (see section 4.2.3.2),
• the object is found, but the estimated object pose is wrong (see section 4.2.3.3 on page 121),
• the object is found in the right pose, but the pose is estimated with low accuracy (see section 4.2.3.4
on page 121), or
• the recognition is successful but very slow (section 4.2.3.5 on page 122).
4.2.3.1

The Model Generation is Very Slow

The computation time of the model generation increases quadratically with the number of faces in the
CAD model. Thus, if the model generation with create_shape_model_3d is very slow, most possibly
the CAD model is too complex. Note that in most cases, a very coarse model is sufficient to locate a
3D object with shape-based 3D matching. Thus, in case of a very slow model generation, you should
eliminate all unimportant details from your model using suitable CAD software. Alternatively, you
can also increase the value of the generic parameter ’lowest_model_level’ to work with a coarser
model. Additionally, only the part of the object that is relevant for the search should be contained in the
model. Thus, sometimes further modifications of the CAD model using suitable CAD software might be
necessary.
4.2.3.2

The Object is not Found

If the object is not found with find_shape_model_3d, typically the reason can be found in at least one
of the following problems:

Pose Estimation

• the model generation is very slow (see section 4.2.3.1),

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• The value of the parameter MinScore was chosen too large. Check if the object can be found if
matches with a smaller score are accepted. But note that the possibility of finding false matches
increases with a decreasing MinScore value.
• The value of the parameter Greediness was chosen too large. A large value leads to a fast but
less robust search. Check if the object can be found with a smaller Greediness value.
• The value of the parameter NumLevels was estimated or chosen too large. Note that the shape
representation of the views on the highest pyramid level must still be recognizable and must contain
enough model points. You can visually check the views on the specific pyramid levels using
get_shape_model_3d_contours.
• The value of the generic parameter ’min_face_angle’ was chosen too small when generating
the 3D shape model with create_shape_model_3d. Thus, the 3D shape model contains also
non-visible object edges. Note that after creating a model, you can visually check the model edges
on the specific pyramid levels using get_shape_model_3d_contours. Before creating a model,
you can display the contours of the underlying 3D object model with a specific minimum face
angle using the operator project_object_model_3d.
• The chosen reference pose or the pose range are not correct. Check whether the created views cover the desired pose range. Note that you can query the number of created
views for each pyramid level using get_shape_model_3d_params (setting GenParamNames to
’num_views_per_level’). Then, you can visually check selected views out of this range of
views using get_shape_model_3d_contours.
• The value of the parameter MinContrast was chosen too large when generating the 3D shape
model with create_shape_model_3d. Thus, edges that belong to the object are not extracted in
the search image and the matching score decreases. Check the value of MinContrast by extracting
edges in the search image using edges_image with the parameter Filter set to ’sobel_fast’
and the parameters Low and High set to the value of MinContrast.
• The camera parameters are not accurate enough. Thus, the projected model and the imaged object
do not accurately fit together. If possible, improve the accuracy of the calibration. Otherwise,
stop the search on a higher pyramid level where the differences between the projected model and
the imaged object are small enough. To stop the search on a higher pyramid level, the parameter
NumLevels is set as a tuple consisting of the highest and lowest used pyramid levels.
• The projected model and the imaged object do not accurately fit together because the CAD file is
not modeled accurately enough or the objects do not exactly correspond to the model, e.g., because
of tolerances at the fabrication. In the first case, if possible, improve the accuracy of the CAD
model. Otherwise, stop the search on a higher pyramid level where the differences between the
projected model and the imaged object are small enough. To stop the search on a higher pyramid
level, the parameter NumLevels is set as a tuple consisting of the highest and lowest used pyramid
levels.
• Some of the object edges are no “sharp” edges. Model the round edges in the CAD model or
stop the search on a higher pyramid level where the differences between the projected model and
the imaged object are small enough. To stop the search on a higher pyramid level, the parameter
NumLevels is set as a tuple consisting of the highest and lowest used pyramid levels.
• The object edges are not visible in the image. To enhance the visibility of the edges during the

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image acquisition, follow the advices for a robust shape-based matching that are given in section 4.2.2.1 on page 116.
4.2.3.3

The Object is Found in a Wrong Pose

If the object is found with find_shape_model_3d, but the estimated pose is wrong, typically the reason
can be found in at least one of the following problems:
• The value of the parameter MinScore was chosen too small so that false matches could be found.
Check if the object still can be found with a larger MinScore value.
• The value of the generic parameter ’min_face_angle’ was chosen too large when generating
the 3D shape model with create_shape_model_3d. Thus, the 3D shape model does not contain
all visible object edges. Note that after creating a model, you can visually check the model edges
on the specific pyramid levels using get_shape_model_3d_contours. Before creating a model,
you can display the contours of the underlying 3D object model with a specific minimum face
angle using the operator project_object_model_3d.

• The background contains too much clutter. To solve this problem for one-channel images, you
can set ’metric’ to ’ignore_part_polarity’. But if possible, you should use another background, which is more homogeneous. If the background cannot be changed, you can try to optimize its appearance in the images by using a diffuse lighting source and by adjusting the light’s
direction. Note that you can check the background by extracting edges in the search image using
edges_image with the parameter Filter set to ’sobel_fast’ and the parameters Low and High
set to the value of MinContrast.
• The pose range contains degenerated views, i.e., views that do not significantly represent the object
anymore. For example, if a cube is viewed exactly orthogonol to one of its faces, the view collapses
from a perspective representation of a 3D cube to a simple square. Such a view may lead to many
false matches. An even more extreme example is an exact side-view on a flat object. There, the
view may collapse to a straight line. Such a view would lead to many matches in any search image,
even if the actual 3D object is not contained. Therefore, you should limit the pose range as much
as possible and ensure that no degenerated views are contained in the pose range.
• There are too many clutter edges around the object, which is typical for some bin picking applications where objects touch or overlap each other. Try to separate the objects, e.g., by dumping the
objects onto a plane and/or shaking the bin or plane to equally distribute the objects.
4.2.3.4

The Object Pose is Estimated With Low Accuracy

If the object is found with find_shape_model_3d, but the accuracy of the estimated object pose is low,
typically the reason can be found in at least one of the following problems:

Pose Estimation

• The value of the parameter MinContrast was chosen too small when generating the 3D shape
model with create_shape_model_3d. This leads to clutter edges in the search image. Check
the value of MinContrast by extracting edges in the search image using edges_image with the
parameter Filter set to ’sobel_fast’ and the parameters Low and High set to MinContrast.

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• Some of the object edges are no “sharp” edges. Model the round edges in the CAD model or
stop the search on a higher pyramid level where the differences between the projected model and
the imaged object are small enough. To stop the search on a higher pyramid level, the parameter
NumLevels is set as a tuple consisting of the highest and lowest used pyramid levels.
• The camera parameters are not accurate enough. Thus, the projected model and the imaged object
do not accurately fit together. If possible, improve the accuracy of the calibration.
• The projected model and the imaged object do not accurately fit together because the CAD file is
not modeled accurately enough or the objects do not exactly correspond to the model, e.g., because
of tolerances at the fabrication. In the first case, if possible, improve the accuracy of the CAD
model.
4.2.3.5

The Object Recognition is Very Slow

If the object is found with find_shape_model_3d, but the recognition is very slow, typically the reason
can be found in at least one of the following problems:
• The value of the parameter Greediness was chosen too small. A larger value speeds up the search,
but because the search becomes less robust, possibly some objects are not recognized. Thus, you
should check if the object can still be found with a larger Greediness value.
• The value of the parameter MinScore was chosen too small so that many false matches could be
found. Check if the object still can be found with a larger MinScore value.
• The value of the parameter MinContrast was chosen too small when generating the 3D shape
model with create_shape_model_3d. This leads to clutter edges in the search image. These
clutter edges generate many matching candidates that must be examined. Check the value of
MinContrast by extracting edges in the search image using edges_image with the parameter
Filter set to ’sobel_fast’ and the parameters Low and High set to the value of MinContrast.
• The CAD model is very complex and ’fast_pose_refinement’ was set to ’false’ when generating the 3D shape model with create_shape_model_3d. As the computation time of the
least-squares refinement increases quadratically with the number of faces in the CAD model, you
can speed up the recognition by eliminating all important details from the model using your CAD
software and setting ’fast_pose_refinement’ to ’true’.
• The image size, especially the extent of the search object in pixels, is very large so that the number
of model views on lower pyramid levels becomes very large, too. If you cannot work with images
of a reduced resolution as is described in section 4.2.2.2 on page 117, we strongly recommend to
restrict the search in the image to an ROI and to increase the value of ’lowest_model_level’
when generating the 3D shape model with create_shape_model_3d.
• The pose range is too large.
When generating the 3D shape model with
create_shape_model_3d, restrict the pose range as much as possible, especially take care
of DistMin. Additionally, for objects with multiple stable poses you should use multiple models.
That is, instead of generating one model that covers the complete pose range, multiple models
each covering one stable pose should be used. Note that you can also restrict the pose range by
exploiting rotational symmetries of the object (see section 4.2.1 on page 114).

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• The background contains too much clutter so that the clutter edges in the search image lead to
many matching candidates that must be examined. To solve this problem for one-channel images,
you can set ’metric’ to ’ignore_part_polarity’. But if possible, you should use another
background, which is more homogeneous. If the background cannot be changed, you can try to
optimize its appearance in the images by using a diffuse light source and by adjusting the light’s
direction. Note that you can check the background by extracting edges in the search image using
edges_image with the parameter Filter set to ’sobel_fast’ and the parameters Low and High
set to the value of MinContrast.

4.3

Pose Estimation Using Surface-Based 3D Matching

Pose Estimation

For the 3D pose estimation with surface-based matching, a surface model is generated from a 3D object
model that was obtained either from a 3D computer aided design (CAD) model or from a 3D reconstruction approach, e.g., stereo vision (chapter 5 on page 139) or sheet of light (chapter 6 on page 177). The
surface model consists of a set of 3D points and the points’ normal vectors. That is, the corresponding
information must be provided (at least implicitly) by the 3D object model. In contrast to the shape-based
3D matching, the instances of the object are not located in images but in a 3D scene, i.e., in a set of 3D
points that is provided as another 3D object model and which can be obtained by a 3D reconstruction
approach, too.

Figure 4.10: Surface-based matching: (left) 3D model of engine cover, (right) model instances found in a
3D scene.

In the following, the general proceeding for surface-based 3D matching is introduced (see section 4.3.1).

4.3.1

General Proceeding for Surface-Based 3D Matching

Surface-based 3D matching consists of the following basic steps:
• access the 3D object model needed for the creation of the surface model,
• create the surface model from it,

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• access the 3D object model that represents the search data,
• use the surface model to search the object in the search data, and
• destroy the handles for the matching result, all 3D object models, and the surface model.
An example for the surface-based 3D matching of engine covers (see figure 4.10 on page 123) is the HDevelop example program hdevelop\3D-Matching\Surface-Based\find_surface_model.hdev.
Step 1:

Access the 3D object model needed for the creation of the surface model

In contrast to shape-based 3D matching, for surface-based matching the 3D object model does not need
to be available as CAD model but can also be derived by a 3D reconstruction. If a 3D object model is
available as a CAD model or if the 3D object model was saved to file after an offline 3D reconstruction,
it can be accessed using read_object_model_3d. Suitable file formats are, e.g., DXF, OFF, PLY, or
OM3. The format OM3 is a HALCON-specific format for 3D object models that can be derived from
the results of HALCON’s 3D reconstruction approaches (see figure 2.23 on page 45 in chapter 1). Tips
on how to obtain a suitable model, including the specification of the requirements the CAD models
must fulfill, are provided with the description of the operator read_object_model_3d in the Reference
Manual.
In the example program, the 3D object model is not read from file but is derived from X, Y, and Z images
that were obtained by a specific 3D sensor: a “time-of-flight” (TOF) camera. To get the 3D object model
of a single engine cover, a region containing a single engine cover is extracted by a blob analysis from
the Z image (see figure 4.11) and the Z image is reduced to the corresponding ROI.
read_image (Image, ImagePath + 'engine_cover_xyz_01')
decompose3 (Image, Xm, Ym, Zm)
threshold (Zm, ModelZ, 0, 650)
connection (ModelZ, ConnectedModel)
select_obj (ConnectedModel, ModelROI, [10,9])
union1 (ModelROI, ModelROI)
reduce_domain (Xm, ModelROI, Xm)

The X, Y, and Z images are now transformed into a 3D object model using the operator
xyz_to_object_model_3d. This 3D object model is needed to create the surface model that will serve
as model for the matching.
xyz_to_object_model_3d (Xm, Ym, Zm, ObjectModel3DModel)

Note that for surface-based matching information about the coordinates of the 3D points and their
normals is needed. Thus, if a 3D object model is obtained from a CAD model or from multi-view
stereo, the normals or alternatively a triangular or polygon mesh should be contained in the 3D object model. If the 3D object model does not already contain normals, e.g., because it has been created
from X, Y, and Z images like in the example program, the normals are automatically determined by
create_surface_model. Note that in this case, the orientation of the normals is ambiguous (see next
step).
Step 2:

Create the surface model

The operator create_surface_model creates a surface model by sampling the 3D object model with
a certain distance. The sampling distance can be adjusted with the parameter RelSamplingDistance.

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Figure 4.11: A single engine cover extracted from the Z image.

Note that a smaller value leads to a slower but more robust matching, whereas a larger value speeds up
the matching but at the same time decreases the robustness.

The correct match will only be found if the normals of the model and of the search object point in
the same direction. If the normals must be derived by create_surface_model because they are not
available in the 3D object model, the orientation of the normals is ambiguous and consequently, a suitable orientation of the normals cannot be ensured. To cope with this, especially if the orientation of
the search object may vary considerably, two models should be created with the generic parameter
’model_invert_normals’ set to ’true’ and ’false’, respectively. Both models should then be used for
matching and the match with the higher score should be selected.
Step 3:

Access the 3D object model that represents the search data

Similar to the 3D object model that was needed for the creation of the surface model, the 3D object models in which the object of interest are searched must be accessed, i.e., they are read from
file using read_object_model_3d or they are derived online by a 3D reconstruction. In the example program, the 3D object models are derived again from X, Y, and Z images using the operator
xyz_to_object_model_3d. Note that a blob analysis is applied to remove the background plane from
the search data.
NumImages := 10
for Index := 2 to NumImages by 1
read_image (Image, ImagePath + 'engine_cover_xyz_' + Index$'02')
decompose3 (Image, X, Y, Z)
threshold (Z, SceneGood, 0, 666)
reduce_domain (X, SceneGood, XReduced)
xyz_to_object_model_3d (XReduced, Y, Z, ObjectModel3DSceneReduced)

Pose Estimation

create_surface_model (ObjectModel3DModel, 0.03, [], [], SFM)

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Step 4:

Find the surface model in the search data

With the surface model that was created by create_surface_model or read from file by
read_surface_model, the object can be searched for in the search data using the operator
find_surface_model.
find_surface_model (SFM, ObjectModel3DSceneReduced, 0.05, 0.3, 0.15, \
'true', 'num_matches', 10, Pose, Score, \
SurfaceMatchingResultID)

Several parameters can be set to control the search process. For detailed information, we recommend
to read the description of the operator in the Reference Manual. For each model instance that could be
located by the matching, the operator returns the pose and a score that describes the quality of the match.
After the matching, the result can be visualized. Here, for each match that exceeds a certain score, the
3D object model is transformed with the corresponding pose and the transformed 3D object models of a
specific search scene are stored in the tuple ObjectModel3DResult.
ObjectModel3DResult := []
for Index2 := 0 to |Score| - 1 by 1
if (Score[Index2] < 0.11)
continue
endif
CPose := Pose[Index2 * 7:Index2 * 7 + 6]
rigid_trans_object_model_3d (ObjectModel3DModel, CPose, \
ObjectModel3DRigidTrans)
ObjectModel3DResult := [ObjectModel3DResult, \
ObjectModel3DRigidTrans]
endfor

The visualization is then applied using visualize_object_model_3d. Actually, the transformed models are displayed together with the original 3D scene (see figure 4.10 on page 123), i.e., with the 3D object
model that is obtained from the X, Y, and Z images from which also the search data was obtained, but
this time no blob analysis is applied to remove the background plane.
xyz_to_object_model_3d (X, Y, Z, ObjectModel3DScene)
NumResult := |ObjectModel3DResult|
tuple_gen_const (NumResult, 'green', Colors)
tuple_gen_const (NumResult, 'circle', Shapes)
tuple_gen_const (NumResult, 3, Radii)
visualize_object_model_3d (WindowHandle, [ObjectModel3DScene, \
ObjectModel3DResult], [], PoseOut, \
['color_' + [0,Indices],'point_size_0'], \
['gray',Colors,1.0], Message, [], \
Instructions, PoseOut)

Step 5:

Destroy the handles for the matching result, all 3D object models, and the surface model

The 3D object models that were accessed or obtained during a search or visualization process
are destroyed from memory before searching the model in another 3D scene. The same applies

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for the handle of the matching result that was returned by the matching. The 3D object models are destroyed with clear_object_model_3d and the handle of the result is destroyed with
clear_surface_matching_result.
clear_object_model_3d (ObjectModel3DSceneReduced)
clear_object_model_3d (ObjectModel3DScene)
clear_object_model_3d (ObjectModel3DResult)
clear_surface_matching_result (SurfaceMatchingResultID)
endfor

When the 3D object model of the model and the surface model that was created from it are not needed
anymore, they are destroyed, too. The surface model is destroyed with clear_surface_model.
clear_object_model_3d (ObjectModel3DModel)
clear_surface_model (SFM)

4.4

Pose Estimation Using Deformable Surface-Based 3D
Matching

Deformable surface-based 3D matching allows to find 3D objects in 3D scenes even if the 3D objects
are deformed to a certain degree (see figure 4.12).
For the 3D pose estimation with deformable surface-based 3D matching, a deformable surface model
is generated from a 3D object model that was obtained either from a 3D computer aided design (CAD)
model, from a 3D reconstruction approach, e.g., stereo vision (chapter 5 on page 139) or sheet of light
(chapter 6 on page 177), or directly by a 3D sensor. The deformable surface model consists of a set of
3D points and the points’ normal vectors. That is, the corresponding information must be provided (at
least implicitly) by the 3D object model. In contrast to the shape-based 3D matching, the instances of the
object are not located in images but in a 3D scene, i.e., in a set of 3D points that is provided as another
3D object model and which can be obtained by one of the above mentioned approaches or sensors.

4.4.1

General Proceeding for Deformable Surface-Based 3D Matching

Deformable surface-based 3D matching consists of the following basic steps:
• provide the 3D object model for the creation of the deformable surface model,
• create the deformable surface model with create_deformable_surface_model,
• optionally define some reference points with
add_deformable_surface_model_reference_point,

Pose Estimation

Besides the basic steps, it is often required to inspect the 3D object model, to re-use the surface model,
or to visualize the result of the matching. These steps are described in the Solution Guide I, chapter 10
on page 145.

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(a)

(b)

(c)

(d)

Figure 4.12: Deformable Surface-based matching: (a) The silicone baking mold used as test object, (b) a
3D object model of the silicone baking mold, (c) a 3D scene that shows a deformed baking
mold, and (d) matching result (deformed object and reference points).

• optionally extend the range of deformations that the deformable surface model is able to cope with,
• provide a 3D scene that represents the search data,
• search for the (possibly) deformed
find_deformable_surface_model, and

object

the

data

with

• clear all handles for matching results, all 3D object models, and the deformable surface model.
The HDevelop example program hdevelop\3D-Matching\Deformable-Surface-Based\
find_deformable_surface_model.hdev shows how to use the deformable surface-based 3D
matching to find even deformed silicone baking molds (see figure 4.12a).
Step 1:

Provide the 3D object model needed for the creation of the deformable surface model

In contrast to shape-based 3D matching, for deformable surface-based matching, the 3D object model
does not need to be available as CAD model but can also be provided by a 3D sensor or be derived by a

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3D reconstruction. For more information on the possible sources for 3D data, see section 4.3.1 on page
123.
In the example program, the 3D object model is provided as CAD model (see figure 4.12b on page 128).
This kind of data source was mainly chosen for visualization reasons. With this, it is easier to distinguish
the deformed model from the search scene.
Note that for deformable surface-based matching information about the coordinates of the 3D points and
their normals is needed. Thus, if a 3D object model is obtained from a CAD model or from multi-view
stereo, the normals or a triangular or polygon mesh should be contained in the 3D object model.
Step 2:

Create the deformable surface model with create_deformable_surfabe_model

The operator create_deformable_surface_model creates a deformable surface model by sampling
the 3D object model with a certain distance. The sampling distance can be adjusted with the parameter
RelSamplingDistance. Note that a smaller value leads to a slower but more robust matching, whereas
a larger value speeds up the matching but at the same time decreases the robustness.

If the 3D object model does not contain normals,
they are determined by
create_deformable_surface_model automatically. Note that in this case, the orientation of
the normals is ambiguous. Internally, they are automatically oriented such that all normals have a
positive z component.
The correct match will only be found if the normals of the model and of the search object point in the
same direction. If the orientation of the search object may vary considerably, some effort should be made
to orient the normals of the search object consistently with those of the model. E.g., it might be helpful
to check whether all normals point away from the center of gravity of the object and to flip those normals
that do not. Of course, this correction of the normals must be performed for both the model and the search
object. Another possibility is to create two models with the generic parameter ’model_invert_normals’
set to ’true’ and ’false’, respectively. Both models can then be used for matching and the match with the
higher score should be selected.
Step 3: Optionally define reference points with
add_deformable_surface_model_reference_point
It is possible to define a set of reference points with
add_deformable_surface_model_reference_point.
add_deformable_surface_model_reference_point (DeformableSurfaceModel, \
ReferencePointX, \
ReferencePointY, \
ReferencePointZ, \
ReferencePointIndex)

Reference points are 3D points that can be placed at any position, i.e., it is not necessary that
they lie on the object’s surface. After a — possibly deformed — object has been found with
find_deformable_surface_model, the respective 3D position of the reference points in the deformed
scene can be determined with get_deformable_surface_matching_result.

Pose Estimation

create_deformable_surface_model (ObjectModel3DReference, 0.03, 'stiffness', \
0.85, DeformableSurfaceModel)

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find_deformable_surface_model (DeformableSurfaceModel, \
ObjectModel3DSearchSceneForDef, 0.03, 0, [], \
[], Score, DeformableSurfaceMatchingResult)
get_deformable_surface_matching_result (DeformableSurfaceMatchingResult, \
'reference_point_x', 'all', \
ReferencePointXDeformed)
get_deformable_surface_matching_result (DeformableSurfaceMatchingResult, \
'reference_point_y', 'all', \
ReferencePointYDeformed)
get_deformable_surface_matching_result (DeformableSurfaceMatchingResult, \
'reference_point_z', 'all', \
ReferencePointZDeformed)

Reference points are, e.g., helpful to determine the grasping points for the deformed 3D object.
Step 4: Optionally extend the range of deformations that the deformable surface model is able
to cope with
The supported range of deformations can be extended if information about the expected deformations is
added to the deformable surface model.
First, provide 3D scenes that contain deformed instances of the 3D object, then, detect those instances with find_deformable_surface_model, and finally, add the found deformed objects to the
deformable surface model with add_deformable_surface_model_sample to extend the range of deformations that the deformable surface model is able to cope with.
To make the matching in the 3D scenes used for the model extension faster and more robust, in the
example program, the background is eliminated from the reconstructed 3D scene by eliminating all
points that are close to the (known) background plane.
gen_plane_object_model_3d (PlanePose, [], [], ObjectModel3DPlane)
distance_object_model_3d (ObjectModel3D, ObjectModel3DPlane, [], 0, [], [])
get_object_model_3d_params (ObjectModel3D, '&distance', Distances)
select_points_object_model_3d (ObjectModel3D, '&distance', \
MinDistanceToPlane, max(Distances), \
ObjectModel3DThresholded)

In the resulting 3D scene, the deformed object is detected with find_deformable_surface_model
and the found deformed object is added to the deformable surface model with
add_deformable_surface_model_sample:
find_deformable_surface_model (DeformableSurfaceModel, \
ObjectModel3DSearchSceneForDef, 0.03, 0, [], \
[], Score, DeformableSurfaceMatchingResult)
get_deformable_surface_matching_result (DeformableSurfaceMatchingResult, \
'deformed_sampled_model', 0, \
ObjectModel3DDeformedSampled)
add_deformable_surface_model_sample (DeformableSurfaceModel, \
ObjectModel3DDeformedSampled)

4.4 Pose Estimation Using Deformable Surface-Based 3D Matching

Step 5:

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Provide a 3D scene that represents the search data

The 3D scene in which the object of interest is searched for must be provided as a 3D object model (see
figure 4.12c), similar to the 3D object model that was used for the creation of the deformable surface
model or for the extension of the supported deformation range. Typically, the 3D object models used for
the search are acquired by 3D sensors or reconstructed from stereo image data.
Step 6:

Use the deformable surface model to search for the object in the search data

With the deformable surface model that was created with create_deformable_surface_model or
read from file with read_deformable_surface_model, the deformed object can be searched for in the
search data using the operator find_deformable_surface_model.
find_deformable_surface_model (DeformableSurfaceModel, \
ObjectModel3DSearchSceneForDef, 0.03, 0, [], \
[], Score, DeformableSurfaceMatchingResult)

Several parameters can be set to control the search process. For detailed information, we recommend
to read the description of the operator find_deformable_surface_modelin the Reference Manual. If
a model instance was found, the operator returns a score that describes the quality of the match and a
handle that contains the results, like the deformed position of the reference points.

get_deformable_surface_matching_result (DeformableSurfaceMatchingResult, \
'deformed_model', 0, \
ObjectModel3DDeformed)

The position of the reference points — transformed to the deformed 3D object that was found — can
be determined with get_deformable_surface_matching_result. It can, e.g., be used to grasp the
deformed object.
get_deformable_surface_matching_result (DeformableSurfaceMatchingResult, \
'reference_point_x', 'all', \
ReferencePointXDeformed)
get_deformable_surface_matching_result (DeformableSurfaceMatchingResult, \
'reference_point_y', 'all', \
ReferencePointYDeformed)
get_deformable_surface_matching_result (DeformableSurfaceMatchingResult, \
'reference_point_z', 'all', \
ReferencePointZDeformed)

In the example, the deformed 3D object model is visualized together with the reference points on top of
the search scene (see figure 4.12d on page 128).
Note that, especially, if the object does not have a distinct 3D shape, e.g., because most of the points have
a similar height, it might be necessary to manually eliminate the background and other objects from the
3D scene to ensure a proper matching result. Otherwise, the object might be found in the background
because a relatively flat object fits perfectly to a relatively flat background, especially if deformations of
the object are allowed.

Pose Estimation

After the matching, the result can be visualized. For this, the deformed reference model can be determined with get_deformable_surface_matching_result:

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3D Position Recognition of Known Objects

Step 7:
model

Clear all handles for matching results, all 3D object models, and the deformable surface

The 3D object models that were accessed or obtained during a search or visualization process should
be cleared with clear_object_model_3d before searching the model in another 3D scene. The same
applies to the matching result handles that were returned by the matching. They must be cleared with
clear_deformable_surface_matching_result.
When the 3D object model and the deformable surface model that was created from it are not
needed anymore, they should be cleared, too. Deformable surface models are cleared with
clear_deformable_surface_model.

4.5

Pose Estimation Using 3D Primitives Fitting

3D primitives fitting determines amongst others the positions or 3D poses of simple 3D shapes, so-called
“3D primitives”, that are fitted into segmented parts of a 3D scene. The 3D scene is a set of 3D points
that is available as 3D object model. It can be obtained, e.g., by stereo vision (chapter 5 on page 139)
or sheet of light (section 6.1 on page 177). The available types of 3D primitives comprise a sphere, a
cylinder, and a plane. When fitting 3D primitives into the segmented 3D data, the results comprise a
radius and position for a sphere, a radius and 3D pose for a cylinder, and a 3D pose for a plane.

Figure 4.13: 3D primitives fitting: (left) segmentation of 3D object model, (right) result of the fitting.

The fitting typically consists of two steps. The first step is the segmentation of the 3D scene into subsets of neighbored 3D points that may correspond to selected types of 3D primitives (see figure 4.13,
left). The segmentation can be applied by different means, depending on the input data. In any case, the
resulting sub-sets of 3D points must be available as 3D object models that contain, at least implicitly,
the coordinates of the 3D points and their meshing. The second step is the actual fitting (see figure 4.13,
right). There, the operator fit_primitives_object_model_3d tries to find the best fitting 3D primitive for an individual 3D object model. The result is another 3D object model from which the parameters
of the successfully fitted primitive can be queried.

4.5 Pose Estimation Using 3D Primitives Fitting

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An example for a 3D primitives fitting that uses a simple 2D segmentation to derive the 3D object model
of a single cylinder from a 3D scene that is provided by X, Y, and Z images is the HDevelop example
program hdevelop\3D-Tools\3D-Segmentation\fit_primitives_object_model_3d.hdev.

Figure 4.14: Height data used for 3D primitives fitting: (from left to right) X, Y, and Z image.

read_image (XYZ, '3d_machine_vision/segmentation/3d_primitives_xyz_02.tif')
access_channel (XYZ, X, 1)
access_channel (XYZ, Y, 2)
access_channel (XYZ, Z, 3)
threshold (Z, Region, 0.0, 0.83)
reduce_domain (X, Region, XTmp)
xyz_to_object_model_3d (XTmp, Y, Z, ObjectModel3DID)

A 3D primitive of the ’primitive_type’ ’cylinder’ is fitted into the 3D object model using the
operator fit_primitives_object_model_3d.
ParFitting := ['primitive_type','fitting_algorithm','output_xyz_mapping']
ValFitting := ['cylinder','least_squares_huber','true']
fit_primitives_object_model_3d (ObjectModel3DID, ParFitting, ValFitting, \
ObjectModel3DOutID)

The result of the fitting is a handle for a 3D object model from which information like the primitive’s
parameters can be queried with get_object_model_3d_params.
Often, a simple 2D segmentation is not suitable, e.g., because the 3D data is not derived from
Y, X, and Z images or because the 3D scene consists of several 3D structures that touch or
overlap. Then, a 3D segmentation using the operator segment_object_model_3d has to be applied as is shown in the HDevelop example program hdevelop\3D-Object-Model\Segmentation\
segment_object_model_3d.hdev. There, the individual objects can not be separated by a simple 2D
segmentation, because they overlap (see figure 4.16). The 3D data is accessed again from X, Y, and Z
images, but this time, the 3D object model is created without a preceeding 2D segmentation.

Pose Estimation

The individual channels of the image are accessed with access_channel (see figure 4.14). Then, a
threshold is applied to the Z image to separate the cylinder from the background (see figure 4.15). The
corresponding ROI is created with reduce_domain. From the reduced image channels, a 3D object
model is created using xyz_to_object_model_3d.

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3D Position Recognition of Known Objects

Figure 4.15: 3D primitives fitting: (left) simple threshold for 2D segmentation, (right) result returned by the
fitting.

read_image (XYZ, '3d_machine_vision/segmentation/3d_primitives_xyz_01.tif')
access_channel (XYZ, X, 1)
access_channel (XYZ, Y, 2)
access_channel (XYZ, Z, 3)
xyz_to_object_model_3d (X, Y, Z, ObjectModel3DID)

Figure 4.16: Height data used for 3D primitives fitting: (from left to right) X, Y, and Z image.

Instead, the derived 3D object model is prepared for a 3D segmentation by calling the operator
prepare_object_model_3d. Note that, if you do not explicitly prepare the 3D object model, the operator is called internally during the segmentation with segment_object_model_3d, which may slow
down the application if the same 3D object model is used several times.
prepare_object_model_3d (ObjectModel3DID, 'segmentation', 'false', \
'max_area_holes', 100)

The operator segment_object_model_3d segments the 3D object model into different sub-sets of 3D
points that have similar characteristics like the same orientation of the normals or the same curvature of

4.5 Pose Estimation Using 3D Primitives Fitting

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the underlying surface. By default, the operator not only segments the 3D scene but already tries to fit
primitives of the selected types into the segments (see figure 4.13 on page 132). Thus, no separate calls
to fit_primitives_object_model_3d are needed. Here, ’primitive_type’ is set to ’all’. That
is, for each sub-set of 3D points the best fitting type of primitive is returned (see figure 4.13 on page
132).

The result of the combined segmentation and fitting is a tuple of handles for the 3D object models
that represent the different sub-sets of 3D points. From each of these 3D object models, information
like the success of the fitting, the primitives’ types, and the primitives’ parameters can be queried with
get_object_model_3d_params. Querying the primitive’s parameters, a tuple is returned. The size and
content of this tuple depends on the type of the primitive. In particular, for a cylinder the tuple contains
seven values (three for the position, three for the orientation, and one for the radius), for a sphere it
contains four values (three for the position and one for the radius), and for a plane it contains again four
values (three for the unit normal vector and one for the orthogonal distance of the plane from the origin
of the coordinate system). How to access a single parameter from such a tuple is shown exemplarily for
the radius of a cylinder.
for Index := 0 to |ObjectModel3DOutID| - 1 by 1
get_object_model_3d_params (ObjectModel3DOutID[Index], \
'has_primitive_data', ParamValue)
if (ParamValue == 'true')
get_object_model_3d_params (ObjectModel3DOutID[Index], \
'primitive_parameter', GenParamValuesP)
get_object_model_3d_params (ObjectModel3DOutID[Index], \
'primitive_type', ParamValue)
if (ParamValue == 'cylinder')
RadiusCylinder := GenParamValuesP[6]
endif
endif
endfor

When the different 3D object models are not needed anymore, they are destroyed.
clear_object_model_3d (ObjectModel3DID)
for Index := 0 to |ObjectModel3DOutID| - 1 by 1
clear_object_model_3d (ObjectModel3DOutID[Index])
endfor

Pose Estimation

ParSegmentation := ['max_orientation_diff','max_curvature_diff', \
'output_xyz_mapping','min_area']
ValSegmentation := [0.13,0.11,'true',150]
ParFitting := ['primitive_type','fitting_algorithm']
ValFitting := ['all','least_squares_huber']
segment_object_model_3d (ObjectModel3DID, [ParSegmentation,ParFitting], \
[ValSegmentation,ValFitting], ObjectModel3DOutID)

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3D Position Recognition of Known Objects

4.6

Pose Estimation Using Calibrated Perspective Deformable Matching

The perspective deformable matching finds and locates objects that are similar to a template model in an
image. This matching approach uses the contours of the object in the images and is independent of the
perspective view on the object, i.e., perspective deformations are considered when searching the model
in unknown images. The perspective deformable matching can be applied either for a calibrated camera,
then the 3D pose of the object is returned, or for an uncalibrated camera, then only the 2D projective
transformation matrix (homography) is returned.
The perspective deformable matching is suitable for all planar objects or planar object parts that are
clearly distinguishable by their contours. Compared to the shape-based 3D matching (see section 4.2
on page 111 or the Solution Guide I, chapter 10 on page 145), there is no need to pregenerate different
views of an object. Thus, it is significantly faster. Hence, if you search for planar perspectively deformed
objects, we recommend the perspective deformable matching.
The Solution Guide II-B, section 3.6 on page 124 shows in detail how to apply the approach. Figure 4.17 shows the poses of engine parts obtained by the HDevelop example hdevelop\Applications\
Position-Recognition-3D\locate_engine_parts.hdev.

Figure 4.17: 3D poses of engine parts obtained by a calibrated perspective deformable matching.

4.7 Pose Estimation Using Calibrated Descriptor-Based Matching

4.7

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Pose Estimation Using Calibrated Descriptor-Based
Matching

Similar to the perspective deformable matching, the descriptor-based matching finds and locates objects
that are similar to a template model in an image. Again, the matching can be applied either for a calibrated camera, then the 3D pose of an object is returned, or for an uncalibrated camera, then only the 2D
projective transformation matrix (homography) is returned.
The essential difference between the descriptor-based and the perspective deformable matching is that the
descriptor-based matching is not based on contours but uses distinctive object points, so-called interest
points, to describe the template and to find the model in the image.

The Solution Guide II-B, section 3.7 on page 136 shows in detail how to apply the approach.
Figure 4.18 shows the 3D pose of a cookie box obtained by the HDevelop example hdevelop\
Applications\Object-Recognition-2D\locate_cookie_box.hdev. A comparison between the
calibrated descriptor-based matching and pose estimation methods that use manually extracted correspondences (see section 4.1 on page 106) is provided by the HDevelop example hdevelop\Matching\
Descriptor-Based\pose_from_point_correspondences.hdev.

Figure 4.18: 3D pose of a cookie box label obtained by a calibrated descriptor-based matching.

Pose Estimation

Note that the calibrated descriptor-based matching is suitable mainly to determine the 3D pose of planar
objects with characteristic texture and distinctive object points. For low-textured objects with rounded
edges you should select one of the other pose estimation approaches. Further, the descriptor-based matching is less accurate than the perspective deformable matching. But on the other hand, it is significantly
faster if a large search space, e.g., caused by a large scale range, is used.

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3D Position Recognition of Known Objects

4.8

Pose Estimation for Circles

HALCON offers an alternative approach to estimate the pose of 3D circles, which can be applied
with less effort than the previously described approaches. It is based on the known geometrical behavior of perspectively distorted circles. In particular, 3D circles are represented as ellipses in the
image. Using the extracted 2D ellipse of a 3D circle together with the internal camera parameters
and the known radius of the circle, the two possible 3D poses of the circle (having the same position but opposite orientations) can be obtained easily using the operator get_circle_pose. The
HDevelop examples hdevelop\Transformations\Poses\get_circle_pose.hdev and hdevelop\
Applications\Position-Recognition-3D\3d_position_of_circles.hdev show in detail how
to apply the approach.

4.9

Pose Estimation for Rectangles

Additionally to the pose estimation for 3D circles, also the poses of 3D rectangles can be estimated
with an approach that can be applied with less effort than the general approach. It is based on the
known geometrical behavior of perspectively distorted rectangles. In particular, a contour is segmented
into four line segments and their intersections are considered as the corners of a quadrangular contour.
Using the extracted 2D quadrangle of the 3D rectangle together with the internal camera parameters
and the known size of the rectangle, the four (or eight in case of a square) possible 3D poses of the
rectangle can be obtained easily using the operator get_rectangle_pose. The HDevelop examples
hdevelop\Applications\Position-Recognition-3D\get_rectangle_pose_barcode.hdev
and hdevelop\Applications\Position-Recognition-3D\3d_position_of_rectangle.hdev
show in detail how to apply the approach.

3D Vision With a Stereo System

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Chapter 5

3D Vision With a Stereo System
With a stereo system, i.e., with two or more cameras, you can derive 3D information of the surface of
arbitrarily shaped objects. Possible results are distance images, 3D coordinates, or 3D surfaces.
Typical applications of stereo vision comprise, but are not limited to, completeness checks, inspection of
ball grid arrays, etc. Reconstructing surfaces can also serve as a preprocessing step for surface-based 3D
matching (see section 4.3 on page 123) or 3D primitives fitting (see section 4.5 on page 132).

Stereo Vision

Figure 5.1 shows a surface reconstructed from four stereo images. Figure 5.2 shows an image of a
binocular stereo camera system, the resulting stereo image pair, and the height map that has been derived
from the reconstructed distance image.

Figure 5.1: Left: images of a 4-camera system; right: reconstructed surface (3D object model).

HALCON actually provides two stereo methods:
• Binocular stereo (section 5.3 on page 148) uses exactly two cameras. The result is a disparity
image, a distance image, or 3D coordinates. The latter are returned either for selected points or for
the complete view.

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3D Vision With a Stereo System

)

PP
PP

PP
q

PP
PP

PP
q

)

P
i
PP
PP

Figure 5.2: Top: stereo camera system; center: stereo image pair; bottom: height map.

5.1 The Principle of Stereo Vision

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• Multi-view stereo (section 5.4 on page 167) can use more than two cameras. Thus, it is able to
image the whole 3D object and not just the surface of a specific view. It returns results in form of
3D surfaces (given as 3D object models, see page 41) or 3D coordinates of selected points.
Before describing the two methods, we first take a look at some general topics. In particular, we
• introduce you to the principle of stereo vision (section 5.1) and
• show how to calibrate a stereo system (section 5.2 on page 144).

5.1

The Principle of Stereo Vision

Assume the simplified configuration of two parallel looking 1D cameras with identical internal parameters as shown in figure 5.3. Furthermore, the basis, i.e., the straight line connecting the two optical
centers of the two cameras, is assumed to coincide with the x-axis of the first camera.

zc
f

f
u

Image 1

u Image 2

Stereo Vision

P
Figure 5.3: Vertical section of a binocular stereo camera system.

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3D Vision With a Stereo System

Then, the image plane coordinates of the projections of the point P (xc , z c ) into the two images can be
expressed by

u1
u2

xc
zc
xc − b
= f
zc
= f

(5.1)
(5.2)

where f is the focal length and b the length of the basis.
The pair of image points that results from the projection of one object point into the two images is often
referred to as conjugate points or homologous points.
The difference between the two image locations of the conjugate points is called the disparity d:

d = (u2 − u1 ) = −

f ·b
zc

(5.3)

Given the camera parameters and the image coordinates of two conjugate points, the z c coordinate of the
corresponding object point P , i.e., its distance from the stereo camera system, can be computed by

zc = −

f ·b
d

(5.4)

Note that the internal camera parameters of both cameras and the relative pose of the second camera in
relation to the first camera are necessary to determine the distance of P from the stereo camera system.
Thus, the tasks to be solved for stereo vision are
1. to determine the camera parameters and
2. to determine conjugate points.
The first task is achieved by the calibration of the stereo camera system, which is described in section 5.2.
This calibration is quite similar to the calibration of a single camera, described in section 3.2 on page 68,
in fact, it even uses the same operators.
The second task is the so-called stereo matching process, which in HALCON is just a call of the operator binocular_disparity (or binocular_distance, respectively) for the correlation-based stereo
and a call of the operator binocular_disparity_mg (or binocular_distance_mg, respectively) for
multigrid stereo. These operators are described in section 5.3.5, together with the operators doing all the
necessary calculations to obtain world coordinates from the stereo images.
The multi-view surface reconstruction described in section 5.4.2.1 on page 170 extends the basic stereo
vision principle to more than one image pair. There, the matching step is “hidden” in the reconstruction
operators, which use the operator binocular_disparity internally to compute disparity images of the
individual stereo image pairs.

5.1 The Principle of Stereo Vision

5.1.1

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The Setup of a Stereo Camera System

The basic stereo camera system consists of two cameras looking at the same object from different positions (see figure 5.4). Multi-view stereo systems have additional cameras, but the principle stays the
same.
right

left

Cameras with
optical centers

yc
zc

yc
zc
c
f
u

P’

Virtual image planes

P’

Object point

Figure 5.4: Stereo camera system (r and y axes point towards the reader).

It is very important to ensure that neither the internal camera parameters (e.g., the focal length) nor the
relative pose between the cameras changes during the calibration process or between the calibration process and the ensuing application of the calibrated stereo camera system, because the calibration remains
valid only as long as the cameras preserve their relative pose. Therefore, it is advisable to mount the
cameras on a stable platform.

Stereo Vision

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3D Vision With a Stereo System

5.1.2

Resolution of a Stereo Camera System

The manner in which the cameras are placed influences the accuracy of the results that is achievable with
the stereo camera system.
The distance resolution ∆z, i.e., the accuracy with which the distance z of the object surface from the
stereo camera system can be determined, can be expressed by

∆z =

z2
· ∆d
f ·b

(5.5)

To achieve a high distance resolution, the setup should be chosen such that the length b of the basis as
well as the focal length f are large, and that the stereo camera system is placed as close as possible
to the object. In addition, the distance resolution depends directly on the accuracy ∆d with which the
disparities can be determined. If the calibration has been performed accurately and the corresponding
calibration error is in the order of 0.1 pixels, the disparities typically can be determined with an accuracy
of 1/5 up to 1/10 pixel, which corresponds to approximately 1 µm for a camera with 7.4 µm pixel size.
Note that generally the disparities cannot be determined more accurately than the calibration error of the
calibration of a stereo camera system.
In figure 5.5, the distance resolutions that are achievable in the ideal case are plotted as a function of the
distance for four different configurations of focal lengths and base lines, assuming ∆d to be 1 µm.
Note that if the ratio between b and z is very large, problems during the stereo matching process may
occur, because the two images of the stereo pair differ too much. The maximum reasonable ratio b/z
depends on the surface characteristics of the object. In general, objects with little height differences can
be imaged with a higher ratio b/z, whereas objects with larger height differences should be imaged with
a smaller ratio b/z.
If this is difficult to ensure in your application when using two cameras, you should consider using a
multi-view stereo system, i.e., more than two cameras.

5.1.3

Optimizing Focus with Tilt Lenses

For a successful stereo reconstruction, it is necessary that corresponding points of the object appear
sharp in both cameras. In some setups, e.g., when using a large magnification or telecentric lenses, this
requirement can be difficult to fulfill. This is because the depth of field is small and the angle between
the two cameras is relatively large. Therefore, the overlapping area, where the object is in focus from
both cameras, may not cover the whole object (see figure 5.6).
In such cases, tilt lenses can be used to adjust the focus planes of each camera to better fit the object
plane. Section 2.2.3 describes the effect of tilt lenses in more detail.

5.2

Calibrating the Stereo Camera System

As mentioned above, the calibration of the stereo camera system is very similar to the calibration of
a single camera (section 3.2 on page 68). The major difference is that it is not sufficient to view the

5.2 Calibrating the Stereo Camera System

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1.4
b = 10cm, f = 8mm
b = 10cm, f = 12.5mm
b = 20cm, f = 8mm
b = 20cm, f = 12.5mm

Distance resolution [mm]

1.2

0.8

0.6

0.4

0.2

0
0

0.2

0.4
0.6
Distance [m]

0.8

Stereo Vision

Figure 5.5: Distance resolution plotted over the distance (∆d = 1 µm).

Figure 5.6: If the depth of field is small while the angle between the two cameras is relatively large,
only a small part of the object plane is in focus (left). With tilt lenses, the focus planes of
the cameras can be aligned with the object plane. This is done, when the condition of the
Scheimpflug principle is met. After that, the whole object surface is in focus (right).

calibration plate from a single camera, but it must be visible in at least some of the overlapping parts of
the images that are taken by each pair of neighbored cameras (see section 5.2.2 on page 147 for details).
There, it must be completely visible for calibration plates with rectangularly arranged marks whereas for

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3D Vision With a Stereo System

calibration plates with hexagonally arranged marks at least one finder pattern must be visible.
In this section only a brief description of the calibration process is given. More details can be found in
section 3.2 on page 68. Here, the stereo-specific parts of the calibration process are described in depth.
In particular, it is shown
• how to create and configure the calibration data model for multiple cameras (section 5.2.1),
• how to acquire suitable calibration images (section 5.2.2),
• how to add the observation data of multiple cameras to the calibration data model (section 5.2.3),
and
• how to perform the calibration (section 5.2.4 on page 148).
The code fragments used for the following descriptions belong to the HDevelop example program
hdevelop\Calibration\Multi-View\calibrate_cameras_multiple_camera_setup.hdev.

5.2.1

Creating and Configuring the Calibration Data Model

As described in section 3.2.1 on page 69, the first step of preparing for a calibration is to create the
calibration data model with the operator create_calib_data. In the example, four cameras are used
with one calibration object.
NumCameras := 4
NumCalibObjects := 1
create_calib_data ('calibration_object', NumCameras, NumCalibObjects, \
CalibDataID)

Then, the initial camera parameters are set with the operator set_calib_data_cam_param (see also
section 3.2.2 on page 69). The parameter CameraIdx is set to ’all’, so that the values are set for all
cameras.
StartCamPar := [0.0085,0.0,0.0,0.0,0.0,0.0,6e-6,6e-6,Width * .5, \
Height * .5,Width,Height]
set_calib_data_cam_param (CalibDataID, 'all', CameraType, StartCamPar)

Finally, the calibration object is described with the operator set_calib_data_calib_object (see
section 3.2.3 on page 74).
CaltabDescr := 'caltab_100mm.descr'
set_calib_data_calib_object (CalibDataID, 0, CaltabDescr)

5.2 Calibrating the Stereo Camera System

5.2.2

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Acquiring Calibration Images

For the calibration of the stereo camera system, each camera acquires multiple images of one or more
calibration objects in different poses. Note that it is not necessary that the calibration object is always
visible in all poses for each camera. The only requirement is that the cameras can be “connected” in
a chain by the calibration object poses, e.g., that camera 0 and 1 observe the pose 0 of the calibration
object, camera 1 and 2 observe pose 3, and camera 2 and 3 observe pose 8 (see figure 5.7).
camera 0

camera 1

camera 2

Note that when taking the images for the calibration, the rules for taking the calibration images for the
single camera calibration (see section 3.2.4.1 on page 78) apply accordingly.
Note that you must not change the camera setup between the acquisition of the calibration images and
the acquisition of the stereo images of the object to be investigated. How to obtain suitable stereo image
pairs for the object to investigate is shown in section 5.3.3 on page 151.

5.2.3

Observing the Calibration Object

As in the case of the single camera calibration, the main input data for the camera calibration are the
observed points of the calibration objects in the camera images (see section 3.2.4 on page 78). Generally, you extract the observations and then add them to the calibration data model with the operator
set_calib_data_observ_points. If you use a standard HALCON calibration plate as calibration
object, you can extract and store the observations in a single step using find_calib_object. In the
example, this operator is called within a double loop over all poses and all cameras. The code raises a
warning if the calibration plate could not be found in an image.

Stereo Vision

Figure 5.7: Cameras are connected in a chain.

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3D Vision With a Stereo System

for PoseIndex := 0 to NumPoses - 1 by 1
for CameraIndex := 0 to NumCameras - 1 by 1
read_image (Image, FileName)
Message := ['Camera ' + CameraIndex,'Pose # ' + PoseIndex]
try
find_calib_object (Image, CalibDataID, CameraIndex, 0, \
PoseIndex, [], [])
catch (Exception)
if (Exception[0] == 8402)
Message := [Message,'No calibration tab found!']
elseif (Exception[0] == 8404)
Message := [Message,'Marks were not identified!']
else
Message := [Message,'Unknown Exception!.']
endif
Message := [Message,'This image will be ignored.']
endtry
disp_message (WindowHandles[CameraIndex], Message, 'window', 12, 12, \
Color, 'true')
endfor
endfor

5.2.4

Calibrating the Cameras

The actual calibration of the stereo camera system is carried out with the operator calibrate_cameras
(see section 3.2.6 on page 81).
calibrate_cameras (CalibDataID, Error)

The calibration stores its results in the calibration data model. How to access them is described in
separate sections for binocular stereo (section 5.3.2 on page 150) and multi-view stereo (section 5.4.1.1
on page 168) because the methods use the results in different forms.

5.3

Binocular Stereo Vision

With the operators for binocular stereo vision, you can compute disparity and distance images and 3D coordinates using two cameras. In fact, HALCON provides three methods for binocular stereo: correlationbased stereo, multigrid stereo, and multi-scanline stereo (see section 5.3.1 for the differences).
The following sections show how to
• access the results of the calibration (section 5.3.2 on page 150),
• rectify the stereo images (section 5.3.4 on page 151), and
• reconstruct 3D information (section 5.3.5 on page 154).

5.3 Binocular Stereo Vision

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If not stated otherwise, the example programs used in this section are
• stereo_calibration.dev,
• height_above_reference_plane_from_stereo.dev, and
• 3d_information_for_selected_points.dev.
They can be found in the directory solution_guide\3d_vision.
As an alternative to fully calibrated stereo, HALCON offers the so-called uncalibrated stereo vision.
Here, the relation between the cameras is determined from the scene itself, i.e., without needing a special
calibration object. Please refer to section 5.3.6 on page 166 for more information about this method.

5.3.1

Comparison of the Stereo Matching Approaches CorrelationBased, Multigrid, and Multi-Scanline Stereo

In HALCON, three approaches for stereo matching are available: the traditionally used correlation-based
stereo matching, the multigrid stereo matching, and the multi-scanline stereo matching.
The correlation-based stereo matching uses correlation techniques to find corresponding points and
thus to determine the disparities or distances for the observed image points. The disparities or distances
are calulated with the operators binocular_disparity or binocular_distance, respectively. The
correlation-based stereo is characterized by the following advantages and disadvantages:
Most important advantages of correlation-based stereo:
• fast,

• is invariant against gray-value changes.
Most important disadvantage of correlation-based stereo:
• works good only for significantly textured areas. Areas without enough texture cannot be reconstructed.
The multigrid stereo matching uses a variational approach based on multigrid methods. This approach returns disparity and distance values also for image parts that contain no texture (as long as
these parts are surrounded by significant structures between which an interpolation of values is possible). The disparities or distances are calculated with the operators binocular_disparity_mg or
binocular_distance_mg, respectively. The multigrid stereo is characterized by the following advantages and disadvantages:
Most important advantages of multigrid stereo:
• interpolates 3D information for areas without texture based on the surrounding areas,

Stereo Vision

• can be automatically parallelized on multi-core or multi-processor hardware, and

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3D Vision With a Stereo System

• in particular for edges, the accuracy in general is higher than for correlation-based stereo, and
• the resolution is higher than for correlation-based stereo, i.e., smaller objects can be reconstructed.
Most important disadvantages of multigrid stereo:
• only partially invariant against gray value changes,
• edges are somewhat blurred, and
• it cannot be parallelized automatically.
The multi-scanline stereo matching uses a multi-scanline optimization. Similar to the multigrid approach, it returns disparity and distance values also for image parts that contain only little texture but tries
to preserve discontinuities, which are blurred by the multigrid approach. The disparities or distances are
calculated with the operators binocular_disparity_ms or binocular_distance_ms, respectively.
The multi-scanline stereo is characterized by the following advantages and disadvantages:
Most important advantages of multi-scanline stereo:
• determines 3D information for areas with little texture and
• preserves discontinuities.
Most important disadvantages of multi-scanline stereo:
• runtime increases significantly with image size and disparity search range and
• high memory consumption.

5.3.2

Accessing the Calibration Results

As described in section 3.2.7 on page 81, you access the results of the calibration with the operator
get_calib_data. In addition to the internal camera parameters, we now use the operator also to get the
relative pose between the two cameras.
get_calib_data (CalibDataID, 'camera', 0, 'params', CamParamL)
get_calib_data (CalibDataID, 'camera', 1, 'params', CamParamR)
get_calib_data (CalibDataID, 'camera', 1, 'pose', cLPcR)

If you want to perform the calibration in an offline step, you can save the camera setup model with
write_camera_setup_model and then destroy the calibration data model with clear_calib_data.
write_camera_setup_model (CameraSetupModelID, 'stereo_camera_setup.csm')
clear_calib_data (CalibDataID)

5.3 Binocular Stereo Vision

5.3.3

C-151

Acquiring Stereo Images

The following rules help you to acquire suitable stereo image pairs. Section 5.3.3.1 and section 5.3.3.2
provide you with background information to understand the rules.
• Do not change the camera setup between the acquisition of the calibration images and the acquisition of the stereo images of the object to be investigated. How to obtain suitable images for the
calibration of a stereo camera system is shown in section 5.2.2 on page 147.
• Ensure a proper illumination of the object, e.g., avoid reflections.
• If the object shows no texture, consider to project texture onto it or use multigrid stereo (see
section 5.3.1 on page 149).
• Place the object such that repetitive patterns are not aligned with the rows of the rectified images.
5.3.3.1

Image Texture

The 3D coordinates of each object point are derived by intersecting the lines of sight of the respective
conjugate image points. The conjugate points are determined by an automatic matching process. This
matching process has some properties that should be accounted for during the image acquisition.

This applies in particular for the correlation-based (binocular) stereo (which is currently the base for
multi-view stereo). The multigrid (binocular) stereo, however, can interpolate values in areas without
texture. But note that even then, a certain amount of texture must be available to enable the interpolation.
In section 5.3.1 on page 149 the differences between both stereo matching approaches are described in
more detail.
5.3.3.2

Repetitive Patterns

If the images contain repetitive patterns, the matching process may be confused, since in this case many
points look alike. In order to make the matching process fast and reliable, the stereo images are rectified
such that pairs of conjugate points always have identical row coordinates in the rectified images, i.e.,
that the search space in the second rectified image is reduced to a line. With this, repetitive patterns can
disturb the matching process only if they are parallel to the rows of the rectified images (figure 5.9).

5.3.4

Rectifying the Stereo Images

With the internal camera parameters and the relative pose, the stereo images can be rectified, so that
conjugate points lie on the same row in both rectified images.
Note that it is assumed that the parameters of the first image of a pair stem from the left image and the
parameters of the second image stem from the right image, whereas the notations ’left’ and ’right’ refer

Stereo Vision

For each point of the first image, the conjugate point in the second image must be determined. This point
matching relies on the availability of texture. The conjugate points cannot be determined correctly in
areas without sufficient texture (figure 5.8).

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3D Vision With a Stereo System

@
@
R
@

Figure 5.8: Rectified stereo images and matching result of the correlation-based stereo in a poorly textured
area (regions where the matching process failed are displayed white).

to the line of sight of the two cameras (see figure 5.4 on page 143). If the images are used in the reverse
order, they will appear upside down after the rectification.
In figure 5.10 the original images of a stereo pair are shown, where the two cameras are rotated heavily
with respect to each other. The corresponding rectified images are displayed in figure 5.11.
The rectification itself is carried out using the operators gen_binocular_rectification_map and
map_image. The operator gen_binocular_rectification_map requires the internal camera parameters of both cameras and the relative pose of the second camera in relation to the first camera.
gen_binocular_rectification_map (MapL, MapR, CamParamL, CamParamR, cLPcR, 1, \
'geometric', 'bilinear', RectCamParL, \
RectCamParR, CamPoseRectL, CamPoseRectR, \
RectLPosRectR)

The parameter SubSampling can be used to change the size and resolution of the rectified images with
respect to the original images. A value of 1 indicates that the rectified images will have the same size as

5.3 Binocular Stereo Vision

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@
@
R
@

Stereo Vision

Figure 5.9: Rectified stereo images with repetitive patterns aligned to the image rows and cutout of the
matching result (regions where the matching process failed are displayed white).

Figure 5.10: Original stereo images.

the original images. Larger values lead to smaller images with a resolution reduced by the given factor,
smaller values lead to larger images.
Reducing the image size has the effect that the following stereo matching process runs faster, but also
that less details are visible in the result. In general, it is proposed not to use values below 0.5 or above 2.
Otherwise, smoothing or aliasing effects occur, which may disturb the matching process.
The rectification process can be described as projecting the original images onto a common rectified
image plane. The method to define this plane can be selected by the parameter Method. So far, only
the method ’geometric’ can be selected, in which the orientation of the common rectified image plane is

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Figure 5.11: Rectified stereo images.

defined by the cross product of the base line and the line of intersection of the two original image planes.
The rectified images can be thought of as being acquired by a virtual stereo camera system, called
rectified stereo camera system, as displayed in figure 5.12. The optical centers of the rectified cameras
are the same as for the real cameras, but the rectified cameras are rotated such that they are looking
parallel and that their x-axes are collinear. In addition, both rectified cameras have the same focal length.
Therefore, the two image planes coincide. Note that the principal point of the rectified images, which is
the origin of the image plane coordinate system, may lie outside the image.
The parameter Interpolation specifies whether bilinear interpolation (’bilinear’) should be applied
between the pixels of the input images or whether the gray value of the nearest pixel (’none’) should be
used. Bilinear interpolation yields smoother rectified images, whereas the use of the nearest neighbor is
faster.
The operator returns the rectification maps and the camera parameters of the virtual, rectified cameras.
Finally, the operator map_image can be applied to both stereo images using the respective rectification
map generated by the operator gen_binocular_rectification_map.
map_image (ImageL, MapL, ImageRectifiedL)
map_image (ImageR, MapR, ImageRectifiedR)

If the calibration was erroneous, the rectification will produce wrong results. This can be checked very
easily by comparing the row coordinates of conjugate points selected from the two rectified images. If the
row coordinates of conjugate points are different within the two rectified images, they are not correctly
rectified. In this case, you should check the calibration process carefully.
An incorrectly rectified image pair may look like the one displayed in figure 5.13.

5.3.5

Reconstructing 3D Information

There are many possibilities to derive 3D information from rectified stereo images.

5.3 Binocular Stereo Vision

xc
yc

zc
f

u
Rectified images

v
r

Object point

P’

P’
r

Figure 5.12: Rectified stereo camera system.

Non-metrical information: If only non-metrical information about the surface of an object is needed, it
may be sufficient to determine the disparities within the overlapping area of the stereo image pair
by using the operator binocular_disparity for correlation-based stereo (section 5.3.5.1 on
page 156) or binocular_disparity_mg for multigrid stereo (section 5.3.5.2 on page 159). The
differences between both stereo matching approaches are described in more detail in section 5.3.1
on page 149.
Distance of the object surface: If
metrical
information
is
required,
the
operator
binocular_distance (section 5.3.5.4 on page 161) or binocular_distance_mg (section 5.3.5.5 on page 163), respectively, can be used to extract the distance of the object surface
from the stereo camera system.
3D coordinate images: Having a disparity image of a rectified binocular stereo system, you can additionally derive the corresponding x, y, and z coordinates using disparity_image_to_xyz (see

Stereo Vision

Rectified cameras with
optical centers

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3D Vision With a Stereo System

Figure 5.13: Incorrectly rectified stereo images.

section 5.3.5.8 on page 164).
3D coordinates for selected points: To derive metrical information for selected points only, the operators disparity_to_distance or disparity_to_point_3d can be used. The first of these two
operators calculates the distance z of points from the stereo camera system based on their disparity (see section 5.3.5.7 on page 164). The second operator calculates the x, y, and z coordinates
from the row and column position of a point in the first rectified image and from its disparity (see
section 5.3.5.8 on page 164).
Alternatively, the operator intersect_lines_of_sight can be used to calculate the x, y, and z
coordinates of selected points (see section 5.3.5.9 on page 165). Then, there is no need to determine the disparities in advance. Only the image coordinates of the conjugate points and the camera
parameters are needed. This operator can also handle image coordinates of the original stereo images. Thus, the rectification can be omitted. In exchange, you must determine the conjugate points
by yourself.
Note that all operators that deal with disparities or distances require all input to be based on the rectified
images. This applies to the image coordinates as well as to the camera parameters.
5.3.5.1

Determining Disparities Using Correlation-Based Stereo

Disparities are an indicator for the distance z of object points from the stereo camera system, since points
with equal disparities also have equal distances z (equation 5.4 on page 142).
Therefore, if it is only necessary to know whether there are locally high objects, it is sufficient to derive
the disparities. For correlation-based stereo, this is done by using the operator binocular_disparity.
binocular_disparity (ImageRectifiedL, ImageRectifiedR, DisparityImage, \
ScoreImageDisparity, 'ncc', MaskWidth, MaskHeight, \
TextureThresh, MinDisparity, MaxDisparity, NumLevels, \
ScoreThresh, 'left_right_check', 'interpolation')

The operator requires the two rectified images as input. The disparities are derived only for those conjugate points that lie within the respective image domain in both images. With this, it is possible to speed

5.3 Binocular Stereo Vision

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up the calculation of the disparities if the image domain of at least one of the two rectified images is
reduced to a region of interest, e.g., by using the operator reduce_domain.
Several parameters can be used to control the behavior of the matching process that is performed by the
operator binocular_disparity to determine the conjugate points:
With the parameter Method, the matching function is selected. The methods ’sad’ (summed absolute
differences) and ’ssd’ (summed squared differences) compare the gray values of the pixels within a
matching window directly, whereas the method ’ncc’ (normalized cross correlation) compensates for the
mean gray value and its variance within the matching window. Therefore, if the two images differ in
brightness and contrast, the method ’ncc’ should be preferred. However, since the internal computations
are less complex for the methods ’sad’ and ’ssd’, they are faster than the method ’ncc’.
The width and height of the matching window can be set independently with the parameters MaskWidth
and MaskHeight. The values should be odd numbers. Otherwise they will be increased by one. A larger
matching window will lead to a smoother disparity image, but may result in the loss of small details. In
contrast, the results of a smaller matching window tend to be noisy but they show more spatial details.
Because the matching process relies on the availability of texture, low-textured areas can be excluded
from the matching process. The parameter TextureThresh defines the minimum allowed variance
within the matching window. For areas where the texture is too low, no disparities will be determined.
The parameters MinDisparity and MaxDisparity define the minimum and maximum disparity values.
They are used to restrict the search space for the matching process. If the specified disparity range does
not contain the actual range of the disparities, the conjugate points cannot be found correctly. Therefore,
the disparities will be incomplete and erroneous. On the other hand, if the disparity range is specified
too large, the matching process will be slower and the probability of mismatches increases.

• If you know the minimum and maximum distance of the object from the stereo camera system
(section 5.3.5.4 on page 161), you can use the operator distance_to_disparity to determine
the respective disparity values.
• You can also determine these values directly from the rectified images. For this, you should display
the two rectified images and measure the approximate column coordinates of the point N , which
image1
image2
is nearest to the stereo camera system (Ncol
and Ncol
) and of the point F , which is the
image1
image2
farthest away (Fcol
and Fcol
), each in both rectified images.
Now, the values for the definition of the disparity range can be calculated as follows:
MinDisparity
MaxDisparity

image2
image1
= Ncol
− Ncol

image2
Fcol

−

image1
Fcol

(5.6)
(5.7)

The operator binocular_disparity uses image pyramids to improve the matching speed. The disparity range specified by the parameters MinDisparity and MaxDisparity is only used on the uppermost
pyramid level, indicated by the parameter NumLevels. Based on the matching results on that level, the
disparity range for the matching on the next lower pyramid levels is adapted automatically.
The benefits with respect to the execution time are greatest if the objects have different regions between
which the appropriate disparity range varies strongly. However, take care that the value for NumLevels is

Stereo Vision

Therefore, it is important to set the parameters MinDisparity and MaxDisparity carefully. There are
several possibilities to determine the appropriate values:

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not set too large, as otherwise the matching process may fail because of lack of texture on the uppermost
pyramid level.
The parameter ScoreThresh specifies which matching scores are acceptable. Points for which the
matching score is not acceptable are excluded from the results, i.e., the resulting disparity image has a
reduced domain that comprises only the accepted points.
Note that the value for ScoreThresh must be set according to the matching function selected via
Method. The two methods ’sad’ (0 ≤ score ≤ 255) and ’ssd’ (0 ≤ score ≤ 65025) return lower matching
scores for better matches. In contrast, the method ’ncc’ (-1 ≤ score ≤ 1) returns higher values for better
matches, where a score of zero indicates that the two matching windows are totally different and a score
of minus one denotes that the second matching window is exactly inverse to the first matching window.
The parameter Filter can be used to activate a downstream filter by which the reliability of the resulting
disparities is increased. Currently, it is possible to select the method ’left_right_check’, which verifies
the matching results based on a second matching in the reverse direction. Only if both matching results
correspond to each other, the resulting conjugate points are accepted. In some cases, this may lead to
gaps in the disparity image, even in well textured areas, as this verification is very strict. If you do not
want to verify the matching results based on the ’left_right_check’, set the parameter Filter to ’none’.
The subpixel refinement of the disparities is switched on by setting the parameter SubDisparity to
’interpolation’. It is switched off by setting the parameter to ’none’.
The results of the operator binocular_disparity are the two images Disparity and Score, which
contain the disparities and the matching score, respectively. In figure 5.14, a rectified stereo image pair
is displayed, from which the disparity and score images that are displayed in figure 5.15 were derived.

Figure 5.14: Rectified stereo images.

Both resulting images refer to the image geometry of the first rectified image, i.e., the disparity for the
point (r,c) of the first rectified image is the gray value at the position (r,c) of the disparity image. The
disparity image can, e.g., be used to extract the components of the board, which would be more difficult
in the original images, i.e., without the use of 3D information.
In figure 5.15, areas where the matching did not succeed, i.e., undefined regions of the images, are
displayed white in the disparity image and black in the score image.

5.3 Binocular Stereo Vision

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Figure 5.15: Disparity image (left) and score image (right).

5.3.5.2

Determining Disparities Using Multigrid Stereo

For multigrid stereo, the disparities can be derived with binocular_disparity_mg. Similar to the
correlation-based approach, the two rectified images are used as input and the disparity image as well as
a score image are returned. But as the disparities are obtained by a different algorithm, the parameters
that control the behavior of the multigrid stereo matching process are completely different. In particular, the multigrid-specific control parameters are GrayConstancy, GradientConstancy, Smoothness,
InitialGuess, CalculateScore, MGParamName, and MGParamValue. They are explained in detail in
the Reference Manual entry for binocular_disparity_mg. The significant differences between the
different stereo matching approaches are listed in section 5.3.1 on page 149.

There, first the two stereo images are rectified.
read_image (ImageL, 'stereo/board/board_l_01')
read_image (ImageR, 'stereo/board/board_r_01')
CamParamL := [0.0130507774353,-665.817817207,1.4803417027e-5,1.48e-5, \
155.89225769,126.70664978,320,240]
CamParamR := [0.0131776504517,-731.860636733,1.47997569293e-5,1.48e-5, \
162.98210144,119.301040649,320,240]
RelPose := [0.153573364258,-0.00373362231255,0.0447351264954, \
0.174289124775,319.843388114,359.894955219,0]
gen_binocular_rectification_map (MapL, MapR, CamParamL, CamParamR, RelPose, \
1, 'geometric', 'bilinear', RectCamParL, \
RectCamParR, CamPoseRectL, CamPoseRectR, \
RectLPosRectR)
map_image (ImageL, MapL, ImageRectifiedL)
map_image (ImageR, MapR, ImageRectifiedR)

Then, binocular_disparity_mg is called four times, each time with a different parameter for the

Stereo Vision

How to determine the disparities of the components on a PCB using binocular_disparity_mg
with different levels of accuracy is shown in the HDevelop example program hdevelop\
3D-Reconstruction\Binocular-Stereo\binocular_disparity_mg.hdev.

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parameter MGParamValue, which sets the accuracy that should be achieved. Note that an increasing
accuracy also leads to an increasing runtime.
DefaultParameters := ['fast','fast_accurate','accurate','very_accurate']
for I := 0 to |DefaultParameters| - 1 by 1
Parameters := DefaultParameters[I]
binocular_disparity_mg (ImageRectifiedL, ImageRectifiedR, Disparity, \
Score, 1, 10, 5, 0, 'false', \
'default_parameters', Parameters)
endfor

5.3.5.3

Determining Disparities Using Multi-Scanline Stereo

For multi-scanline stereo, the disparities can be derived with binocular_disparity_ms. Similar
to the correlation-based approach, the two rectified images are used as input and the disparity image as well as a score image are returned. But as the disparities are obtained by a different algorithm, the parameters that control the behavior of the multi-scanline stereo matching process are completely different. In particular, the control parameters specific for the multi-scanline approach are
SurfaceSmoothing and EdgeSmoothing. They are explained in detail in the Reference Manual entry for binocular_disparity_ms. The significant differences between the different stereo matching
approaches are listed in section 5.3.1 on page 149.
How to determine the disparities of the components on a PCB using binocular_disparity_ms
with different amounts of smoothing is shown in the HDevelop example program hdevelop\
3D-Reconstruction\Binocular-Stereo\binocular_disparity_ms.hdev.
There, first the two stereo images are rectified.
read_image (ImageL, 'stereo/board/board_l_01')
read_image (ImageR, 'stereo/board/board_r_01')
CamParamL := [0.0130507774353,-665.817817207,1.4803417027e-5,1.48e-5, \
155.89225769,126.70664978,320,240]
CamParamR := [0.0131776504517,-731.860636733,1.47997569293e-5,1.48e-5, \
162.98210144,119.301040649,320,240]
RelPose := [0.153573364258,-0.00373362231255,0.0447351264954, \
0.174289124775,319.843388114,359.894955219,0]
gen_binocular_rectification_map (MapL, MapR, CamParamL, CamParamR, RelPose, \
1, 'geometric', 'bilinear', RectCamParL, \
RectCamParR, CamPoseRectL, CamPoseRectR, \
RectLPosRectR)
map_image (ImageL, MapL, ImageRectifiedL)
map_image (ImageR, MapR, ImageRectifiedR)

Then, binocular_disparity_ms is called four times, each time with a different parameter settings for
the parameters SurfaceSmoothing and EdgeSmoothing, which control the amount of smoothing.

5.3 Binocular Stereo Vision

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SurfaceSmoothing := [0,10,20,30,40,50,0,0,0,0,0,30]
EdgeSmoothing := [0,0,0,0,0,0,10,20,30,40,50,20]
for I := 0 to |SurfaceSmoothing| - 1 by 1
binocular_disparity_ms (ImageRectifiedL, ImageRectifiedR, Disparity, \
Score, 20, 40, SurfaceSmoothing[I], \
EdgeSmoothing[I], [], [])
endfor

5.3.5.4

Determining Distances Using Correlation-Based Stereo

The distance of an object point from the stereo camera system is defined as its distance from the xy-plane of the coordinate system of the first rectified camera. For correlation-based stereo, it can
be determined by the operator binocular_distance, which is used analogously to the operator
binocular_disparity described in section 5.3.5.1 on page 156.
binocular_distance (ImageRectifiedL, ImageRectifiedR, DistanceImage, \
ScoreImageDistance, RectCamParL, RectCamParR, \
RectLPosRectR, 'ncc', MaskWidth, MaskHeight, \
TextureThresh, MinDisparity, MaxDisparity, NumLevels, \
ScoreThresh, 'left_right_check', 'interpolation')

The three additional parameters, namely the camera parameters of the rectified cameras as well as the
relative pose of the second rectified camera in relation to the first rectified camera can be taken directly
from the output of the operator gen_binocular_rectification_map.

Stereo Vision

Figure 5.16 shows the distance image and the respective score image for the rectified stereo pair of
figure 5.14 on page 158. Because the distance is calculated directly from the disparities and from the
camera parameters, the distance image looks similar to the disparity image (figure 5.15). What is more,
the score images are identical, since the underlying matching process is identical.

Figure 5.16: Distance image (left) and score image (right).

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3D Vision With a Stereo System

It can be seen from figure 5.16 that the distance of the board changes continuously from left to right.
The reason is that, in general, the x-y-plane of the coordinate system of the first rectified camera will be
tilted with respect to the object surface (see figure 5.17).

fied c

lan
x−y−p

e recti
e of th

amera

Rectified cameras
yc

ce
Distan

Object surface
Figure 5.17: Distances of the object surface from the x-y-plane of the coordinate system of the first
rectified camera.

If it is necessary that one reference plane of the object surface has a constant distance value of, e.g.,
zero, the tilt can be compensated easily: First, at least three points that lie on the reference plane must be
defined. These points are used to determine the orientation of the (tilted) reference plane in the distance
image. Therefore, they should be selected such that they enclose the region of interest in the distance
image. Then, a distance image of the (tilted) reference plane can be simulated and subtracted from the
distance image of the object. Finally, the distance values themselves must be adapted by scaling them
with the cosine of the angle between the tilted and the corrected reference plane.
These calculations are carried out in the procedure tilt_correction, which is part of the example program solution_guide\3d_vision\height_above_reference_plane_from_stereo.hdev (appendix A.4 on page 280).
procedure tilt_correction (DistanceImage, RegionDefiningReferencePlane,
DistanceImageCorrected):::

5.3 Binocular Stereo Vision

C-163

In principle, this procedure can also be used to correct the disparity image, but note that you must not
use the corrected disparity values as input to any operators that derive metric information.
If the reference plane is the ground plane of the object, an inversion of the distance image generates an
image that encodes the heights above the ground plane. Such an image is displayed on the left hand side
in figure 5.18.
Objects of different height above or below the ground plane can be segmented easily using a simple
threshold with the minimal and maximal values given directly in units of the world coordinate system,
e.g., meters. The image on the right hand side of figure 5.18 shows the results of such a segmentation,
which can be carried out based on the corrected distance image or the image of the heights above the
ground plane.

5.3.5.5

Determining Distances Using Multigrid Stereo

For multigrid stereo, the distance of an object point from the stereo camera system can be determined with the operator binocular_distance_mg, which is used analogously to the operator
binocular_disparity_mg described in section 5.3.5.2 on page 159, but with the three additional parameters described also for binocular_distance. These are the camera parameters of the rectified
cameras as well as the relative pose of the second rectified camera in relation to the first rectified camera,
which can be taken directly from the output of the operator gen_binocular_rectification_map.
How to determine the depths of the components on a PCB with high accuracy is shown
in the HDevelop example program hdevelop\3D-Reconstruction\Binocular-Stereo\
binocular_distance_mg.hdev.
binocular_distance_mg (ImageRectifiedL, ImageRectifiedR, Distance, Score, \
RectCamParL, RectCamParR, RectLPosRectR, 1, 10, 5, \
0, 'false', 'default_parameters', 'accurate')

Stereo Vision

Figure 5.18: Left: Height above the reference plane; Right: Segmentation of high objects (white: 00.4 mm, light gray: 0.4-1.5 mm, dark gray: 1.5-2.5 mm, black: 2.5-5 mm).

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3D Vision With a Stereo System

5.3.5.6

Determining Distances Using Multi-Scanline Stereo

For multi-scanline stereo, the distance of an object point from the stereo camera system can be determined with the operator binocular_distance_ms, which is used analogously to the operator
binocular_disparity_ms described in section 5.3.5.3 on page 160, but with the three additional parameters described also for binocular_distance. These are the camera parameters of the rectified
cameras as well as the relative pose of the second rectified camera in relation to the first rectified camera,
which can be taken directly from the output of the operator gen_binocular_rectification_map.
How to determine the depths of the components on a PCB with high accuracy, especially at
discontinuities, is shown in the HDevelop example program hdevelop\3D-Reconstruction\
Binocular-Stereo\binocular_distance_ms.hdev.
binocular_distance_ms (ImageRectifiedL, ImageRectifiedR, Distance, Score, \
RectCamParL, RectCamParR, RelPose, 20, 40, 30, 20, \
[], [])

5.3.5.7

Determining Distances for Selected Points from the Disparity Image

If only the distances of selected points should be determined, the operator disparity_to_distance
can be used. It simply transforms given disparity values into the respective distance values. For example,
if you want to know the distances of two given points from the stereo camera system you can determine
the respective disparities from the disparity image and transform them into distances.
get_grayval (Disparity, RowL, ColumnL, DisparityOfSelectedPoints)
disparity_to_distance (RectCamParL, RectCamParR, RectLPosRectR, \
DisparityOfSelectedPoints, DistanceOfPoints)

This transformation is constant for the entire rectified image, i.e., all points having the same disparity
have the same distance from the x-y-plane of the coordinate system of the first rectified camera. Therefore, besides the camera parameters of the rectified cameras, only the disparity values need to be given.
5.3.5.8

Determining 3D Coordinates from the Disparity Image

If the x, y, and z coordinates of points have to be calculated, different operators are available that derive
the corresponding information from the disparity image:
disparity_to_point_3d computes the 3D coordinates for specified image coordinates and returns
them in three tuples.
disparity_to_point_3d (RectCamParL, RectCamParR, RectLPosRectR, RowL, \
ColumnL, DisparityOfSelectedPoints, \
X_CCS_FromDisparity, Y_CCS_FromDisparity, \
Z_CCS_FromDisparity)

5.3 Binocular Stereo Vision

C-165

disparity_image_to_xyz computes 3D coordinates for the complete image and returns them in three
images (see the HDevelop example program hdevelop\3D-Reconstruction\Binocular-Stereo\
disparity_image_to_xyz.hdev).
disparity_image_to_xyz (DisparityImage, X, Y, Z, RectCamParL, RectCamParR, \
RectLPosRectR)

The operators require the camera parameters of the two rectified cameras and the relative pose of the
cameras.
The x, y, and z coordinates are returned in the coordinate system of the first rectified camera.
5.3.5.9

Determining 3D Coordinates for Selected Points from Point Correspondences

If only the 3D coordinates of selected points should be determined, you can alternatively use the operator intersect_lines_of_sight to determine the x, y, and z coordinates of points from the image
coordinates of the respective conjugate points. Note that you must determine the image coordinates of
the conjugate points yourself.
intersect_lines_of_sight (RectCamParL, RectCamParR, RectLPosRectR, RowL, \
ColumnL, RowR, ColumnR, X_CCS_FromIntersect, \
Y_CCS_FromIntersect, Z_CCS_FromIntersect, \
Dist)

The x, y, and z coordinates are returned in the coordinate system of the first (rectified) camera.

It is possible to transform the x, y, and z coordinates determined by the latter two operators from the
coordinate system of the first (rectified) camera into a given world coordinate system (WCS), e.g., a
coordinate system with respect to the building plan of a factory building. For this, a homogeneous transformation matrix, which describes the transformation between the two coordinate systems, is needed.
This homogeneous transformation matrix can be determined in various ways. The easiest way is to
take an image of a HALCON calibration plate with the first camera only. If the 3D coordinates refer to the rectified camera coordinate system, the image must be rectified as well. Then, the pose of
the calibration plate in relation to the first (rectified) camera can be determined using the operators
find_calib_object and get_calib_data_observ_points (note that before applying these operators a HALCON calibration data model must be created and initialized).
find_calib_object (ImageRectifiedL, CalibDataID, 0, 0, 0, [], [])
get_calib_data_observ_points (CalibDataID, 0, 0, 0, RCoordL, CCoordL, \
IndexLnew, PoseL)

The resulting pose can be converted into a homogeneous transformation matrix.
pose_to_hom_mat3d (PoseL, HomMat3D_WCS_to_RectCCS)

Stereo Vision

The operator can also handle image coordinates of the original stereo images. Thus, the rectification can
be omitted. In this case, the camera parameters of the original stereo cameras have to be given instead of
the parameters of the rectified cameras.

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3D Vision With a Stereo System

If necessary, the transformation matrix can be modified with the operators hom_mat3d_rotate_local,
hom_mat3d_translate_local, and hom_mat3d_scale_local.
hom_mat3d_translate_local (HomMat3D_WCS_to_RectCCS, 0.01125, -0.01125, 0, \
HomMat3DTranslate)
hom_mat3d_rotate_local (HomMat3DTranslate, rad(180), 'y', \
HomMat3D_WCS_to_RectCCS)

The homogeneous transformation matrix must be inverted in order to represent the transformation from
the (rectified) camera coordinate system into the WCS.
hom_mat3d_invert (HomMat3D_WCS_to_RectCCS, HomMat3D_RectCCS_to_WCS)

Finally, the 3D coordinates can be transformed using the operator affine_trans_point_3d.
affine_trans_point_3d (HomMat3D_RectCCS_to_WCS, X_CCS_FromIntersect, \
Y_CCS_FromIntersect, Z_CCS_FromIntersect, X_WCS, \
Y_WCS, Z_WCS)

The homogeneous transformation matrix can also be determined from three specific points. If
the origin of the WCS, a point on its x-axis, and a third point that lies in the x-y-plane, e.g.,
directly on the y-axis, are given, the transformation matrix can be determined using the procedure gen_hom_mat3d_from_three_points, which is part of the HDevelop example program
solution_guide\3d_vision\3d_information_for_selected_points.hdev.
procedure gen_hom_mat3d_from_three_points (Origin, PointOnXAxis,
PointInXYPlane, HomMat3d):::

The resulting homogeneous transformation matrix can be used as input for the operator
affine_trans_point_3d, as shown above.

5.3.6

Uncalibrated Stereo Vision

Similar to uncalibrated mosaicking (see chapter 10 on page 247), HALCON also provides an “uncalibrated” version of stereo vision, which derives information about the cameras from the scene itself by
matching characteristic points. The main advantage of this method is that you need no special calibration
object. The main disadvantage of this method is that, without a precisely known calibration object, the
accuracy of the result is highly dependent on the content of the scene, i.e., the accuracy of the result
degrades if the scene does not contain enough 3D information or if the extracted characteristic points in
the two images do not precisely correspond to the same world points, e.g., due to occlusion.
In fact, HALCON provides two versions of uncalibrated stereo: Without any knowledge about the
cameras and about the scene, you can rectify the stereo images and perform a segmentation similar to the method described in section 5.3.5.4 on page 161. For this, you first use the operator
match_fundamental_matrix_ransac, which determines the so-called fundamental matrix. This matrix models the cameras and their relation. But in contrast to the model described in section 5.1 on page

5.4 Multi-View Stereo Vision

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141, internal and external parameters are not available separately. Thus, no metric information can be
derived.
The fundamental matrix is then passed on to the operator gen_binocular_proj_rectification,
which is the “uncalibrated” version of the operator gen_binocular_rectification_map. With
the output of this operator, i.e., the rectification maps, you can then proceed to rectify the images as described in section 5.3.4 on page 151. Because the relative pose of the cameras is not
known, you cannot generate the distance image and segment it as described in section 5.3.5.4 on
page 161. The HDevelop example program hdevelop\Applications\Object-Recognition-2D\
board_segmentation_uncalib.hdev shows an alternative approach that can be used if the reference
plane appears dominant in the images, i.e., if many correspondences are found on it.
Because no calibration object is needed, this method can be used to perform stereo vision with a single camera. Note, however, that the method assumes that there are no lens distortions in the images.
Therefore, the accuracy of the results degrades if the lens has significant distortions.
If the internal parameters of the camera are known, you can determine the relative pose between the
cameras using the operator match_rel_pose_ransac and then use all the stereo methods described for
the fully calibrated case. There is, however, a limitation: The determined pose is relative in a second
sense, because it can be determined only up to a scale factor. The reason for this is that without any
knowledge about the scene, the algorithm cannot know whether the points in the scene are further away
or whether the cameras are further apart because the effect in the image is the same in both cases. If
you have additional information about the scene, you can solve this ambiguity and determine the “real”
relative pose. This method is shown in the HDevelop example program hdevelop\Applications\
3D-Reconstruction\Binocular-Stereo\uncalib_stereo_boxes.hdev.

5.4

Multi-View Stereo Vision

• to use more than two cameras and thus to reconstruct 3D information from all around an object
and
• to reconstruct surfaces and 3D coordinates of selected points in form of 3D object models.
Internally, it is based on binocular stereo. However, by default it does not return the disparity image as a
result.
The following sections show
• how to initialize the stereo model (section 5.4.1) and
• how to reconstruct 3D information (section 5.4.2 on page 170).

5.4.1

Initializing the Stereo Model

The operators for multi-view stereo use a so-called stereo model to encapsulate all needed data. The
following sections show

Stereo Vision

In comparison to binocular stereo, multi-view stereo allows

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• how to access the calibration results (section 5.4.1.1),
• how to specify the world coordinate system (section 5.4.1.2), and
• how to create the stereo model (section 5.4.1.3 on page 170).
5.4.1.1

Accessing the Calibration Results

In contrast to binocular stereo (see section 3.2.7 on page 81), for multi-view stereo you access the
results of the calibration not separately but in form of a so-called camera setup model, which contains the internal camera parameters as well as the relative poses between the cameras. To derive the
camera setup model from the calibration data model, which must have been calibrated before with
calibrate_cameras as is described in section 5.2 on page 144, you call the operator get_calib_data
as follows:
get_calib_data (CalibDataID, 'model', 'general', 'camera_setup_model', \
CameraSetupModelID)

If you want to perform the calibration in an offline step, you can save the camera setup model
with the operator write_camera_setup_model and then destroy the calibration data model with
clear_calib_data and the camera setup model with clear_camera_setup_model.
write_camera_setup_model (CameraSetupModelID, 'four_camera_setup_model.csm')
clear_calib_data (CalibDataID)
clear_camera_setup_model (CameraSetupModelID)

5.4.1.2

Specifying the Coordinate System of the Camera Setup

The coordinate system of the stereo camera setup is identical to the coordinate system of the so-called
reference camera of the setup, which is typically the camera with the index 0 (see the upper part of
figure 5.19). The poses of the other cameras and the reconstructed coordinates are computed relative to
this camera.
You can change the setup’s coordinate system with the operator set_camera_setup_param in two
ways. In particular, you can
• select another camera as reference camera by setting CameraIdx to ’general’ and
ParamName to ’reference_camera’ and passing the index of the camera in ParamValue or
• specify the pose of the desired setup coordinate system (relative to the current one) by setting
CameraIdx to ’general’ and ParamName to ’coord_transf_pose’ and passing the pose in
ParamValue.
The latter case is shown at the bottom of figure 5.19. There, the desired coordinate system is marked by
the calibration plate (typically, you would add a rotation to let the z axis point upwards).

5.4 Multi-View Stereo Vision

camera 1

camera 0 (reference camera)

Cameras with
optical centers and
coordinate systems
(x ,y , z )
c

x c1
y c1

C-169

camera 2

x c0
y c0
ys

y c2

z c0
zs

x c2
z

Camera setup
coordinate system
( xs , y s , z s )

Object

default: reference camera 0

camera 1

Cameras with
optical centers and
coordinate systems

camera 0

x c1
y c1

( x c, y c , z c)

camera 2

x c0
y c2

y c0

z c1

x c2

Camera setup
coordinate system
( xs , y s , z s )

Object
ys

Figure 5.19: Coordinate systems of a multi-view camera setup: Top: default setup coordinate system
located in the reference camera 0; bottom: setup coordinate system moved to the coordinate
system of the calibration plate.

How to change the pose of the setup’s coordinate system is shown,
e.g.,
in
the
HDevelop
example
program
hdevelop\Calibration\Multi-View\
calibrate_cameras_multiple_camera_setup.hdev. There, it is moved from the reference camera
to the calibration plate with pose 0. For that, pose 0 of the calibration plate relative to the reference
camera is accessed with get_calib_data using the parameter ’calib_obj_pose’ and, to consider
the thickness of the calibration plate, the z coordinate of the pose is modified with set_origin_pose.
Then, the setup’s coordinate system is moved into this pose with set_camera_setup_param.

Stereo Vision

moved to calibration plate pose

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3D Vision With a Stereo System

RefPoseIndex := 0
get_calib_data (CalibDataID, 'calib_obj_pose', [0,RefPoseIndex], 'pose', \
PoseCam0Indx0)
set_origin_pose (PoseCam0Indx0, 0, 0, CaltabThickness, ReferencePose)
set_camera_setup_param (CameraSetupModelID, 'general', 'coord_transf_pose', \
ReferencePose)

5.4.1.3

Creating the Stereo Model

After adapting the camera setup model to your requirements, you pass it to the operator
create_stereo_model, which creates the stereo model. Note that at this point you must already
specify whether you want to reconstruct points or surfaces! To reconstruct surfaces, use the parameter ’surface_pairwise’.
create_stereo_model (CameraSetupModelID, 'surface_pairwise', [], [], \
StereoModelID)

To reconstruct points, call the operator with the parameter ’points_3d’.
create_stereo_model (CameraSetupModelID, 'points_3d', [], [], StereoModelID)

5.4.2

Reconstructing 3D Information

With multi-view stereo, you can reconstruct
• the surface of an object (section 5.4.2.1) or
• 3D coordinates for selected points (section 5.4.2.2 on page 172).
5.4.2.1

Reconstructing Surfaces

The main functionality of multi-view stereo is the reconstruction of surfaces. The following code
fragments belong to the HDevelop example program hdevelop\Applications\Robot-Vision\
locate_pipe_joints_stereo.hdev, which reconstructs pipe joints using four cameras (and afterwards performs surface-based 3D matching to estimate the pose of the individual pipe joints, see Solution Guide I, section 10.3.2 on page 155). Figure 5.1 on page 139 shows the four camera images and the
reconstructed surface.
After creating and initializing the stereo model, the model can be configured for the stereo reconstruction
using set_stereo_model_param. With this operator, several parameters are adjusted that control, e.g.,
the subsampling that is used for the reconstruction. Additionally, as the multi-view surface reconstruction
is based on computing binocular disparity images (see section 5.3.5.1 on page 156), several parameters
are used to configure the binocular image rectification and the internally called binocular stereo operators.

5.4 Multi-View Stereo Vision

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* -> Subsampling X, Y, Z
set_stereo_model_param (StereoModelID, 'sub_sampling_step', 3)
* -> Interpolation aliasing by binocular image rectification.
set_stereo_model_param (StereoModelID, 'rectif_interpolation', 'bilinear')
set_stereo_model_param (StereoModelID, 'rectif_sub_sampling', 1.2)
* -> Binocular disparity parameters.
set_stereo_model_param (StereoModelID, 'binocular_filter', \
'left_right_check')

Furthermore, the surface reconstruction must be restricted to a specific part of the 3D space, which is
realized by the definition of a bounding box. This box is built by the coordinates of its front lower left
corner and its back upper right corner. In the program, a camera setup model is used for which the
coordinate system was moved to the object, or more precisely, to a calibration plate that was used for the
calibration of the scene in which the object was placed. Relative to this calibration plate, the coordinates
are specified in meters.
set_stereo_model_param (StereoModelID, 'bounding_box', [-0.2,-0.07,-0.075, \
0.2,0.07,-0.004])

Before
calling
the
reconstruction
operator,
you
must
call
the
operator
set_stereo_model_image_pairs to specify which cameras form pairs, i.e., between which
camera images the disparity images are to be computed. In the example, the cameras 0 and 1 and the
cameras 2 and 3, respectively, form pairs.
set_stereo_model_image_pairs (StereoModelID, [0,2], [1,3])

Then, the actual surface reconstruction is applied with the operator reconstruct_surface_stereo.

If the reconstruction fails,
please refer to the Reference Manual entry
reconstruct_surface_stereo, which contains detailed information about troubleshooting.

The reconstructed surface is returned as a 3D object model (see page 41), which by default consists of
points and their normals. If you need a surface that contains meshing information, e.g., because you
want to apply a 3D primitives fitting (see section 4.5 on page 132) to the 3D object model, you have to
additionally set the parameter ’point_meshing’ within set_stereo_model_param before building
the camera pairs for the reconstruction.
In the example, the 3D object model is visualized by the procedure visualize_object_model_3d,
which allows to interactively rotate, move, and zoom into the model.

Stereo Vision

reconstruct_surface_stereo (Images, StereoModelID, PipeJointPileOM3DID)

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PoseIn := [0.0,0.0,0.5,-30,0,180,0]
if (Index == 1)
visualize_object_model_3d (WindowHandle1, PipeJointPileOM3DID, \
CamParam0, PoseIn, ['color', \
'point_size'], ['yellow',1], \
'Reconstructed scene in ' + ReconsTime$'.3' + ' s', \
[], Instructions, PoseOut)
endif

5.4.2.2

Reconstructing 3D Points

Multi-view stereo also allows to reconstruct the 3D coordinates of selected points. The main advantages
of using the multi-view approach in comparison to the binocular variant (see, section 5.3.5.9 on page
165) are that the reconstruction is more accurate when more than two lines of sight can be taken into
account and that points located on different sides of an object can be reconstructed.
After creating and initializing the stereo model, you can directly use multi-view stereo to reconstruct the 3D coordinates from point correspondences. This is shown in the HDevelop example program hdevelop\3D-Reconstruction\Multi-View\reconstruct_points_stereo.hdev, where
four cameras are used to reconstruct the coordinates of the calibration marks of a calibration plate in
three different poses (see figure 5.20).
The main input for the reconstruction operator reconstruct_points_stereo are the corresponding
points from the multi-view images. They must be passed as tuples in the parameters
• Row (row coordinate of the point),
• Column (column coordinate of the point),
• CameraIdx (index of the camera), and
• PointIdx (index of the point).
Generally, you must extract the corresponding points by yourself. In the example, this task is
easy, because here the points correspond to the marks of a calibration plate, which can be easily extracted using an initialized calibration data model and the operators find_calib_object and
get_calib_data_observ_points. For other objects than the HALCON calibration plate, the extraction may be a bit more challenging.
In the example, the extraction is realized as follows: First, caltab_points is used to derive the number
of calibration marks from the description of the used calibration plate. As with find_calib_object
and get_calib_data_observ_points the calibration marks are always extracted in the same order,
the tuple with the point indices of the correspondence information for a single image can be created
using tuple_gen_sequence. As the parameter CameraIdx must be a tuple of the same lenght as Row,
Column, and PointIdx, a tuple with the same lenght for which each element is ’1’ is created, which is
used later to assign the correct correspondence values to the camera indices.

5.4 Multi-View Stereo Vision

pose 0

pose 1

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pose 2

caltab_points (CaltabDescr, X, Y, Z)
tuple_gen_sequence (0, |X| - 1, 1, Indices)
Ones := gen_tuple_const(|X|,1)

Now, a calibration data model is created and prepared for the extraction of the calibration marks.
Then, for each pose of the calibration plate, empty tuples for the correspondence information are created and filled with the values obtained for each camera that images the calibration plate under this
pose. In particular, for each camera, the calibration marks are extracted with find_calib_object and
get_calib_data_observ_points. The resulting row and column coordinates are added to the tuples
AllRow and AllColumn. The tuple with the point indices is added to the tuple AllIndices and the
corresponding elements with the specific camera index are added to the tuple AllCams.

Stereo Vision

Figure 5.20: Top: images of four cameras of the calibration plate with extracted marks in three different
poses; Bottom: reconstructed points.

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3D Vision With a Stereo System

Objects3D := []
create_calib_data ('calibration_object', 4, 1, CalibDataID)
set_calib_data_calib_object (CalibDataID, 0, CaltabDescr)
for PoseIndex := 0 to 2 by 1
AllRow := []
AllColumn := []
AllIndices := []
AllCams := []
for CameraIndex := 0 to 3 by 1
get_camera_setup_param (CameraSetupModelID, CameraIndex, 'params', \
CameraParam)
ImageFile := ImgPath + 'multi_view_calib_cam_' + CameraIndex + '_' \
+ (13 + PoseIndex)$'02'
read_image (Image, ImageFile)
set_calib_data_cam_param (CalibDataID, CameraIndex, \
'area_scan_polynomial', CameraParam)
find_calib_object (Image, CalibDataID, CameraIndex, 0, 0, [], [])
get_calib_data_observ_points (CalibDataID, CameraIndex, 0, 0, Row, \
Column, Index, Pose)
AllRow := [AllRow,Row]
AllColumn := [AllColumn,Column]
AllIndices := [AllIndices,Indices]
AllCams := [AllCams,CameraIndex * Ones]
endfor

After accumulating the correspondence information for all cameras that image the calibration plate under
the specific pose, the reconstruction is applied with reconstruct_points_stereo.
reconstruct_points_stereo (StereoModelID, AllRow, AllColumn, [], \
AllCams, AllIndices, X, Y, Z, CovWP, \
PointIndexOut)

It returns tuples with the x, y, and z coordinates and with the index of those points that could be reconstructed, i.e., which were extracted in two or more images. In the example, the coordinates are
transformed via x, y, and z images into a 3D object model and the models of all three calibration plate
poses are interactively visualized with visualize_object_model_3d, which allows to rotate, move,
and zoom into the model.

5.4 Multi-View Stereo Vision

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gen_image_const (ImageX, 'real', 1, |X|)
gen_image_const (ImageY, 'real', 1, |X|)
gen_image_const (ImageZ, 'real', 1, |X|)
...
set_grayval (ImageX, Indices, 0 * Indices, X)
set_grayval (ImageY, Indices, 0 * Indices, Y)
set_grayval (ImageZ, Indices, 0 * Indices, Z)
xyz_to_object_model_3d (ImageX, ImageY, ImageZ, ObjectModel3DID)
Objects3D := [Objects3D,ObjectModel3DID]
disp_continue_message (WindowHandles[3], 'black', 'true')
stop ()
endfor
visualize_object_model_3d (WindowHandle, Objects3D, CameraParam, [], \
'color_attrib', 'coord_z', [], [], Instructions, \
PoseOut)

Stereo Vision

Please note that the example shows only the main functionality of the stereo point reconstruction. For
more information, e.g., about input and output covariances or the influence of the bounding box of the
stereo model, please refer to the Reference Manual entry of reconstruct_points_stereo.

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Laser Triangulation with Sheet of Light

C-177

Chapter 6

Laser Triangulation with Sheet of
Light
Laser triangulation can be used to reconstruct the surface of a 3D object by approximating it via a set
of height profiles. HALCON provides operators for a special type of laser triangulation that is called
sheet-of-light technique.

6.1

The Principle of Sheet of Light

The sheet-of-light technique can be applied either to a calibrated measurement setup or to the uncalibrated setup. If the setup is calibrated, the measurement returns the disparities, the x, y, and z coordinates
of the points that build the profiles in the world coordinate system (WCS, see figure 6.1), and a 3D object
model that is derived from the x, y, and z coordinates. The disparities are returned in form of a disparity
image, i.e., the disparities of each measured profile are stored in one row of the disparity image (see
figure 6.2 and note that the camera must be oriented such that the profiles are roughly parallel to the rows
of the image). The x, y, and z coordinates are also not explicitly returned as values but are expressed
as values of pixels within images. That is, three images are returned, one for the x coordinates (X), one
for the y coordinates (Y), and one for the z coordinates (Z). The 3D object model contains information

Sheet of Light

The basic idea of the sheet-of-light technique is to project a thin luminous straight line, e.g., generated
by a laser line projector, onto the surface of the object that is to be reconstructed and then image the
projected line with a camera. As shown in figure 6.1 the projection of the laser line builds a plane that is
called light plane or sheet of light. The optical axis of the camera and the light plane form an angle α,
which is called angle of triangulation. The points of intersection between the laser line and the camera
view depend on the height of the object. Thus, if the object onto which the laser line is projected differs
in height, the line is not imaged as a straight line but represents a profile of the object. Using this profile,
we can obtain the height differences of the object. To reconstruct the whole surface of an object, i.e.,
to get many height profiles, the object must be moved relative to the measurement system, i.e., the unit
built by the laser line projector and the camera.

Laser Triangulation with Sheet of Light

Camera
Laser line projector

P’
r

xc
yc

P’
r

α
P

yw
xw

ag
e

Camera

Laser line projector

C-178

α
P

Object

Object
xw
Scanning direction

Light plane pose
Camera coordinate system
World coordinate system
Image coordinate system
Angle of triangulation

x y z
c c c
x y z
w w w
x y z
r (row) c (column)
α

Figure 6.1: Basic principle of sheet of light (light plane marked in gray).

about the 3D coordinates and the corresponding 2D mapping. If the setup is uncalibrated, only the disparity image and a score that describes how reliable the measurement result is can be returned by the
measurement.

Image 0
Image 1
Image 2

Disparity image
Row 0 (profile of image 0)
Row 1 (profile of image 1)
Row 2 (profile of image 3)

...

Image 511

Row 511 (profile of image 511)

Figure 6.2: Disparity image: the disparity obtained from each profile (or image, respectively) is stored in a
row of the disparity image.

Note that the disparity image for sheet of light has not exactly the same meaning as the disparity image
described for stereo matching in section 5.1 on page 141 and section 5.3.5.1 on page 156. For stereo,

6.2 The Measurement Setup

C-179

the disparity describes the difference between the row coordinates of the left and right stereo images,
whereas for sheet of light, the disparity is built by the subpixel row values at which the profile was
detected.

6.2

The Measurement Setup

The hardware needed for a sheet-of-light measurement consists of a projector that is able to project a thin
luminous line, a camera, a positioning system, and the object to measure. In the following, we assume
the projector to be a laser line projector and the positioning system to be linear (e.g., a conveyor belt), as
these are very common in laser triangulation applications.
The relation between the projector, the camera, and the linear positioning system must not be changed,
whereas the position of the object that is transported by the positioning system changes in relation to the
projector-camera unit. Since the profile images are processed column by column, the profiles must be
oriented roughly horizontal, i.e., roughly parallel to the rows of the image.
The relation between the laser line projector, the camera, and the object to measure can be described
by different measurement setups. Figure 6.3 shows the three apparent configurations for the three components. In the first case, the camera view is orthogonal to the object and the light plane is tilted. The
second case shows a tilted camera view and an orthogonal light plane. For the third case, both the camera
view and the laser line are tilted.

Camera

P
a)

Laser line
projector

Object
b)

Laser line
projector

P
Object

Object
c)

Figure 6.3: Basic configurations possible for a sheet-of-light measurement setup.

Figure 6.4 exemplarily shows a measurement setup as it is used for the examples that will be discussed
in the following sections.
Which measurement configuration to use depends on the goal of the measurement and the geometry of
the object. The configuration in figure 6.3 (a), e.g., is especially suitable if an orthogonal projection
of the object in the image is needed for any reason. Then, a cuboid is imaged as a rectangle. For all
configurations in which the camera is not placed orthogonal, it would be imaged as a trapezoid because
of the perspective deformations (see figure 6.5).

Sheet of Light

Laser line
projector

C-180

Laser Triangulation with Sheet of Light

Figure 6.4: Examplary setup for a sheet-of-light measurement consisting of a camera, a laser line projector, and a positioning system.
Laser line
projector

Camera
Camera

Object

Laser line
projector

Object

Figure 6.5: With the camera being orthogonal to the object, an orthogonal projection of the object is
possible: (left) orthogonal camera view, (right) perspective view.

The most important criterion for the selection of the measurement setup is the geometry of the object.
The setup should be selected such that the amount of shadowing effects and occlusions is minimized.
Occlusions occur if an object point is illuminated by the laser line but is not visible in the image, because
other parts of the object lie between the line of sight of the camera and the object point (see figure 6.6 on
page 181, top). Shadowing effects occur if an object point is visible in the image but is not illuminated
by the laser line, because other parts of the object lie between the laser projection and the imaged object

6.3 Calibrating the Sheet-of-Light Setup

C-181

point (see figure 6.6 on page 181, bottom).
Camera

Laser line
projector

Camera

Laser line
projector

Occlusion
P
Object

Object
Object point is illuminated
and visible in the image

Laser line
projector

Camera

Occluded
area
P

Object point is illuminated
but occluded in the image

Laser line
projector

Camera

Shadowing effect
P
Object
Object point is illuminated
and visible in the image

Object

Shadowed
area
P

Object point is visible in the
image but is not illuminated

Figure 6.6: Problems that have to be considered before selecting the measurement setup: (top) occlusions
and (bottom) shadowing effects.

For all three setup configurations, the angle of triangulation, i.e., the angle between the light plane and
the optical axis of the camera, should be in a range of 30° to 60° to get a good measurement accuracy.
If the angle is smaller, the accuracy decreases. If the angle is larger, the accuracy increases, but more
problems because of occlusions and shadowing effects are to be expected. Thus, you have to find a
trade-off between the accuracy and the completeness of the measurement.

6.3

Calibrating the Sheet-of-Light Setup

A sheet-of-light setup can be calibrated in two different ways:
Calibration using a standard HALCON calibration plate
To calibrate a sheet-of-light setup using a standard HALCON calibration plate (see section 6.3.1),
first the camera is calibrated conventionally. Then, the pose of the light plane and the movement

Sheet of Light

Additionally, the accuracy decreases if the light plane is out of focus of the camera. That may happen,
if the angle of triangulation is too small, i.e., the angle between the light plane and the focus plane is
too large, and therefore only a small part near the intersection of both planes is in focus. To overcome
this problem, a tilt lens can be used to align the focus plane with the light plane using the Scheimpflug
principle (see section 2.2.3 and section 5.1.3 for details).

C-182

Laser Triangulation with Sheet of Light

of the objects to be measured must be determined based on additional images of the calibration
plate.
Calibration using a special 3D calibration object
For the calibration of a sheet-of-light setup with a special 3D calibration object (see section 6.3.2),
first the 3D calibration object must be provided. The calibration itself requires only one (uncalibrated) reconstruction of the 3D calibration object, i.e., its disparity image.

6.3.1

Calibrating the Sheet-of-Light Setup using a standard HALCON
calibration plate

This section describes how to calibrate the sheet-of-light measurement setup. If the uncalibrated case is
sufficient for your application, you can skip this section and proceed with section 6.4 on page 190.
The calibration of the sheet-of-light setup is applied to get the internal and external camera parameters,
the orientation of the light plane in the WCS, and the relative movement of the object between two
successive measurements. The calibration consists of the following steps:
1. Calibrate the camera.
2. Determine the orientation of the light plane with respect to the WCS.
3. Calibrate the movement of the object relative to the measurement setup.
The camera is calibrated by a standard camera calibration as described in section 3.2 on page 68. As
result, the camera calibration returns the internal camera parameters and the pose of the WCS relative to
the camera coordinate system (external camera parameters).
To determine the light plane and its pose, we need at least three corresponding points (see figure 6.7), in
particular two points in the plane of the WCS with ’z=0’ (P1, P2) and one point that differs significantly
in z direction (P3). Thus, you place the calibration object, e.g., the standard HALCON calibration plate,
once or twice so that it lies in the plane of the WCS with ’z=0’, and once so that a higher position
can be viewed, i.e., the plate is either translated in z direction or it is placed in a tilted position. For
each position of the calibration plate, you take two images, one showing the calibration plate and one
showing the laser line. Note that you have to adapt the lighting in between to get one image with a clearly
represented calibration plate and one image that shows a well-defined laser line. The translated or tilted
position of the calibration plate should be selected so that the plane that is built by the points P1, P2, and
P3 becomes as large as possible. The height difference should be at least as big as the height differences
expected for the objects to measure.
Note that the laser line has to be projected onto the same plane in which the calibration plate is placed.
But if possible, it should not be directly projected onto the calibration plate but only near to it. This is
because the used standard HALCON calibration plate is made of a ceramic that shows a strong volume
scattering. This leads to a broadened profile (see figure 6.8), which results in a poor accuracy. If you
use a calibration object that consists of a material with different reflection properties, this might be no
problem.

6.3 Calibrating the Sheet-of-Light Setup

Camera
Laser line projector
z

Camera
Laser line projector

C-183

P3
P1

yw
xw

w
w

y
P2

Figure 6.7: Position of the third point (P3) obtained by (left) tilted calibration plate or (right) translated
calibration plate.

Note further that the three corresponding points described above represent the minimum number of points
needed to obtain a plane. To enhance the precision of the calibration, redundancy is needed; thus, we
recommended to measure more than three corresponding points. Then, the light plane is approximated
by fitting a plane into the obtained point cloud.
In the final step, the pose describing the movement of the object must be calibrated using two images
containing a calibration plate that was moved by the positioning system by a known number of movement
steps.
Summarized, we have to acquire:

Sheet of Light

Figure 6.8: The white parts of the used HALCON calibration plate show a very broad laser line because
of volume scattering.

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Laser Triangulation with Sheet of Light

• a set of images for the camera calibration,
• at least two images that clearly show the laser line in different planes and which correspond to
images that were used for the calibration, and
• at least two images that show the calibration plate in the plane of the linear positioning system.
Between the acquisition of the first and the second image, the calibration plate must be moved by
the positioning system by a known number of movement steps.
The HDevelop example program
hdevelop\Applications\Measuring-3D\calibrate_sheet_of_light_calplate.hdev shows
in detail how to calibrate a sheet-of-light measurement setup with a standard HALCON calibration plate.
For the first step, i.e., the camera calibration, initial values for the internal camera parameters and for the
thickness of the calibration plate are set.
StartParameters := [0.0125,0.0,0.0,0.0,0.0,0.0,0.000006,0.000006,376.0, \
120.0,752,240]
CalTabDescription := 'caltab_30mm.descr'
* Note that the thickness of the calibration target used for this example \
* is 0.63 mm.
* If you adapt this example program to your application, it is necessary \
* to determine
* the thickness of your specific calibration target and to use this value \
* instead.
CalTabThickness := .00063

Then, the calibration images are read. These should fulfill the requirements that are described for a
camera calibration in section 3.2.4.1 on page 78. Now, for each image, the calibration plates are searched,
the contours and centers of their marks are extracted, and the pose of the calibration plate is estimated.
The obtained information is stored in the calibration data model.
NumCalibImages := 20
for Index := 1 to NumCalibImages by 1
read_image (Image, 'sheet_of_light/connection_rod_calib_' + Index$'.2')
find_calib_object (Image, CalibDataID, 0, 0, Index, [], [])
endfor

With the obtained data, the actual camera calibration is performed, so that the internal camera parameters
(CameraParameters) and the external camera parameters (camera poses) for all calibration images can
be obtained. The internal camera parameters and the camera pose for one of the calibration images
are the first two variables that we need for the sheet-of-light measurement that is described in the next
section.
calibrate_cameras (CalibDataID, Errors)
get_calib_data (CalibDataID, 'camera', 0, 'params', CameraParameters)

Note that by selecting the camera pose of one of the calibration images you define the origin of the WCS
used for the measurement.

6.3 Calibrating the Sheet-of-Light Setup

C-185

For the second step, i.e., the orientation of the light plane in relation to the WCS, the poses of two of
the calibration images are needed. The images show the calibration plates in different heights. The pose
of one image is used to define the WCS and the pose of the other image is used to define a temporary
coordinate system (TCS). For both images, the origins of the poses are shifted with set_origin_pose
to consider the thickness of the calibration plate.
Index := 19
get_calib_data (CalibDataID,
CalTabPose)
set_origin_pose (CalTabPose,
Index := 20
get_calib_data (CalibDataID,
CalTabPose)
set_origin_pose (CalTabPose,

'calib_obj_pose', [0,Index], 'pose', \
0.0, 0.0, CalTabThickness, CameraPose)
'calib_obj_pose', [0,Index], 'pose', \
0.0, 0.0, CalTabThickness, TmpCameraPose)

For each of the two calibration images a corresponding laser line image was acquired. There, the laser
line is clearly projected onto the same plane that contained the calibration plate in the calibration image.
With the two laser line images and the poses obtained from the two corresponding calibration images
the procedure compute_3d_coordinates_of_light_line calculates the 3D coordinates of the points
that build the laser lines. The obtained point cloud consists of the points of the light plane in the plane
of the WCS with ’z=0’ (see P1 and P2 in figure 6.7 on page 183) and the points of the light plane in the
plane of the TCS with ’z=0’ (see P3 in figure 6.7 on page 183).
read_image (ProfileImage1, \
'sheet_of_light/connection_rod_lightline_019.png')
compute_3d_coordinates_of_light_line (ProfileImage1, MinThreshold, \
CameraParameters, [], CameraPose, \
X19, Y19, Z19)
read_image (ProfileImage2, \
'sheet_of_light/connection_rod_lightline_020.png')
compute_3d_coordinates_of_light_line (ProfileImage2, MinThreshold, \
CameraParameters, TmpCameraPose, \
CameraPose, X20, Y20, Z20)

procedure fit_3d_plane_xyz (X, Y, Z, Ox, Oy, Oz, Nx, Ny, Nz, MeanResidual)
get_light_plane_pose (Ox, Oy, Oz, Nx, Ny, Nz, LightPlanePose)

In the third step, i.e., the calibration of the movement of the object in relation to the measurement setup,
the calibration plate is moved in discrete steps by the linear positioning system that will be used also for
the following measurement. To calibrate the movement of the linear positioning system, two images with
different movement states are needed. To enhance the accuracy, we do not use images of two succeeding
movement steps but use images with a known number of movement steps between them. Here, the
number of movement steps between both images is 19.

Sheet of Light

Now, the procedure fit_3d_plane_xyz fits a plane into the point cloud. This plane is the light plane,
for which the pose is needed as the third variable for the calibrated sheet-of-light measurement. This
pose (LightPlanePose) is calculated from the plane using the procedure get_light_plane_pose.

C-186

Laser Triangulation with Sheet of Light

read_image (CaltabImagePos1, 'sheet_of_light/caltab_at_position_1.png')
read_image (CaltabImagePos20, 'sheet_of_light/caltab_at_position_2.png')
StepNumber := 19

Now, for both images the poses of the calibration plates are derived.
find_calib_object (CaltabImagePos1, CalibDataID, 0, 0, NumCalibImages + 1, \
[], [])
get_calib_data_observ_points (CalibDataID, 0, 0, NumCalibImages + 1, Row1, \
Column1, Index1, CameraPosePos1)
find_calib_object (CaltabImagePos20, CalibDataID, 0, 0, NumCalibImages + 2, \
[], [])
get_calib_data_observ_points (CalibDataID, 0, 0, NumCalibImages + 2, Row1, \
Column1, Index1, CameraPosePos20)

Then, the pose that describes the transformation between these two poses, i.e., the transformation needed
for 19 movement steps, is calculated (MovementPoseNSteps). Note that a rotation is not assumed and
therefore all rotational elements are set to 0.
pose_to_hom_mat3d (CameraPosePos1, HomMat3DPos1ToCamera)
pose_to_hom_mat3d (CameraPosePos20, HomMat3DPos20ToCamera)
pose_to_hom_mat3d (CameraPose, HomMat3DWorldToCamera)
hom_mat3d_invert (HomMat3DWorldToCamera, HomMat3DCameraToWorld)
hom_mat3d_compose (HomMat3DCameraToWorld, HomMat3DPos1ToCamera, \
HomMat3DPos1ToWorld)
hom_mat3d_compose (HomMat3DCameraToWorld, HomMat3DPos20ToCamera, \
HomMat3DPos20ToWorld)
affine_trans_point_3d (HomMat3DPos1ToWorld, 0, 0, 0, StartX, StartY, StartZ)
affine_trans_point_3d (HomMat3DPos20ToWorld, 0, 0, 0, EndX, EndY, EndZ)
MovementPoseNSteps := [EndX - StartX,EndY - StartY,EndZ - StartZ,0,0,0,0]

To get the pose for a single movement step (MovementPose), the elements of MovementPoseNSteps
that describe a translation are divided by the number of steps. MovementPose, together with the internal
and external camera parameters and the pose of the light plane can now be used to apply a calibrated
sheet-of-light measurement.
MovementPose := MovementPoseNSteps / StepNumber

For details about the proceedings inside the stated procedures, we refer to the example.

6.3.2

Calibrating the Sheet-of-Light Setup Using a Special 3D Calibration
Object

Figure 6.9 shows a sheet-of-light setup together with a special 3D calibration object. To calibrate the
sheet-of-light setup, one disparity image of the 3D calibration object is acquired with the sheet-oflight setup. The sheet-of-light setup is then calibrated using this disparity image with the operator
calibrate_sheet_of_light.

6.3 Calibrating the Sheet-of-Light Setup

C-187

Camera
Laser line projector

3D calibration object
zw xw
yw

Figure 6.9: Sheet-of-light setup with 3D calibration object.

The calibration of a sheet-of-light setup with a 3D calibration object is simpler but slightly less accurate than the calibration of a sheet-of-light setup with a standard HALCON calibration plate, which
is described in section 6.3.1 on page 182. Nevertheless, first a suitable 3D calibration object must be
provided.
In the following, the steps that are necessary for the calibration are described.
The
HDevelop
example
program
hdevelop\3D-Reconstruction\Sheet-Of-Light\
calibrate_sheet_of_light_3d_calib_object.hdev shows in detail how to calibrate a sheet-oflight setup with a special 3D calibration object.

A special 3D calibration object must be provided. This calibration object must correspond to the CAD
model created with create_sheet_of_light_calib_object. The 3D calibration object has an inclined plane on which a truncated pyramid is located. It has a thinner side, which is hereinafter referred
to as front side. The thicker side is referred to as back side of the calibration object. Figure 6.10 shows
the 3D calibration object.
The dimensions of the calibration object should be chosen such that the calibration object covers the complete measuring volume. Be aware, that only parts on the 3d calibration object
above the minimum height of the tilted plane (see the parameter HeightMin of the operator
create_sheet_of_light_calib_object) are taken into account.

Sheet of Light

Supply of a 3D Calibration Object

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Laser Triangulation with Sheet of Light

Figure 6.10: 3D calibration object.

The CAD model, which is written as a DXF file by create_sheet_of_light_calib_object, also
serves as description file of the calibration object. This CAD model can then be used to manufacture the individual calibration object. Note that MVTec does not offer such 3D calibration objects. More information on how to manufacture a calibration object is provided by the description of
create_sheet_of_light_calib_object.
Preparation of the Sheet-Of-Light Model
To prepare a sheet-of-light model for the calibration, the following steps must be performed.
1. Create a sheet-of-light model with create_sheet_of_light_model.
2. Set the initial parameters of the camera with set_sheet_of_light_param. So far, only pinhole
cameras with the division model are supported.
3. Set
the
description
file
of
the
calibration
object
(created
create_sheet_of_light_calib_object) with set_sheet_of_light_param.

with

create_sheet_of_light_model (Domain, [], [], SheetOfLightModelID)
set_sheet_of_light_param (SheetOfLightModelID, 'camera_parameter', \
CameraParam)
set_sheet_of_light_param (SheetOfLightModelID, 'calibration_object', \
'calib_object.dxf')

Uncalibrated Reconstruction of the 3D Calibration Object
The 3D calibration object must be reconstructed with the (uncalibrated) sheet-of-light model prepared
above, i.e., a disparity image of the 3D calibration object must be created (see figure 6.11).

6.3 Calibrating the Sheet-of-Light Setup

C-189

for Index := 1 to 1000 by 1
measure_profile_sheet_of_light (ProfileImage, SheetOfLightModelID, [])
endfor

Figure 6.11: Disparity image of the 3D calibration object.

For this, the calibration object must be oriented such that either its front side or its back side intersect the
light plane first (i.e., the movement vector should be parallel to the Y axis of the calibration object, see
figure 6.9 on page 187 or create_sheet_of_light_calib_object). As far as possible, the domain
of the disparity image of the calibration object should be restricted to the calibration object. Besides,
the domain of the disparity image should have no holes on the truncated pyramid. All four sides of the
truncated pyramid must be clearly visible.
provided

the

camera,

can

set

with

set_profile_sheet_of_light (CalibObjectDisparity, SheetOfLightModelID, [])

Calibration of the Sheet-Of-Light Setup
The calibration is then performed with calibrate_sheet_of_light.
calibrate_sheet_of_light (SheetOfLightModelID, Error)

The returned Error is the RMS of the distance of the reconstructed points to the calibration object in
meters.

Sheet of Light

If the disparity image is already
set_sheet_of_light_param directly.

C-190

Laser Triangulation with Sheet of Light

For sheet-of-light models calibrated with calibrate_sheet_of_light, it is not possible to obtain
values for the calibrated camera parameters, camera pose, lightplane pose, and movement pose.

6.4

Performing the Measurement

A sheet-of-light measurement is applied to get height information for the object to measure. This height
information is presented by a disparity image in which each row contains the disparities of one measured
profile of the object (see figure 6.2 on page 178), by the images X, Y, and Z that express the x, y, and
z coordinates of the measured profiles as values of pixels within images, or by a 3D object model that
contains the coordinates of the object’s 3D points and the corresponding 2D mapping. The images X, Y,
and Z as well as the 3D object model can be obtained only for a calibrated measurement setup, whereas
the disparity image can be obtained also for the uncalibrated case. A sheet-of-light measurement consists
of the following basic steps:
1. Calibrate the measurement setup (if a calibrated measurement is needed) as described in the previous section.
2. Create a sheet-of-light model with create_sheet_of_light_model and set additional parameters with successive calls to set_sheet_of_light_param.
3. Acquire images for each profile to measure, e.g., using grab_image_async.
4. Measure the profile of each image with measure_profile_sheet_of_light.
5. Get the results of the measurement with successive calls to get_sheet_of_light_result
or,
if
a
3D
object
model
is
required,
with
a
single
call
to
get_sheet_of_light_result_object_model_3d.
6. If only the uncalibrated case was applied and a disparity image was obtained, but the x, y, and
z coordinates or the 3D object model are still needed, you can subsequently apply a calibration. Then, you have to calibrate the measurement setup like described in the previous section
and add the obtained camera parameters to the model with set_sheet_of_light_param. Afterwards you call the operator apply_sheet_of_light_calibration with the disparity image
and the adapted sheet-of-light model. The resulting images that contain the coordinates X, Y,
and Z or the 3D object model are queried from the model with get_sheet_of_light_result
or get_sheet_of_light_result_object_model_3d, respectively. How to subsequently apply a sheet-of-light calibration to a disparity image is shown in the HDevelop example program
hdevelop\Applications\Measuring-3D\calibrate_sheet_of_light_calplate.hdev.
7. Clear the model from memory with clear_sheet_of_light_model.
Optionally, you can query all parameters that you have already set for a specific model or that were set by
default using get_sheet_of_light_param. To query all parameters that can be set for a sheet-of-light
model you call query_sheet_of_light_params.

6.4 Performing the Measurement

6.4.1

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Calibrated Sheet-of-Light Measurement

How to apply a calibrated sheet-of-light measurement is shown in the HDevelop example program
hdevelop\Applications\Measuring-3D\reconstruct_connection_rod_calib.hdev, which
measures the object shown in figure 6.12.

Figure 6.12: Object to measure.

The first step is to assign the information obtained by the calibration of the sheet-of-light measurement
setup (see previous section) to a set of variables.
CamParam := [0.0126514,640.275,-2.07143e+007,3.18867e+011,-0.0895689, \
0.0231197,6.00051e-006,6e-006,387.036,120.112,752,240]
CamPose := [-0.00164029,1.91372e-006,0.300135,0.575347,0.587877,180.026,0]
LightplanePose := [0.00270989,-0.00548841,0.00843714,66.9928,359.72, \
0.659384,0]
MovementPose := [7.86235e-008,0.000120112,1.9745e-006,0,0,0,0]

Now, some parameters are set with set_sheet_of_light_param. As a calibrated measurement is
applied, the parameter ’calibration’ is set to ’xyz’. For an uncalibrated measurement, it would
be ’none’, which is the default. Further, the variables with the calibration information are passed as
values to the corresponding parameters for the internal camera parameters (’camera_parameter’), the
external camera parameters (’camera_pose’), the pose of the light plane (’lightplane_pose’), and
the movement of the object relative to the measurement setup (’movement_pose’).

Sheet of Light

Then, a sheet-of-light model is created for a rectangular region of interest using
create_sheet_of_light_model. The ROI should be selected as large as necessary but as
small as possible. That is, it should approximately be some pixels larger than the width of the object in
width and the maximal expected displacement of the laser line caused by the height of the object, i.e.,
the largest expected disparity, in height.

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Laser Triangulation with Sheet of Light

gen_rectangle1 (ProfileRegion, 120, 75, 195, 710)
create_sheet_of_light_model (ProfileRegion, ['min_gray','num_profiles', \
'ambiguity_solving'], [70,290,'first'], \
SheetOfLightModelID)
set_sheet_of_light_param (SheetOfLightModelID, 'calibration', 'xyz')
set_sheet_of_light_param (SheetOfLightModelID, 'scale', 'mm')
set_sheet_of_light_param (SheetOfLightModelID, 'camera_parameter', CamParam)
set_sheet_of_light_param (SheetOfLightModelID, 'camera_pose', CamPose)
set_sheet_of_light_param (SheetOfLightModelID, 'lightplane_pose', \
LightplanePose)
set_sheet_of_light_param (SheetOfLightModelID, 'movement_pose', \
MovementPose)

Then, for each profile to measure an image is acquired (see, e.g., figure 6.13) to apply the actual measurement. Here, the images for each movement step are read from file with read_image. In practice, you will
most probably grab the images directly from your image acquisition device using grab_image_async
(see Solution Guide II-A for details about image acquisition). For each image, the profile within the rectangular region of interest is measured with measure_profile_sheet_of_light, i.e., the disparities
for the profile are determined and stored in the sheet-of-light model.
for Index := 1 to 290 by 1
read_image (ProfileImage, 'sheet_of_light/connection_rod_' + Index$'.3')
measure_profile_sheet_of_light (ProfileImage, SheetOfLightModelID, [])
endfor

Figure 6.13: Measure profile inside a rectangular ROI.

The default for the number of profiles to measure is 512. You can change it with the parameter
’num_profiles’ within create_sheet_of_light_model or set_sheet_of_light_param. If you
measure more than the specified number of profiles, the value of ’num_profiles’ is automatically
adapted in the model. Nevertheless, this adaptation requires additional runtime. Thus, we recommend
to set ’num_profiles’ to a suitable value before starting the measurement. Note that the number of
measured profiles defines the number of rows and the width of the ROI used for the measurement defines
the number of columns for the result images (i.e., the disparity image, the X, Y, and Z images, and the
score image).
After all measurements were performed, the results of the sheet-of-light measurement are queried with
calls to get_sheet_of_light_result and get_sheet_of_light_result_object_model_3d.
Here, we query the disparity image (ResultName set to ’disparity’), the images X, Y, and Z
(ResultName set to ’x’, ’y’, and ’z’, respectively), and the 3D object model. The images X, Y,

6.4 Performing the Measurement

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and Z are shown in figure 6.14. The 3D object model is interactively displayed using the procedure
visualize_object_model_3d as shown in figure 6.15 on page 194.
The interpretation of the gray values of the disparity image and the images X, Y, and Z is as follows:
black parts are outside of the domain of the resulting image, i.e., they indicate parts for which no 3D
information could be reconstructed. For the pixels inside the domain of the image bright parts show
low object parts and dark parts show higher object parts. Note that in this example the images are not
visualized by their default gray values but are converted additionally by a look-up table so that the images
are colored. This is done because the human eye can separate much more colors than gray values. Thus,
details can be better distinguished during a visual inspection.

Figure 6.14: Result of calibrated sheet-of-light measurement: images representing the (from top to bottom) x, y, and z coordinates of the object.

At the end of the measurement, the model is cleared from memory.
clear_sheet_of_light_model (SheetOfLightModelID)

Sheet of Light

get_sheet_of_light_result (Disparity, SheetOfLightModelID, 'disparity')
get_sheet_of_light_result (X, SheetOfLightModelID, 'x')
get_sheet_of_light_result (Y, SheetOfLightModelID, 'y')
get_sheet_of_light_result (Z, SheetOfLightModelID, 'z')
get_sheet_of_light_result_object_model_3d (SheetOfLightModelID, \
ObjectModel3DID)

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Laser Triangulation with Sheet of Light

Figure 6.15: Result of calibrated sheet-of-light measurement: 3D object model.

6.4.2

Uncalibrated Sheet-of-Light Measurement

The uncalibrated sheet-of-light measurement is shown in the HDevelop example program hdevelop\
Applications\Measuring-3D\reconstruct_connection_rod_uncalib.hdev. Here, no calibration results are needed, so we simply create the model for the specified region of interest and set the few
needed parameters directly within create_sheet_of_light_model.
gen_rectangle1 (ProfileRegion, 120, 75, 195, 710)
create_sheet_of_light_model (ProfileRegion, ['min_gray','num_profiles', \
'ambiguity_solving','score_type'], [70,290, \
'first','width'], SheetOfLightModelID)

The actual measurement is applied by the same process used for the calibrated measurement.
for Index := 1 to 290 by 1
read_image (ProfileImage, 'sheet_of_light/connection_rod_' + Index$'.3')
measure_profile_sheet_of_light (ProfileImage, SheetOfLightModelID, [])
endfor

As result, we can only query the disparity image (see figure 6.16) and the score (see section 6.5) of the
measurement (ResultName set to ’score’).
get_sheet_of_light_result (Disparity, SheetOfLightModelID, 'disparity')
get_sheet_of_light_result (Score, SheetOfLightModelID, 'score')

At the end of the program, the model is cleared from memory.
clear_sheet_of_light_model (SheetOfLightModelID)

6.5 Using the Score Image

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Figure 6.16: Result of uncalibrated sheet-of-light measurement: disparity image.

6.5

Using the Score Image

Caused by the specific characteristics of a laser line projector and the general principle of triangulation
the results of a sheet-of-light measurement, i.e., the disparities or the calibrated coordinates, sometimes
show disturbing artifacts. The score image can be used to detect and partially remove artifacts.
There are two types of artifacts. The first type is caused by the geometry of the surface that is to be
reconstructed. As illustrated in figure 6.17, compared to flat and smooth surfaces (e.g., the object in
figure 6.16), curved surfaces with a small radius of curvature and surfaces with a significant slope lead
to a broadening of the light line. Furthermore, the light distribution within the profile might be no longer
symmetric, which leads to a reduced measurement accuracy.
By using the width of the profile stored in the score image (for each pixel of the disparity, the
score value is set to the number of pixels used to determine the disparity value) it is possible
to detect artifacts and to reject the corresponding disparities or the corresponding calibrated coordinates. Figure 6.18 shows the score image obtained by the uncalibrated sheet-of-light measurement performed in the HDevelop example program hdevelop\Applications\Measuring-3D\
reconstruct_connection_rod_uncalib.hdev. The gray values inside the score image indicate the
widths of the laser line in each pixel. Thus, artifacts, in this case parts with a significantly broadened
laser line, can be recognised easily by their brightness.

threshold (Score, ScoreRegion, 1.5, 7.5)
reduce_domain (Disparity, ScoreRegion, DisparityReduced)

The second type of artifacts is caused by the interaction of the coherent laser light with the surface of
the object. Laser light produces disturbing interference patterns when it is projected on a rough textured
surface. Those interference patterns are called speckle and can be considered as a non-additive noise,
which means that this noise can not be reduced by averaging during the image acquisition. In this case,
the only way to increase the measurement accuracy is to use a higher aperture for the image acquisition
or a low-speckle line projector. Note that enlarging the aperture for the image acquisition device will
also reduce the depth of field which might be an undesired side effect. If your application requires high
accuracy, we strongly recommend to use low-speckle projection devices. Note that speckle in most cases
is the limiting factor for the accuracy of laser triangulation systems.

Sheet of Light

In the example, the artifacts are rejected from the disparity image by applying a threshold to the score
image (pixels with a value larger than 7.5 are rejected) and reducing the disparity image to the obtained
region.

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Laser Triangulation with Sheet of Light

Camera

Image of laser line

Gray value distribution
for the profile

Image of laser line

Gray value distribution
for the profile

Laser line
projecor

light−scatter
Object

Camera
Laser line
projecor

light−scatter
Object

Figure 6.17: Curved surfaces with a small radius of curvature and surfaces with a significant slope lead
to a broadening of the light line and thus to a low score: (top) small influence of curvature,
(bottom) large influence of curvature.

Figure 6.18: Result of uncalibrated sheet-of-light measurement: score image (score_type set to ’width’,
i.e., bright parts indicate artifacts).

6.6

3D Cameras for Sheet of Light

The proceeding described in the previous sections works for any standard 2D camera that is suitable for
machine vision. An alternative is to use specific 3D cameras for which the sheet-of-light measurement
is applied inside the camera. These cameras are more expensive than standard 2D cameras, but the
sheet-of-light measurement becomes significantly faster because of the reduced CPU load. Using one of
these cameras, you simply access the camera with HALCON and basically leave the measurement to the

6.6 3D Cameras for Sheet of Light

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camera.
Generally, we distinguish between cameras with an inbuilt laser, i.e., the camera and the laser are integrated in a single unit, and cameras for which the laser is mounted separately.
If the camera and the laser are integrated in a single unit, the measurement setup is restricted to a fixed
angle of triangulation and should be oriented in a defined way. For example, the SICK Ruler camera
should be oriented so that the laser is perpendicular to the linear positioning system. Because of the
preset measurement setup, the camera and the orientation of the light plane with respect to the world
coordinate system are already calibrated. Thus, no further processing with HALCON is needed to obtain
calibrated height profiles.
If the camera and the laser are mounted separately, any configuration of the measurement setup is possible (see section 6.2 on page 179), but by default the result of the measurement is uncalibrated. If the
result of the measurement is needed in world coordinates, you can either query the uncalibrated data
from the camera and subsequently apply a calibration with HALCON as described in section 6.4 on page
190, or, before performing the actual measurement, you apply a calibration that is provided specifically
for the selected camera. For the SICK Ranger cameras, e.g., the camera-specific calibration needs the
software provided with the camera (the SICK Coordinator tool) and a specific calibration object that has
to be purchased separately.

Sheet of Light

Note that in contrast to the proceeding described in the previous sections, the movement of the linear
positioning system is mostly assumed to be known, because the measurement of each profile is triggered
by a signal coming from an encoder on the linear positioning system. That is, when working with an
encoder and if the thus obtained accuracy is sufficient, it is not necessary to calibrate the distance between
two profiles.

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Laser Triangulation with Sheet of Light

Chapter 7

Depth from Focus
Depth from focus (DFF) is a method that enables the reconstruction of 3D surface information from
several images taken at different focus distances between camera and object. It allows a highly accurate
non-destructive 3D measurement of surfaces. The example shown in figure 7.1 was done using microscopic optics with 10x magnification and reaches an accuracy of about 5 micrometers. DFF is even more
precise than the methods stereo chapter 5 on page 139 and laser triangulation with sheet of light chapter 6
on page 177. Furthermore, the setup requires only a single camera, therefore, it is possibly more compact
than, e.g., a stereo setup. DFF requires, however, cameras with telecentric or microscope lenses in order
to achieve a (nearly) parallel projection. Therefore, DFF is only suitable for small objects. Examples for
suitable objects in semiconductors industry are a ball grid array (BGA) (the result of DFF on a single ball
of a BGA can be viewed in figure 7.1) or solder paste inspection, another application in the engineering
sector is the inspection of indexable inserts.

7.1

The Principle of Depth from Focus

With depth from focus, you can reconstruct the surface of a 3D object based on the knowledge that object
points have different distances to the camera and the camera has a limited depth of field. Depending on
the distance and the focus, object points are displayed more or less sharply in the image, i.e., only those
pixels within the correct distance to the camera are focused. Taking images with various object distances,
each object point can be displayed sharply in at least one image. Such a sequence of images is called
“focus stack”. By determining in which image an object point is in focus, i.e., sharply imaged, the
distance of each object point to the camera can be calculated. This principle is clarified in figure 7.2.
For more information about determining sharpness, please refer to hdevelop\Applications\
General\determine_sharpness.hdev. In this example, a flat object is imaged, and the global sharpness of the image is determined. In the case of DFF, the sharpness will be determined for each single
pixel. If you need real three-dimensional information, e.g., if you want to further use your DFF results
for any 3D measurements or surface-based matching (see section 4.3 on page 123 you need to telecentrically calibrate your system. Information on how to perform a calibration can be found in the chapter
about 3D camera calibration (section 3.2 on page 68).

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Depth from Focus

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Depth from Focus

Figure 7.1: This image shows the 3D surface reconstruction of a single ball on a ball grid array.
depth
column
row
image n
image n−l

best focus point
image
focus stack

focus value for each pixel

Figure 7.2: This figure shows the focus stack of images on the left side and the corresponding focus value
- that is determined for each pixel - on the right side. The best focus point is the image where
a pixel has the highest sharpness.

Depth of field (DOF) is a similar term which, however, is not a method but a technical term concerning
the camera. The depth of field is the range of distance within which the image is sharp as opposed
to the best focus point, which is the point with perfect sharpness in the image. The DOF depends on
the pixel size, the aperture (f-number), the focal length in the case of a non-telecentric lens, and the
focusing distance. A low depth of field means that only a small slice of the object is sharply imaged (see
figure 7.3), whereas a high depth of field means that a big part or maybe the whole image is sharp. For
DFF, a low depth of field leads to a higher precision. In order to obtain a low depth of field, use lenses
with a high aperture which is a small f-number on your lens.

7.1 The Principle of Depth from Focus

CCD

lens

CCD

object

F’

object

F’

depth of field

depth of field
a)

Figure 7.3: a) Only a small slice of the image is sharp. Therefore point P1 is mapped to a single point
and is therefore sharp, whereas P2 is mapped to a spot and is consequently blurry. b) If the
object is very flat, the whole object can be sharp at the same time, even if the depth of field is
not very high.

A method for 3D surface measurement, similar to DFF, is depth from defocus. This method requires one
sharp image of the foreground and one sharp image of the background. The distance of all points that
lie between foreground and background is interpolated by their amount of blur. As it only depends on
two images, it may be faster but it is also not as precise as depth from focus. Depth from defocus is not
available in HALCON.

7.1.1

Speed vs. Accuracy

As mentioned before, depth from focus allows highly accurate 3D measurements. However, this accuracy
comes at the price of a longer runtime.
7.1.1.1

Depth from Focus

DFF may consume more processing time due to the number of images that may need to be processed.
This does, however, depend on the actual number of images that are necessary for the specific task. Generally speaking, the more images need to be processed, the higher the accuracy, the longer the runtime.
Therefore, for some applications the runtime will naturally not be very high, because the required accuracy may be lower. Nevertheless, the runtime can also be improved for applications that require a high as
well as for those which require a low accuracy. Internally, the speed of HALCON’s depth_from_focus
is sped up using parallelization (only available in the ’local’ mode).

Depth from Focus

lens

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Depth from Focus

7.1.1.2

Depth from Focus vs. Other Methods

Due to the high number of images that have to be processed, the data volume is also high. This leads
to a reduced speed compared to other methods. DFF has proved to be very useful in the microscopical
range (small objects that are magnified more than once) and often easier to realize than other methods.
In many cases, the setup can be more compact than stereo (chapter 5 on page 139) which needs quite
a lot of space due to the two cameras. Nevertheless, the choice of a suitable lens is very important for
DFF, and it can counter the advantage of using a single camera. For small objects having dimensions
smaller than some ten millimeters, Sheet of Light (chapter 6 on page 177) might become very expensive
because of the thin laser line that is needed and less precise due to speckle. Photometric stereo (see
“3D Reconstruction . Photometric Stereo”) is usually more precise and easier to realize for flat objects
without steep geometric edges. For macroscopic measurements (measurement range up to 100 mm),
stereo or sheet of light would be quicker as they require less images.

7.2

Setup

Before actually starting your application with depth from focus, it is important to first set up your application environment properly. This section concentrates on the camera and application object setup to
prepare the image acquisition. Additional information about equipment and image acquisition can be
read in section 7.3.1 on page 208.

7.2.1

Camera

7.2.1.1

Recommended Camera Setup and Adjustments

As performance of depth from focus for your application depends on the used lens, the depth of field and
the precision of the movements, it is important to use the right camera with the right adjustments. Please
read our recommended camera setup and adjustments for the best results:
1. If possible, use a camera with a telecentric lens. In order to perform depth from focus, a camera
with a telecentric lens or an almost telecentric lens, e.g., a microscopic imaging system, will
produce the best results. Only a telecentric lens enables you to take images with exactly the same
field of view at different focus positions. This is important because depth from focus uses image
coordinates and for each pixel finds the image in which it is displayed sharply. Only in images
taken with a telecentric lens, those pixel are comparable, i.e., in the same position. Therefore, DFF
is a good method for measuring small objects, like, e.g., microelectronic workpieces. Good results
have been achieved, for example, when performing DFF with the XENOPLAN telecentric lens
series by Schneider-Kreuznach. Note however that even when using a telecentric lens, aberration the effect that not all pixels are in focus on a planar surface - can occur. Aberration influences the
accuracy of DFF and should be calibrated. How to calibrate aberration is described in section 7.4.1
on page 211. It is also possible, though not recommended due to the reasons mentioned before, to
use DFF with a standard lens. For performing DFF with a standard lens, please read section 7.6 on
page 213. A problem that is related to the lens is the correspondence problem. This refers to effect
that the position of the pixels outside the optical axis shifts when the distance is changed during

image acquisition. Those planes that lie close to the border of the field of view as well as those
that lie diagonal to the optical axis are affected most. A small depth of field reduces the effect
for planes with other directions. The correspondence problem results in wrong focus distances in
the depth image. This effect is minimal when using a telecentric lens. For telecentric lenses, it is
important that the movement causing a change of focus is applied parallel to the optical axis. For
more information about this effect see figure 7.5.
2. Use mirrors to obtain focus images. For DFF, mirrors are a very good solution because they
can be moved very quickly and accurately and - if necessary - can be replaced easily and inexpensively. Moving the camera may be harmful to the camera sensor which suffers under the
vibrations. Moving the object might be difficult as the object is probably located on a conveyor
belt. All movements for DFF have to be performed very precisely. What such a setup with camera
and mirrors can look like is shown in figure 7.4. Note that the focus should only be moved along
the optical axis.
3. Use a low depth of field to achieve a higher accuracy. This requires, however, more images at
different focus positions. In contrast, large distances between the images require a higher depth
of field and lead to a less precise height reconstruction. A low depth of field requires - as stated
before - images at various focus positions.
4. Use a high aperture. The aperture needs to be as open as possible as this reduces the depth of field
which is responsible for a higher precision as mentioned before. The highest possible precision is,
therefore, limited by the depth of field.

7.2.1.2

Acquire Measurement Range

There are four rules that define how the measurement range can be acquired.
First rule: The distance range has to exceed the height of the measure object.
Second rule: The range in which the image part changes from sharp over blurred to sharp should be
approximately five images (as depicted in figure 7.6). Otherwise, areas of the object’s surface cannot be
determined correctly and, therefore, cannot be measured precisely.
Third rule: As the depth of field of the used lens is fixed, the minimum number of focus positions has
to exceed:
f ocus range in m
depth of f ield of the used lens
The reason for the shifting of the depth of field area of the lens. This shift needs to be smaller than the
depth of field of the lens so that the depth of field areas from two successive images can overlap. The
more those image overlap, the higher the achieved precision. The downside is an increased runtime due
to the high number of images. You cannot, however, increase precision indefinitely as it is also limited
by the camera noise. The smaller the overlap, the smaller the achieved precision, the shorter the runtime.
Fourth rule: If there is a limit to the number of images that can be acquired, for example, because
you have limited runtime, the depth of field of the used lens needs to be increased (e.g., by closing the
aperture of the lens).
Images from a focus stack can be viewed in figure 7.7.
Depending on the direction of image acquisition, i.e., from the camera or towards the camera, the result
of the measurement is either a distance image or a height image.

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Depth from Focus

7.2 Setup

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Depth from Focus

linear
piezo
stage
mirror

semi−transparent
mirror

object

Figure 7.4: Rays from the object first cross a semi-transparent mirror and are then reflected by a mirror
mounted on a linear piezo stage. The rays then reach the semi-transparent mirror again and
are this time reflected back to the camera. By moving the mirror that is mounted on the linear
piezo stage, the distance between the object and the camera can be varied in a controlled
way, which makes it possible to acquire a sequence of images with varying focus.

7.2.2

Illumination

7.2.2.1

Choosing Illumination

In order to enhance the surface texture of the object, direct illumination is needed. At the same time,
reflections have to be minimized. Therefore, coaxial and light-field lighting would not work. A suitable
lighting could be an illumination from various directions, as depicted in figure 7.8, because lighting that
comes from different directions enhances the structure very well and also causes few reflections. As
an alternative dark-field lighting is also possible, because the low angle enhances the surface structure
- it does, however, result in an image that has quite dark parts in some areas. Other illumination setups
can be used as long as the lighting enhances the surface texture and causes as few reflections as possible. A suitable illumination, therefore, highly depends on the application object’s features. For more
information about lighting, please refer to the Solution Guide II-A, appendix C.1 on page 59.
7.2.2.2

Overexposure

Overexposure is an illumination-connected problem that may occur and reduce the accuracy of the resulting image. It leads to a loss of information in the overexposed region which then reaches saturation
(a gray value of 255). Furthermore, overexposure causes the detected false sharp pixels in blurry areas

7.2 Setup

offset
Depth from Focus

parallel to the image

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a) o.k.

b) not o.k.

Figure 7.5: This figure illustrates the correspondence problem. In figure a), a change of focus is applied
parallel to the optical axis, which is good. If the focus is not changed parallel to the optical
axis, a lateral offset occurs, as depicted in figure b).
y

measurement range
object
sharp pixel

depth of field

Figure 7.6: This image depicts an object from which images are taken at different focusing distances
(indicated by the dashed lines) for a focus sequence. It also exemplarily shows the depth of
field, consisting of five images in which the image with the sharpest pixel is marked by a dot.

which are the result of high frequencies between saturated image areas and blurry areas. The results
of overexposure are visualized in figure 7.9. In order to avoid overexposure, it is recommended to take
darker images for your measurements and improve visualization with the operators scale_image and
scale_image_max as presented in the example section 7.3.2.1 on page 209.

7.2.3

Object

In order to successfully perform DFF, it is important to know your application object.
First of all, DFF is a practical method for any small object with textured surface. For objects that are

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Depth from Focus

Figure 7.7: The images above are part of a focus sequence. They show the same object but are taken
and different focus positions. Image a) focuses on the top of the object, image b) is sharp at
a medium hight and image c) is sharp in the background.

object
Figure 7.8: Illumination from various directions leads to good results for depth from focus. It can, for
example, be produced with lights that are arranged within a dome.

larger than that, there are several other 3D methods available like binocular stereo (chapter 5 on page
139). DFF is can, however, still be used for very high objects that would usually require a very high
number of images and thus slow down the measuring process a lot. If the height of such an object is
required there are two possibilities of gaining these measuring results.
• On possibility is to take a certain number (maybe 10) images that depict the bottom of the object,
then also take a certain number images showing the top of the object. Because you know the
distance between the highest image that is taken of the bottom of the object and the first image
taken of the top of the object, you can just add this height to the images taken of the top of the
object.
• Another possibility is to acquire two sequences: one has to be taken around the bottom of the
object and another one at the top of the objects. Within each of the sequences the sharpest level

7.3 Working with Depth from Focus

depth

column
row

focus stack

best focus point

0
255
focus value for each pixel
a)

false edges

255

Figure 7.9: a) The sharp pixel in this figure is overexposed. b) If wider parts of the image are overexposed,
instead of detecting the correct location of a sharp pixel, wrong sharp pixels are detected at
the border between the saturated image areas and the blurry image areas.

is detected. As the movement of the motor that determines the acquisition of images at different
focus levels is known, the distance between those images can be calculated.
d, two image sequences are acquired, one taken around the highest point of measurement and one sequence around the lowest point of measurement. Once, the sharpest pixels at the highest and lowest point
are found, the height can be calculated.
Regular (passive) DFF can be used for objects with a textured surface. Fortunately, small objects mostly
have a structured surface when observed at a suitable magnification. The exception are perfectly polished, i.e., specular surfaces.
If there is very little texture on the object, it is possible to still perform the so-called active depth from
focus. Active DFF compensates for the missing surface structure by projecting a texture on the object.
When performing active DFF, it is possible to produce focus images with the projector. It is, however,
necessary that - for every image that is taken - the depth of field of the camera is conform with the depth
of field of the projector. Active DFF does work for objects with little texture on their surfaces, it does
not work for reflecting surfaces, though.

7.3

Working with Depth from Focus

Depth from focus requires a focus stack of images, i.e., images that have each a different distance to the
object and in which, therefore, different pixels are sharp. Depth from focus is then applied to this focus
stack returning a sharp image, i.e., an image containing only gray values from focused, sharp pixels as
these are the relevant pixels for DFF, as well as a depth image showing the three-dimensional shape of
the object that is inspected. Those two results can be viewed in figure 7.10.

Depth from Focus

depth

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Depth from Focus

7.3.1

Rules for Taking Images

There are several rules that should be followed when taking images for measuring with depth from focus.
Those rules concern image quality as well as the system setup.
7.3.1.1

Rules for Achieving a High Image Quality

1. Avoid overexposure.
2. Avoid reflections.
3. Use a camera with a wide dynamic range.
4. Use direct illumination - preferably from several directions.
5. Use a camera with low noise.
6. The maximum number of images is limited by camera noise. Taking more images to cover
the whole object only improves the precision of the reconstruction to a certain degree which is,
amongst others, limited by the camera noise.
7. As a rule of thumb, the minimum number of images that should be acquired for DFF is 10
and the maximum number is about 150. While 10 images can be just sufficient for measuring
(depending on the required accuracy), more than 150 images will usually not improve accuracy
any more. The corresponding formula would be:
object height
×5
depth of f ield
7.3.1.2

Rules for the Best Results with your System Setup

1. Cover the whole distance range.
2. Let the depth of field areas overlap (see figure 7.6).
3. The distance between the light source and the object should remain constant.
4. The optical system should enable an orthographic projection (parallel projection), i.e., use a
telecentric lens or a microscope in your system setup. If this is impossible, please read section 7.6
on page 213.
5. The axis of the focusing displacement must be parallel to the optical axis of the lens, otherwise
the object shifts laterally within the focus sequence.
6. Do not move the camera or object in x or y direction. DFF does not work on an object that is,
e.g., moving on a conveyor belt between images or an object that is moved by any kind of vibration
or agitation.
Note that the quality of your measuring results does depend on the quality of the input image and you
should therefore aim to achieve the highest possible quality.

7.3.2

Practical Use of Depth from Focus

7.3.2.1

Example Application: Inspecting a PCB with DFF

This section describes an example application where the task is to test if a PCB board is covered by an appropriate amount of soldering paste. This inspection is performed with DFF
and can be viewed in the HDevelop example program hdevelop\Applications\Measuring-3D\
measure_solder_paste_dff.hdev (for more information also refer to the example hdevelop\
Applications\Measuring-3D\measure_bga_dff.hdev which is a similar program showing how
a ball grid array is measured with DFF). An image sequence is acquired and a height map of the single
circuits and pads is calculated. This way parts that have no soldering paste can be identified as well as
those that are covered sufficiently.
First all necessary images for the focus series have to be acquired and combined to a multi-channel image
with the operator channels_to_image.
read_image (ImageArray, 'dff/focus_pcb_solder_paste_' + Sequence$'02')
channels_to_image (ImageArray, Image)

Then, depth from focus is performed, a depth map is calculated and all sharp gray values are selected.
Both results can be used for further processing. Depth from focus is performed with the HALCON
operator depth_from_focus. The channel number is returned for each pixel together with a confidence
value, which is an indicator for the quality of the distance value. Pixels with the best focus are chosen.
The method can be selected with the parameters Filter and Selection.
It is striking that the images are very dark, when looking at the images in the example hdevelop\
Applications\Measuring-3D\measure_solder_paste_dff.hdev. They have been acquired like
this on purpose to avoid overexposure. However, to improve the visibility of the object’s surface in the
image, the operator scale_image enhances the sharpness, scale_image_max spreads the gray values
in the image to improve visibility despite of the darkness. For more information on depth from focus and
overexposure please refer to section 7.2.2.2 on page 204. median_rect suppresses unwanted outliers.
depth_from_focus (Image, Depth, Confidence, 'bandpass', 'next_maximum')
select_grayvalues_from_channels (Image, Depth, SharpenedImage)
scale_image (SharpenedImage, ImageScaled, 4, 0)
scale_image_max (Depth, ImageScaleMax)
median_rect (ImageScaleMax, DepthMean, 25, 25)

Finally, the results - the sharp image as well as the 3D plot - are displayed as shown in figure 7.10. A
sharp image is reconstructed by selecting the gray value of each pixel that is in focus for each coordinate
using the depth image as index table. The focus stack and the depth image are used as input to reconstruct
a focused image using the parameters MultiChannelImage, IndexImage and Selected. Sharp gray
values can be identified by high frequencies, i.e., high edge amplitudes in the image, i.e., where the grayvalue information changes quickly. For each sharp pixel, a confidence score is returned.The amount of
sharpness defines the score. Furthermore, a 3D plot of the object in the image is calculated which is
helpful for detecting defects.

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7.3 Working with Depth from Focus

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Figure 7.10: Results of the PCB solder inspection: a) a synthetic sharp image and b) a 3D plot of the
object.

dev_open_window (0, Width * 0.7 + 5, Width * 0.7, Height * 0.7, 'black', \
WindowHandle3D)
dev_set_paint (['3d_plot','texture'])
compose2 (DepthMean, ImageScaled, MultiChannelImage)
dev_display (MultiChannelImage)

7.3.3

Volume Measurement with Depth from Focus

In contrast to stereo, the height information is not calibrated for depth from focus. The values in the
height image are indices of input images. To measure a real world height or volume, the distance in
between these images must be known. The easiest case is that images are taken with the same movement
z. If two coordinates differ by an index value of n, the real world distance will be z × n, with the unit of
z. The unit in x and y, i.e., the size of a pixel must be known. Make sure that all dimensions are given in
the same unit. Volume can be determined with the operator area_center_gray by adding up the pixel
values which are equal to the height values. The resulting value must be multiplied by x, y and z.

7.4

Solutions for Typical Problems With DFF

There are two main problems that occur with depth from focus: overexposure and reflections.
1. Overexposure leads to extreme values (255) and results in peaks within the constant gray-value
range of an image. This means that even within parts of the images that are blurry, the difference
between neighboring gray values is so dominant that false edges might be detected (this effect
is visualized in figure 7.9 on page 207). Adjusting your illumination to minimize reflections can

reduce overexposure For more information on suitable illumination and handling overexposure,
please refer to section 7.2.2 on page 204.
2. Object surfaces may reflect too much for measuring with depth from focus. One advantage of
depth from focus is, however, that you are very close to the object. Therefore, some objects that
seem to have a reflecting surfaces might under the microscope show some surface texture after all.
Try to minimize reflections with diffuse illumination as described in section 7.2.2 on page 204.
Furthermore, aberration occurs for DFF especially when images were taken with a standard lens
but it also cannot totally be avoided for telecentric lenses either.

7.4.1

Calibrating Aberration

7.4.1.1

Aberration

Aberration is the effect that, when looking perpendicularly on a planar surface, not all pixels are in focus
at the same time. Either the center or the outer part of the image are completely in focus. This effect is
illustrated in figure 7.11.

Figure 7.11: Image a) is an example for aberration. It is sharp in the middle and pixels become more
blurry the further away they are from the center. Image b) shows a graphic that clarifies
what happens when aberration occurs.

Aberration is the curvature of field that effects images taken with cameras using standard lenses but also
cannot be completely avoided when using cameras with telecentric lenses. It results in an error in the
depth image.
There are two main kinds of optical aberration:
1. Spherical aberration occurs when light rays have different focus points depending on their distance to the optical axis, i.e., the center part and the border of the image are not simultaneously
sharp, even though a planar object is imaged.

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7.4 Solutions for Typical Problems With DFF

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2. Coma is the asymmetric accumulation of light intensity for off-axis points. It effects the periphery
of the field of view. This kind of aberration is, however, not relevant here and is just mentioned for
the sake of completeness.
The following paragraph describes how to calibrate spherical aberration and therefore avoid errors.
7.4.1.2

Setup Aberration Calibration

In order to get an accurate result from your depth from focus application, aberration has to be calibrated.
This enables the calculation of correct sharp depth images and also enable further processing of the DFF
results.
Rules for setting up your calibration to correct aberration.
1. A planar surface with reasonable texture is needed as reference plane.
2. The camera has to be mounted perpendicular to the surface (angles can be determined with
HALCON’s camera calibration if necessary).
3. It is important that the same distance and camera setting are used as will be applied during
application!
4. Using depth from focus, the “curvature on the surface”, i.e., the aberration is determined.
5. The extracted reference surface is used to correct the later measurements.
6. It is recommended to store the reference image to file for further use.
7. It is recommended to store the used parameters to file for further use.
8. The aberration can be approximated by a paraboloidal function.
9. The approximation has the advantage of reducing noise effects that influence the measurements.
10. The parameters of the paraboloidal function can be used to generate a reference image.
11. By subtracting the reference image from the depth measurement the error caused by the
aberration can be corrected.

7.5

Special Cases

Depth from focus may not always be used to reconstruct the three-dimensional surface of an object in
order to subsequently measure it.
It may also be used to obtain a sharp image of the object if this is not otherwise possible due to the setup
and continue working with this image.
Another possibility of using the DFF method is simply checking whether an object is present or not.
Therefore, the lens is focused on a certain depth. If sharp pixels are found within an image at this certain
depth, the object is present, otherwise it is missing. In this case, only one image can be sufficient.
Similarly, it can be used to check whether an object is tilted or not.

7.6

Performing Depth from Focus with a Standard Lens

Even though not recommended, it is possible to perform depth from focus with standard lenses. Using
a non-telecentric lens does, however, require some adaptations of the DFF measuring process to ensure
the best results.
Note that the accuracy of DFF measurements with a standard lens instead of a telecentric lens is reduced.
For DFF with standard lenses, the focal length needs to be as long as possible in order to keep the
perspective shift of the points in the field of view small. Furthermore, the depth of field needs to be
small.
If the points shift too much within the field of view, measuring with DFF produces an erroneous result.
Calibrate aberration as described in section 7.4.1 on page 211.
Note that the rules that have been described for DFF with a telecentric lens are also valid for DFF with a
standard lens.

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7.6 Performing Depth from Focus with a Standard Lens

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Chapter 8

Robot Vision
A typical application area for 3D vision is robot vision, i.e., whenever robots are equipped with cameras
that supply information about the parts to be handled. Such systems are also called “hand-eye systems”
because the robotic “hand” is guided by mechanical “eyes”.
In order to use the information extracted by the camera, it must be transformed into the coordinate system
of the robot. Thus, besides calibrating the camera(s) you must also calibrate the hand-eye system, i.e.,
determine the transformation between camera and robot coordinates. The following sections explain
how to perform this hand-eye calibration with HALCON.
Please note that in order to use HALCON’s hand-eye calibration, the camera must observe the
workspace of the robot. If the camera does not observe the workspace of the robot, e.g., if the camera
observes parts on a conveyor belt, which are then handled by a robot further down the line, you must
determine the relative pose of robot and camera with different means.
The calibration result can be used for different tasks. Typically, the results of machine vision, e.g., the
position of a part, are to be transformed from camera into robot coordinates to create the appropriate
robot commands, e.g., to grasp the part. Section 8.7 on page 227 describes such an application. Another
possible application for hand-eye systems is to transform the information extracted from different camera
poses of a moving camera into a common coordinate system.

8.1

Supported Configurations

The hand-eye calibration of HALCON supports different configurations, i.e., kinds of robots and sensors
and ways, the sensor is mounted. In the following, these configurations are briefly discussed.

8.1.1

Articulated Robot vs. SCARA Robot

The arm of an articulated robot (figure 8.1a) typically has three rotary joints covering 6 degrees of freedom (3 translations and 3 rotations). In contrast, SCARA (selective compliance articulated robot arm)

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(a)

(b)

Figure 8.1: Different kinds of robots: (a) Articulated robot and (b) SCARA robot.

robots (figure 8.1b) have typically three parallel rotary joints and one parallel prismatic joint covering
only 4 degrees of freedom (3 translations and 1 rotation).
Articulated (antropomorphic) robots can pick up a part, no matter how it is oriented and then insert it into
a package that may require a special angle under which it is approached. Therefore, they can be used in
a very flexible manner. In contrast, SCARA robots are restricted in their movements. They cannot tilt
the tool. But they offer faster and more precise performance. They are best suited for high-speed pick
and place, packaging, and assembly applications. Because of their compact structure, SCARA robots
are often preferred if only limited space is available.
The hand-eye calibration of articulated robots and of SCARA robots is very similar, with the exceptions
that for SCARA robots the camera must be calibrated prior to the actual hand-eye calibration and that
it is necessary to resolve one ambiguity by manually moving the tool of the robot to a known position.
These points where the calibration of SCARA robots differs from the calibration of articulated robots
are described in section 8.3 on page 220 and section 8.6 on page 227.

8.1.2

Camera and Calibration Plate vs. 3D Sensor and 3D Object

Depending on the application, either an optical camera or a 3D sensor is used to identify the objects and
to determine their pose. If a camera is used, for the calibration of the hand-eye system, a calibration plate
or calibration object is required (figure 8.2a). Note that the automatic detection of the calibration object
works only if one of the HALCON calibration plates (section 3.2.3.1 on page 74) is used. If a 3D sensor
is used (figure 8.2b), the pose of known 3D objects can be determined, e.g., with surface based matching
(Solution Guide I, chapter 10 on page 145). These poses can then directly be used for the calibration of
the hand-eye system.
Note that, although in the following only systems with a single camera are described, you can of course
also use a stereo camera system. In this case, you typically calibrate only the relation of the robot to one

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8.1 Supported Configurations

(a)

(b)

Figure 8.2: Different kinds of sensors: (a) Camera with HALCON calibration plate and (b) 3D sensor with
3D object.

of the cameras, because the relation between the cameras is determined by the stereo calibration (see
section 5.2.4 on page 148).

8.1.3

Moving Camera vs. Stationary Camera

There are two possible scenarios for mounting the camera:
Moving camera
The camera is mounted at the tool and is moved to different positions by the robot.
Stationary camera
The camera is mounted externally and does not move with respect to the robot base.
Figure 8.3 depicts these two scenarios. Note that different poses must be determined by the hand-eye
calibration with a moving camera compared to the hand-eye calibration with a stationary camera. In the
following, the paragraphs that affect only one of these scenarios are marked respectively.
Note that HALCON’s hand-eye calibration is not restricted to systems with a “hand”, i.e., a manipulator.
You can also use it to calibrate cameras mounted on a pan-tilt head or surveillance cameras that rotate
to observe a large area. Both systems correspond to a camera mounted on a robot; the calibration then
allows you to combine visual information from different camera poses.

8.1.4

Calibrating the Camera in Advance vs. Calibrating It During HandEye Calibration

For articulated robots that are used together with one camera, it is possible to calibrate the camera
together with the hand-eye setup. For SCARA robots and all systems that use stereo or multi-view

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Figure 8.3: Robot vision scenarios: (a) moving camera, (b) stationary camera.

setups, the camera(s) must be calibrated in advance.

8.2

The Principle of Hand-Eye Calibration

Like the camera calibration (see section 3.2 on page 68), the hand-eye calibration is based on providing
multiple images of a known calibration object. But in contrast to the camera calibration, here, the
calibration object is not moved manually. Either it is moved by the robot in front of a stationary camera
or the robot moves the camera over a stationary calibration object. The pose, i.e., the position and
orientation, of the robot tool in robot base coordinates for each calibration image must be known with
high accuracy!
This results in a chain of coordinate transformations (see figure 8.4 and figure 8.5). In this chain, two
transformations (poses) are known: the pose of the robot tool in robot base coordinates base Htool and the
pose of the calibration object in camera coordinates cam Hcal , which is determined from the calibration
images. The hand-eye calibration then estimates the other two poses, i.e., the relation between the robot
and the camera and between the robot and the calibration object, respectively. Note that the chain consists
of different poses depending on the used scenario.
Moving camera
For a moving camera, the pose of the robot tool in camera coordinates and the pose of the calibration object in robot base coordinates are determined (see figure 8.4):
cam

Hcal = cam Htool · tool Hbase · base Hcal

(8.1)

8.2 The Principle of Hand-Eye Calibration

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zoom
base

H tool

cam
tool

H tool

z
x cam
cam

camera
y

H cal
y

y
x

base

H tool

Robot Vision

cal. object

x
z
base

H cal

Figure 8.4: Chain of transformations for a moving camera system.

In this chain the inverse input pose, i.e., the pose of the robot base in tool coordinates, is used.
tool

Hbase = (base Htool )

-1

(8.2)

However, the inversion of the pose is done internally by the hand-eye calibration algorithm.
Stationary camera
For a stationary camera, the pose of the robot base in camera coordinates and of the calibration
object in robot tool coordinates are determined (see figure 8.5):
cam

base

Hcal = cam Hbase · base Htool · tool Hcal

H tool

cam

(8.3)

H cal
camera

zoom

tool

y
base

z
y

H cal

cal. object

cam

H base

Figure 8.5: Chain of transformations for a stationary camera system.

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Once all input data has been set as detailed in section 8.4, the hand-eye calibration is performed with a
single call of the operator calibrate_hand_eye.
calibrate_hand_eye (CalibDataID, Errors)

Let’s have a brief look at the parameters. The referenced sections contain more detailed information:
• CalibDataID
As for the camera calibration, you have to prepare a calibration input data model by setting all
required input data. Specifically, the observed poses of the calibration object in camera coordinates
and the corresponding poses of the robot tool in robot base coordinates have to be set. This is
described in detail in section 8.4.
• Errors
These errors represent the quality assessment of the transformations determined during the handeye calibration. How to use the transformations and how to interprete the quality values in robot
vision applications is described in section 8.7 on page 227.
Besides the coordinate systems described above, two others may be of interest in a robot vision application: First, sometimes results must be transformed into a reference (world) coordinate system. You can
define such a coordinate system easily based on a calibration image. Secondly, especially if the robot
system uses different tools (grippers), it might be useful to place the tool coordinate system used in the
calibration at the mounting point of the tools and introduce additional coordinate systems at the gripper
(tool center point). The example application in section 8.7 on page 227 shows how to handle both cases.

8.3

Calibrating the Camera in Advance

This step is not required for 3D sensors and it is optional for articulated robots that are used together
with one camera, but it is obligatory for SCARA robots and for all systems with a stereo or multi-view
camera setup.
Articulated robot
For hand-eye systems with an articulated robot and one camera, typically, the camera is calibrated
together with the whole hand-eye system. Therefore, it is not necessary to calibrate the camera in
advance.
If a stereo or multi-view camera setup is used instead of one camera, these cameras must be
calibrated in advance by the method described in section 5.2 on page 144. If such a calibrated
camera setup is used for hand-eye calibration (or if one already calibrated camera is used whose
camera parameters should be preserved), the internal camera parameters must be excluded from
the optimization with
set_calib_data (CalibDataID, 'camera', 'general', 'excluded_settings', \
'params')

8.4 Preparing the Calibration Input Data

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SCARA robot
For hand-eye systems with a SCARA robot, the camera must be calibrated in advance (see, e.g.,
section 3.2 on page 68). The internal camera parameters are atuomatically excluded from the
optimization.

Preparing the Calibration Input Data

Below, we show how to prepare the data in the calibration data model CalibDataID that is then
used by calibrate_hand_eye. The code examples mostly stem from the HDevelop programs
solution_guide\3d_vision\handeye_movingcam_calibration.hdev and solution_guide\
3d_vision\handeye_stationarycam_calibration.hdev, which perform the calibration of handeye systems with a moving and a stationary camera, respectively. The programs solution_guide\
3d_vision\handeye_movingcam_calibration_poses.hdev and solution_guide\3d_vision\
handeye_stationarycam_calibration_poses.hdev perform the hand-eye calibration by directly
setting the poses of the calibration object obtained with a 3D sensor.

8.4.1

Creating the Data Model

First, a calibration data model for the hand-eye calibration has to be created. The following types are
supported:
Robot type (see section 8.1.1 on page 215):
• Articulated robot
• SCARA robot
Sensor type (see section 8.1.2 on page 216):
• Camera (with calibration plate)
• 3D sensor (with 3D object)
Camera mounting type (see section 8.1.3 on page 217):
• Stationary camera
• Moving camera
Note that in connection with the mounting type (stationary or moving camera (=’eye’)), always the term
camera is used, even if a 3D sensor is used.
The robot type and the camera mounting type are defined in the parameter CalibSetup of
create_calib_data. For example, ’hand_eye_moving_cam’ defines a system of an articulated robot

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8.4

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with a moving camera, while ’hand_eye_scara_stationary_cam’ defines a system of a SCARA robot
with a stationary camera.
The sensor type is defined by the two parameters NumCameras and NumCalibObjects of
create_calib_data. If a camera with a calibration plate is used, both parameters must be set to 1.
If a 3D sensor is used, both parameters must be set to 0.
In the following, the respective calls of create_calib_data are shown for all eight combinations of
the above three categories:
Articulated robot — Camera — Stationary
create_calib_data ('hand_eye_stationary_cam', 1, 1, CalibDataID)

Articulated robot — Camera — Moving
create_calib_data ('hand_eye_moving_cam', 1, 1, CalibDataID)

Articulated robot — 3D sensor — Stationary
create_calib_data ('hand_eye_stationary_cam', 0, 0, CalibDataID)

Articulated robot — 3D sensor — Moving
create_calib_data ('hand_eye_moving_cam', 0, 0, CalibDataID)

SCARA robot — Camera — Stationary
create_calib_data ('hand_eye_scara_stationary_cam', 1, 1, CalibDataID)

SCARA robot — Camera — Moving
create_calib_data ('hand_eye_scara_moving_cam', 1, 1, CalibDataID)

SCARA robot — 3D sensor — Stationary
create_calib_data ('hand_eye_scara_stationary_cam', 0, 0, CalibDataID)

SCARA robot — 3D sensor — Moving
create_calib_data ('hand_eye_scara_moving_cam', 0, 0, CalibDataID)

When a camera with a calibration plate is used, i.e., if the two parameters NumCameras and
NumCalibObjects of create_calib_data are both set to 1, the camera parameters and the calibration
plate must be set in the calibration data model:

8.4 Preparing the Calibration Input Data

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set_calib_data_cam_param (CalibDataID, 0, 'area_scan_division', \
StartCamParam)
set_calib_data_calib_object (CalibDataID, 0, CalTabFile)

The optimization method that results in the most accurately calibrated poses is the non-linear algorithm.
It is the method of choice and is therefore the default method used by calibrate_hand_eye. It can
also be set explicitly as follows.

8.4.2

Robot Vision

set_calib_data (CalibDataID, 'model', 'general', 'optimization_method', \
'nonlinear')

Poses of the Calibration Object

Camera with HALCON calibration plate
When using a camera and the HALCON calibration plate, the pose of the calibration plate is
determined in each calibration image and saved in the calibration data model for the hand-eye
calibration using find_calib_object. Please note that for the hand-eye calibration we strongly
recommend to use the calibration plate with hexagonally arranged calibration marks introduced with HALCON 12 (see section 3.2.3.1 on page 74). This calibration plate is much easier
to handle than the one with rectangularly arranged calibration marks because it may be partly occluded or placed partly outside the image. Do not use the old calibration plate with rectangularly
arranged calibration marks and without the asymmetric pattern in one of its corners because if even
in a single calibration image the pose of the old, symmetric calibration plate is estimated wrongly
because it is rotated by more than 90 degrees, the calibration will fail!

for I := 0 to NumImages - 1 by 1
read_image (Image, ImageNameStart + I$'02d')
find_calib_object (Image, CalibDataID, 0, 0, I, [], [])

Camera with generic calibration object
If a different calibration plate is used, the extracted marker position can be set with
set_calib_data_observ_points. These are then used internally to determine the pose of the
calibration plate. For more details on using generic calibration plates refer to section 3.2.3.2 on
page 77.
3D sensor
If a generic 3D sensor is used, the observed poses are set explicitly in the calibration data model.
set_calib_data_observ_pose (CalibDataID, 0, 0, I, ObjInCamPose)

8.4.3

Poses of the Robot Tool

For each of the calibration images or observed poses, the corresponding pose of the robot must be specified. Note that the accuracy of the poses is critical to obtain an accurate hand-eye calibration. There

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are two ways to “feed” the poses into HALCON: In many cases, you will simply read them from the
robot control unit and then enter them into your HALCON program manually. For this, you can use the
HDevelop example program solution_guide\3d_vision\handeye_create_robot_poses.hdev,
which lets you input the poses in a text window and writes them into files.
As an alternative, if the robot has a serial or socket interface, you can also send them via this connection
to your HALCON program (see the sections “System . Serial” and “System . Sockets” in the Reference
Manual for more information).
In both cases, you then convert the data into HALCON 3D poses using the operator create_pose. As
described in section 2.1.5 on page 25 (and in the Reference Manual entry for create_pose), you can
specify a pose in more than one way, because the orientation can be described by different sequences of
rotations. Therefore, you must first check which sequence is used by your robot system. In many cases,
it will correspond to
Rabg = Rz (RotZ) · Ry (RotY) · Rx (RotX)
(8.4)
If this is the case, select the value ’abg’ for the parameter OrderOfRotation of create_pose. For the
inverse order, select ’gba’.
If your robot system uses yet another sequence, you cannot use create_pose but must create a corresponding homogeneous transformation matrix and convert it into a pose using hom_mat3d_to_pose.
If, e.g., your robot system uses the following sequence of rotations where the rotations are performed
around the z-axis, then around the y-axis, and finally again around the z-axis
Rzyz = Rz (Rl) · Ry (Rm) · Rz (Rr)

(8.5)

the pose can be created with the following code:
hom_mat3d_identity (HomMat3DIdentity)
hom_mat3d_translate (HomMat3DIdentity, Tx, Ty, Tz, HomMat3DTranslate)
hom_mat3d_rotate_local (HomMat3DTranslate, rad(Rl), 'z', HomMat3DT_Rl)
hom_mat3d_rotate_local (HomMat3DT_Rl, rad(Rm), 'y', HomMat3DT_Rl_Rm)
hom_mat3d_rotate_local (HomMat3DT_Rl_Rm, rad(Rr), 'z', HomMat3D)
hom_mat3d_to_pose (HomMat3D, Pose)

Note that the rotation operators expect angles to be given in radians, whereas create_pose expects
them in degrees!
The example program solution_guide\3d_vision\handeye_create_robot_poses.hdev allows
you to enter poses of the three types described above. If your robot system uses yet another sequence of
rotations, you can easily extend the program by modifying (or copying and adapting) the code for ZYZ
poses.
The
HDevelop
example
programs
solution_guide\3d_vision\
handeye_movingcam_calibration.hdev
and
solution_guide\3d_vision\
handeye_stationarycam_calibration.hdev read the robot pose files in the loop of processing the
calibration images.
For each pose of the calibration object, the pose of the robot tool in robot base coordinates that was used
for its observation is read from file using read_pose and accumulated in the calibration data model
CalibDataID.

8.5 Performing the Calibration

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read_pose (DataNameStart + 'robot_pose_' + I$'02d' + '.dat', ToolInBasePose)
set_calib_data (CalibDataID, 'tool', I, 'tool_in_base_pose', ToolInBasePose)

The procedure of setting the robot tool pose in the calibration data model is identical for a hand-eye
system with a moving camera and a system with a stationary camera.

Performing the Calibration

Similar to the camera calibration, the main effort lies in collecting the input data. The calibration itself
is performed with a single operator call.
calibrate_hand_eye (CalibDataID, Errors)

Of course, you should check whether the calibration was successful by looking at the output parameter
Errors, which is a measure for the accuracy of the pose parameters. It contains the pose error of the
complete chain of transformations in form of a tuple with the following four elements:
• the root-mean-square error of the translational part in meter
• the root-mean-square error of the rotational part in degree
• the maximum error of the translational part in meter
• the maximum error of the rotational part in degree
In a pose, typically, the translation is entered in meter and the rotation is entered in degree, therefore
the respective error has the same unit. The error have to be interpreted in the context of the hand-eye
calibration setup, i.e., the size of the robot and the distance between the camera and the calibration object.
Stationary camera
For a hand-eye system with a stationary camera, the pose of the robot base in camera coordinates and the pose of the calibration object in robot tool coordinates is computed by the hand-eye
calibration. These poses can be queried from the calibration data model as follows:
get_calib_data (CalibDataID, 'camera', 0, 'base_in_cam_pose', BaseInCamPose)
get_calib_data (CalibDataID, 'calib_obj', 0, 'obj_in_tool_pose', \
ObjInToolPose)

Moving camera
For a hand-eye system with a moving camera, the pose of the robot tool in camera coordinates and
the pose of the calibration object in robot base coordinates is computed by the hand-eye calibration.
These poses can be queried from the calibration data model as follows:
get_calib_data (CalibDataID, 'camera', 0, 'tool_in_cam_pose', ToolInCamPose)
get_calib_data (CalibDataID, 'calib_obj', 0, 'obj_in_base_pose', \
CalObjInBasePose)

Robot Vision

8.5

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Typically, you then save the calibrated poses in files so that your robot vision application can read them
at a later time. The following code does so for a system with a moving camera:
write_pose (ToolInCamPose, DataNameStart + 'final_pose_cam_tool.dat')
write_pose (CalObjInBasePose, \
DataNameStart + 'final_pose_base_calplate.dat')

The example programs then visualize the calibrated poses by displaying the coordinate system of the
calibration plate in each calibration image. For this, they compute the pose of the calibration plate in
camera coordinates based on the calibrated poses.
Moving camera
For a moving camera system, this corresponds to the following code (compare equation 8.1 on
page 218).
* CalibObjInCamPose = cam_H_calplate
*
= cam_H_tool * tool_H_base * base_H_calplate
*
= ToolInCamPose * BaseInToolPose * CalibrationPose
pose_invert (ToolInBasePose, BaseInToolPose)
pose_compose (ToolInCamPose, BaseInToolPose, BaseInCamPose)
pose_compose (BaseInCamPose, CalibObjInBasePose, CalibObjInCamPose)

This code is encapsulated in a procedure, which is called in a loop over all images.
for I := 0 to NumImages - 1 by 1
read_image (Image, ImageNameStart + I$'02d')

The pose of the robot tool that was set in the calibration data model is queried and the corresponding pose of the calibration object in the camera coordinates is computed and visualized.
get_calib_data (CalibDataID, 'tool', PoseIds[I], 'tool_in_base_pose', \
ToolInBasePose)
* Compute the pose of the calibration object relative to the camera
calc_calplate_pose_movingcam (CalObjInBasePose, ToolInCamPose, \
ToolInBasePose, CalObjInCamPose)
* Display the coordinate system
disp_3d_coord_system (WindowHandle, CamParam, CalObjInCamPose, 0.01)
endfor

Stationary camera
The corresponding procedure for a stationary camera system is listed in appendix A.6 on page 281.
An observation can be deleted using the operator remove_calib_data_observ (compare section 3.2.8
on page 85). The corresponding pose of the robot tool has to be deleted using the operator
remove_calib_data. To determine the effect of a deleted observation on the hand-eye calibration,
the calibration has to be performed again.
Once the calibration data model is not needed anymore, it has to be deleted to free the occupied memory.
clear_calib_data (CalibDataID)

8.6 Determine Translation in Z Direction for SCARA Robots

8.6

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Determine Translation in Z Direction for SCARA Robots

When calibrating SCARA robots, it is not possible to determine the Z translation of ’obj_in_base_pose’
(moving camera) or ’obj_in_tool_pose’ (stationary camera). To eliminate this ambiguity the Z translation is internally set to 0.0 in these poses and the poses ’tool_in_cam_pose’ (moving camera) and
’base_in_cam_pose’ (stationary camera), respectively, are calculated accordingly. It is necessary to determine the true translation in Z after the calibration by moving the robot to a pose of known height in
the camera coordinate system. For this, the following approach can be applied:
Moving camera
The calibration plate is placed at an arbitrary position. The robot is then moved such that the
camera can observe the calibration plate. Now, an image of the calibration plate is acquired and
the current robot pose is queried (ToolInBasePose1). From the image, the pose of the calibration
plate in the camera coordinate system can be determined (ObjInCamPose1). Afterwards, the tool
of the robot is manually moved to the origin of the calibration plate and the robot pose is queried
again (ToolInBasePose2). These three poses and the result of the calibration (ToolInCamPosePre)
can be used to fix the Z ambiguity by using the following lines of code:
pose_invert (ToolInCamPosePre, CamInToolPose)
pose_compose (CamInToolPose, ObjInCamPose1, ObjInToolPose1)
pose_invert (ToolInBasePose1, BaseInToolPose1)
pose_compose (BaseInToolPose1, ToolInBasePose2, Tool2InTool1Pose)
ZCorrection := ObjInToolPose1[2] - Tool2InTool1Pose[2]
set_origin_pose (ToolInCamPosePre, 0, 0, ZCorrection, ToolInCamPose)

Stationary camera
A calibration plate (that is not attached to the robot) is placed at an arbitrary position such that it
can be observed by the camera. The pose of the calibration plate must then be determined in the
camera coordinate system (ObjInCamPose). Afterwards the tool of the robot is manually moved to
the origin of the calibration plate and the robot pose is queried (ToolInBasePose). The two poses
and the result of the calibration (BaseInCamPosePre) can be used to fix the Z ambiguity by using
the following lines of code:
pose_invert (BaseInCamPosePre, CamInBasePose)
pose_compose (CamInBasePose, ObjInCamPose, ObjInBasePose)
ZCorrection := ObjInBasePose[2] - ToolInBasePose[2]
set_origin_pose (BaseInCamPosePre, 0, 0, ZCorrection, BaseInCamPose)

8.7

Using the Calibration Data

Typically, the result of the hand-eye calibration is used to transform the results of machine vision from
camera coordinates into robot base coordinates (cam Hobj → base Hobj ) to generate the appropriate
robot commands, e.g., to grasp an object whose position has been determined in an image as in the
application described in section 8.7.3 on page 230.

Robot Vision

This step is not required for articulated robots, but it is obligatory for SCARA robots.

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Stationary camera
For a stationary camera, this transformation corresponds to the following equation written using
homogeneous transformation matrices (compare figure 8.5 on page 219):

base

Hobj

base

Hcam · cam Hobj

(8.6)

This equation can also be implemented using the corresponding 3D HALCON poses.
pose_invert (BaseInCamPose, CamInBasePose)
pose_compose (CamInBasePose, ObjInCamPose, ObjInBasePose)

Moving camera
For a moving camera system, the equation also contains the pose of the robot tool when acquiring
the image of the object base Htool (acq. pos.) (compare figure 8.4 on page 219):
base

Hobj

base

Htool (acq. pos.) · tool Hcam · cam Hobj

(8.7)

This equation can also be implemented by composing the corresponding 3D HALCON poses.
pose_invert (ToolInCamPose, CamInToolPose)
pose_compose (ToolInBasePose, CamInToolPose, CamInBasePose)
pose_compose (CamInBasePose, ObjInCamPose, ObjInBasePose)

8.7.1

Using the Hand-Eye Calibration for Grasping (3D Alignment)

Grasping an object corresponds to a very simple equation that says “move the robot gripper to the pose
of the object” (“grasping pose”). This is also called 3D alignment.
Note that if the tool coordinate system used during hand-eye calibration is not placed at the gripper (tool
center point), the equation also contains the transformation between tool and gripper coordinate system.
This transformation cannot be calibrated with the hand-eye calibration but must be measured or taken
from the CAD model or technical drawing of the gripper. To grasp an object, the gripper pose in robot
base coordinates has to be identical to the pose of the object in robot base coordinates.

tool = gripper:

base

tool 6= gripper:

base
base

Htool (grip. pos.)

Htool (grip. pos.) ·
Htool (grip. pos.)

base

tool

Hobj

(8.8)

Hgripper =

base

tool

Hobj ·(

Hobj

Hgripper )

-1

Stationary camera
If we replace base Hobj according to equation 8.6, we get the “grasping equation” for a stationary
camera:

8.7 Using the Calibration Data

base

h
-1 i
Htool (grip. pos.) = (base Hcam ) · cam Hobj ·(tool Hgripper )

C-229

(8.9)

h
-1 i
The notation ·(tool Hgripper )
indicates that this part is only necessary if the tool coordinate
system is not identical with the gripper coordinate system.

pose_invert(BaseInCamPose, CamInBasePose)
pose_compose(CamInBasePose,ObjInCamPose, GripperInBasePose)
pose_invert (GripperInToolPose, ToolInGripper)
pose_compose (GripperInBasePose,ToolInGripper,ToolInBasePose)

Moving camera
For a moving camera, base Hobj is replaced by equation 8.7 on page 228 resulting in the following
equation:
base

h
-1 i
Htool (grip. pos.) = base Htool (acq. pos.) · (tool Hcam ) · cam Hobj ·(tool Hgripper )
(8.10)

h
-1 i
The notation ·(tool Hgripper )
indicates that this part is only necessary if the tool coordinate
system is not identical with the gripper coordinate system.
Using the HALCON poses, the above equation appears as follows:
pose_invert(ToolInCamPose, CamInToolPose)
pose_compose(ToolInBasePose_acq,CamInToolPose, CamInBasePose)
pose_invert (GripperInToolPose, ToolInGripper)
pose_compose (CamInBasePose,ToolInGripper,ToolInBasePose_grip)

8.7.2

How to Get the 3D Pose of the Object

The 3D pose of the object in camera coordinates (cam Hobj ) or more general in sensor coordinates can
stem from different sources:
• With a binocular stereo system or a gerneric 3D sensor, you can determine the 3D pose of unknown objects directly. More information on binocular stereo vision is provided in (see chapter 5
on page 139).
• For single camera systems, HALCON provides multiple methods. The most powerful one is
shape-based 3D Matching (see section 4.2 on page 111 or the Solution Guide I, chapter 10 on
page 145), which performs a full object recognition, i.e., it not only estimates the pose but first
locates the object in the image. If only one, planar side of the object is visible, fast alternatives
are the calibrated perspective deformable matching (section 4.6 on page 136) and the calibrated
descriptor-based matching (section 4.7 on page 137).

Robot Vision

Accordingly using the HALCON poses, the pose of the robot tool in robot base coordinates
(ToolInBasePose) that is needed for grasping an object is computed. When the gripper grasps
an object GripperInCamPose has to equal:

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Robot Vision

x
tool

base

x
camera

gripper

camera

tool

x
z

calibration plate

Figure 8.6: Example hand-eye system with a stationary camera: coordinate systems (a) of robot and
camera, (b) with calibration plate.

• If a full object recognition is not necessary, you can use pose estimation to determine the 3D pose
of known objects (see section 8.7.3 for an example application and chapter 4 on page 105 for more
details on pose estimation).
• Finally, you can determine the 3D coordinates of unknown objects if object points lie in a known
plane (see section 8.7.3 for an example application and section 3.3 on page 86 for more details on
determining 3D points in a known plane).
Please note that if you want to use the 3D pose for grasping the object, the extracted pose, in particular
the orientation, must be identical to the pose of the gripper at the grasping position.

8.7.3

Example Application with a Stationary Camera: Grasping a Nut

This section describes an example application realized with the hand-eye system depicted in
figure 8.6.
The task is to localize a nut and determine a suitable grasping pose for the
robot tool (see figure 8.7). The HDevelop example program solution_guide\3d_vision\
handeye_stationarycam_grasp_nut.hdev performs the machine vision part and transforms the
resulting pose into robot coordinates using the calibration data determined with solution_guide\
3d_vision\handeye_stationarycam_calibration.hdev as described in the previous sections. As
you will see, using the calibration data is the shortest part of the program, its main part is devoted to
machine vision.
Step 1:

Read calibration data

First, the calibrated poses are read from files; for later computations, the poses are converted into homogeneous transformation matrices.

8.7 Using the Calibration Data

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C1 − C4: corner points
G1 & G2: grasping points

reference coordinate system

Figure 8.7: (a) Determining the 3D pose for grasping a nut; (b) robot tool at grasping pose.

read_cam_par (DataNameStart + 'final_campar.dat', CamParam)
read_pose (DataNameStart + 'final_pose_cam_base.dat', BaseInCamPose)
pose_to_hom_mat3d (BaseInCamPose, cam_H_base)
read_pose (DataNameStart + 'final_pose_tool_calplate.dat', \
CalplateInToolPose)
pose_to_hom_mat3d (CalplateInToolPose, tool_H_calplate)

In the used hand-eye system, the tool coordinate system used in the calibration process is located at the
mounting point of the tool; therefore, an additional coordinate system is needed between the fingers of
the gripper (see figure 8.6a on page 230). Its pose in tool coordinates is also read from file.
read_pose (DataNameStart + 'pose_tool_gripper.dat', GripperInToolPose)
pose_to_hom_mat3d (GripperInToolPose, tool_H_gripper)

Step 2:

Define reference coordinate system

Now, a reference coordinate system is defined based on one of the calibration images. In this image,
the calibration plate has been placed into the plane on top of the nut. This allows to determine the
3D coordinates of extracted image points on the nut with a single camera and without knowing the
dimensions of the nut. The code for defining the reference coordinate system is contained in a procedure,
which is listed in appendix A.7 on page 281.
define_reference_coord_system (ImageNameStart + 'calib3cm_00', CamParam, \
CalplateFile, WindowHandle, PoseRef)
pose_to_hom_mat3d (PoseRef, cam_H_ref)

Step 3:

Extract grasping points on the nut

The following code extracts grasping points on two opposite sides of the nut. The nut is found with
simple blob analysis; its boundary is converted into XLD contours.

Robot Vision

grasping pose

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threshold (Image, BrightRegion, 60, 255)
connection (BrightRegion, BrightRegions)
select_shape (BrightRegions, Nut, 'area', 'and', 500, 99999)
fill_up (Nut, NutFilled)
gen_contour_region_xld (NutFilled, NutContours, 'border')

The contours are then processed to find long, parallel straight line segments; their corners are accumulated in tuples.
segment_contours_xld (NutContours, LineSegments, 'lines', 5, 4, 2)
fit_line_contour_xld (LineSegments, 'tukey', -1, 0, 5, 2, RowBegin, \
ColBegin, RowEnd, ColEnd, Nr, Nc, Dist)
gen_empty_obj (Lines)
for I := 0 to |RowBegin| - 1 by 1
gen_contour_polygon_xld (Contour, [RowBegin[I],RowEnd[I]], [ColBegin[I], \
ColEnd[I]])
concat_obj (Lines, Contour, Lines)
endfor
gen_polygons_xld (Lines, Polygon, 'ramer', 2)
gen_parallels_xld (Polygon, ParallelLines, 50, 100, rad(10), 'true')
get_parallels_xld (ParallelLines, Row1, Col1, Length1, Phi1, Row2, Col2, \
Length2, Phi2)
CornersRow := [Row1[0],Row1[1],Row2[0],Row2[1]]
CornersCol := [Col1[0],Col1[1],Col2[0],Col2[1]]

Step 4:

Determine the grasping pose in camera coordinates

The grasping pose is calculated in 3D coordinates. For this, the 3D coordinates of the corner points in
the reference coordinate system are determined using the operator image_points_to_world_plane.
The origin of the grasping pose lies in the middle of the corners.
image_points_to_world_plane (CamParam, PoseRef, CornersRow, CornersCol, 'm', \
CornersX_ref, CornersY_ref)
CenterPointX_ref := sum(CornersX_ref) * 0.25
CenterPointY_ref := sum(CornersY_ref) * 0.25

The grasping pose is oriented almost like the reference coordinate system, only rotated around the z-axis
so that it is identical to the gripper coordinate system, i.e., so that the gripper “fingers” are parallel to the
sides of the nut. To calculate the rotation angle, first the grasping points in the middle of the sides are
determined. Their angle can directly be used as the rotation angle.
GraspPointsX_ref := [(CornersX_ref[0] + CornersX_ref[1]) * 0.5, \
(CornersX_ref[2] + CornersX_ref[3]) * 0.5]
GraspPointsY_ref := [(CornersY_ref[0] + CornersY_ref[1]) * 0.5, \
(CornersY_ref[2] + CornersY_ref[3]) * 0.5]
GraspPhiZ_ref := atan((GraspPointsY_ref[1] - GraspPointsY_ref[0]) /
(GraspPointsX_ref[1] - GraspPointsX_ref[0]))

With the origin and rotation angle, the grasping pose is first determined in the reference coordinate
system and then transformed into camera coordinates.

8.7 Using the Calibration Data

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Alternatively, the example also shows how to calculate the grasping pose using pose estimation (see
chapter 4 on page 105 for a detailed description). This method can be used when points on the object are
known. In the example, we specify the 3D coordinates of the corners of the nut.
NX := [0.009,-0.009,-0.009,0.009]
NY := [0.009,0.009,-0.009,-0.009]

The grasping pose is then calculated by simply calling the operator vector_to_pose. Before, however,
the image coordinates of the corners must be sorted such that the first one lies close to the x-axis of the
reference coordinate system. Otherwise, the orientation of the reference coordinate system would differ
too much from the grasping pose and the pose estimation would fail.
sort_corner_points (CornersRow, CornersCol, WindowHandle, NRow, NCol)
vector_to_pose (NX, NY, NZ, NRow, NCol, CamParam, 'iterative', 'error', \
PoseCamNut, Quality)
disp_3d_coord_system (WindowHandle, CamParam, GripperInCamPose, 0.01)

The result of both methods is displayed in figure 8.7a on page 231.
Step 5:

Transform the grasping pose in robot coordinates

Now comes the moment to use the results of the hand-eye calibration: The grasping pose is transformed
into robot coordinates with the formula shown in equation 8.9 on page 229.
hom_mat3d_invert (cam_H_base, base_H_cam)
hom_mat3d_compose (base_H_cam, cam_H_grasp, base_H_grasp)

As already mentioned, the tool coordinate system used in the calibration process is placed at the mounting
point of the tool, not between the fingers of the gripper. Thus, the pose of the tool in gripper coordinates
must be added to the chain of transformations to obtain the pose of the tool in base coordinates.
hom_mat3d_invert (tool_H_gripper, gripper_H_tool)
hom_mat3d_compose (base_H_grasp, gripper_H_tool, base_H_tool)

Step 6:

Transform pose type

Finally, the pose is converted into the type used by the robot controller.
hom_mat3d_to_pose (base_H_tool, PoseRobotGrasp)
convert_pose_type (PoseRobotGrasp, 'Rp+T', 'abg', 'point', \
PoseRobotGrasp_ZYX)

Figure 8.7b on page 231 shows the robot at the grasping pose.

Robot Vision

hom_mat3d_identity (HomMat3DIdentity)
hom_mat3d_rotate (HomMat3DIdentity, GraspPhiZ_ref, 'z', 0, 0, 0, \
HomMat3D_RZ_Phi)
hom_mat3d_translate (HomMat3D_RZ_Phi, CenterPointX_ref, CenterPointY_ref, 0, \
ref_H_grasp)
hom_mat3d_compose (cam_H_ref, ref_H_grasp, cam_H_grasp)

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Calibrated Mosaicking

C-235

Chapter 9

Some objects are too large to be covered by one single image. Multiple images that cover different parts
of the object must be taken in such cases. You can measure precisely across the different images if
the cameras are calibrated and their external parameters are known with respect to one common world
coordinate system.
It is even possible to merge the individual images into one larger image that covers the whole object. This
is done by rectifying the individual images with respect to the same measurement plane (see section 3.4.1
on page 91). In the resulting image, you can measure directly in world coordinates.
Note that the 3D coordinates of objects are derived based on the same principle as described in chapter 3
on page 65, i.e., a measurement plane that coincides with the object surface must be defined. Although
two or more cameras are used, this is no stereo approach. For more information on 3D vision with a
binocular stereo system, please refer to chapter 5 on page 139.
If the resulting image is not intended to serve for high-precision measurements in world coordinates, you
can generate it using the mosaicking approach described in chapter 10 on page 247. With this approach,
it is not necessary to calibrate the cameras.
A setup for generating a high-precision mosaic image from two cameras is shown in figure 9.1. The
cameras are mounted such that the resulting pair of images has a small overlap. The cameras are first
calibrated and then the images are merged together into one larger image. All further explanations within
this section refer to such a two-camera setup.
Typically, the following steps must be carried out:
1. Determination of the internal camera parameters for each camera separately.
2. Determination of the external camera parameters, using one calibration object, to facilitate that the
relation between the cameras can be determined.
3. Merge the images into one larger image that covers the whole object.
The HDevelop example program solution_guide\3d_vision\two_camera_calibration.hdev
shows how to determine the external camera parameters and how to merge two images into one larger
image.

Mosaicking I

Calibrated Mosaicking

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Calibrated Mosaicking

Calibration

Figure 9.1: Two-camera setup.

9.1

Setup

Two or more cameras must be mounted on a stable platform such that each image covers a part of the
whole scene. The cameras can have an arbitrary orientation, i.e., it is not necessary that they are looking
parallel or perpendicular onto the object surface.
To setup focus, illumination, and overlap appropriately, use a big reference object that covers all fields
of view. To permit that the images are merged into one larger image, they must have some overlap (see
figure 9.2 for an example). The overlapping area can be even smaller than depicted in figure 9.2, since
the overlap is only necessary to ensure that there are no gaps in the resulting combined image.

9.2

Calibration

The calibration of the images can be broken down into two separate steps.
The first step is to determine the internal camera parameters for each of the cameras in use. This can be
done for each camera independently, as described in section 3.2 on page 68.

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{
{

9.2 Calibration

Figure 9.2: Overlapping images.

The second step is to determine the external camera parameters for all cameras. Because the final coordinates should refer to one world coordinate system for all images, a big calibration object that appears in
all images has to be used. For calibration plates with rectangularly arranged marks, which must be completely visible in the image, we propose to use a calibration object like the one displayed in figure 9.3,
which consists of as many calibration plates as the number of cameras that are used. As calibration
plates with hexagonally arranged marks may protrude beyond the rim of the image as long as at least one
finder pattern is completely visible in the image, there a single calibration plate would be sufficient for
the mosaicking of two images.
For the determination of the external camera parameters, it is sufficient to use one calibration image from
each camera only. Note that the calibration object must not be moved in between the acquisition of the
individual images. Ideally, the images are acquired simultaneously. The pose of the calibration plates
relative to the cameras is then extracted using an initialized calibration data model and the operators
find_calib_object and get_calib_data_observ_points.
CaltabName := 'caltab_30mm.descr'
create_calib_data ('calibration_object', 2, 1, CalibDataID)
set_calib_data_calib_object (CalibDataID, 0, CaltabName)
set_calib_data_cam_param (CalibDataID, 0, 'area_scan_division',
set_calib_data_cam_param (CalibDataID, 1, 'area_scan_division',
*
* Find and display the calibration plate in the images.
find_calib_object (Image1, CalibDataID, 0, 0, 0, [], [])
get_calib_data_observ_points (CalibDataID, 0, 0, 0, RowCoord1,
Index1, Pose1)
*
find_calib_object (Image2, CalibDataID, 1, 0, 0, [], [])
get_calib_data_observ_points (CalibDataID, 1, 0, 0, RowCoord2,
Index2, Pose2)

CamParam1)
CamParam2)

ColumnCoord1, \

ColumnCoord2, \

Mosaicking I

Overlapping area

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Calibrated Mosaicking

Here, calibration plates with rectangularly arranged marks are used. The calibration is nevertheless easy
if standard HALCON calibration plates mounted on some kind of carrier plate are used such that in
each image one calibration plate is completely visible. An example for such a calibration object for a
two-camera setup is given in figure 9.3. The respective calibration images for the determination of the
external camera parameters are shown in figure 9.4. Note that the relative position of the calibration
plates with respect to each other must be known precisely. This can be done with the pose estimation
described in chapter 4 on page 105.
Note also that only the relative position of the calibration marks among each other shows the high accuracy stated in section 3.2.3.1 on page 74 but not the borders of the calibration plate. The rows of
calibration marks may be slanted with respect to the border of the calibration plate and even the distance of the calibration marks from the border of the calibration plate may vary. Therefore, aligning the
calibration plates along their boundaries may result in a shift in x- and y-direction with respect to the
coordinate system of the calibration plate in its initial position.

Figure 9.3: Calibration object for two-camera setup.

Figure 9.4: Calibration images for two-camera setup.

9.3

Merging the Individual Images into One Larger Image

At first, the individual images must be rectified, i.e, transformed so that they exactly fit together. This
can be achieved by using the operators gen_image_to_world_plane_map and map_image. Then, the

9.3 Merging the Individual Images into One Larger Image

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mosaic image can be generated by the operator tile_images, which tiles multiple images into one
larger image. These steps are visualized in figure 9.5.

map_image

Mosaicking I

map_image

tile_images

Figure 9.5: Image rectification and tiling.

The operators gen_image_to_world_plane_map and map_image are described in section 3.4.1 on
page 91. In the following, we will only discuss the problem of defining the appropriate image detail, i.e.,
the position of the upper left corner and the size of the rectified images. Again, the description is based
on the two-camera setup.

9.3.1

Definition of the Rectification of the First Image

For the first (here: left) image, the determination of the necessary shift of the pose is straightforward.
You can define the upper left corner of the rectified image in image coordinates, e.g., interactively or, as

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Calibrated Mosaicking

in the example program, based on a preselected border width.
UpperRow := HeightImage1 * BorderInPercent / 100.0
LeftColumn := WidthImage1 * BorderInPercent / 100.0

Then, this point must be transformed into world coordinates.
image_points_to_world_plane (CamParam1, Pose1, UpperRow, LeftColumn, 'm', \
LeftX, UpperY)

The resulting coordinates can be used directly, together with the shift that compensates the thickness of
the calibration plate (see section 3.2.7.1 on page 82) to modify the origin of the world coordinate system
in the left image.
set_origin_pose (Pose1, LeftX, UpperY, DiffHeight, PoseNewOrigin1)

This means that we shift the origin of the world coordinate system from the origin of the calibration plate
to the position that defines the upper left corner of the rectified image (figure 9.6).

Rectified image 1
Image 1

Image 2

Figure 9.6: Definition of the upper left corner of the first rectified image.

The size of the rectified image, i.e., its width and height, can be determined from points originally defined
in image coordinates, too. In addition, the desired pixel size of the rectified image must be specified.
PixelSize := 0.0001

9.3 Merging the Individual Images into One Larger Image

C-241

For the determination of the height of the rectified image we need to define a point that lies near the
lower border of the first image.
LowerRow := HeightImage1 * (100 - BorderInPercent) / 100.0

Again, this point must be transformed into the world coordinate system.
image_points_to_world_plane (CamParam1, Pose1, LowerRow, LeftColumn, 'm', \
X1, LowerY)

The height can be determined as the vertical distance between the upper left point and the point near the
lower image border, expressed in pixels of the rectified image.

Analogously, the width can be determined from a point that lies in the overlapping area of the two images,
i.e., near the right border of the first image.
RightColumn := WidthImage1 * (100 - OverlapInPercent / 2.0) / 100.0
image_points_to_world_plane (CamParam1, Pose1, UpperRow, RightColumn, 'm', \
RightX, Y1)
WidthRect := int((RightX - LeftX) / PixelSize)

Note that the above described definitions of the image points, from which the upper left corner and the
size of the rectified image are derived, assume that the x- and y-axes of the world coordinate system are
approximately aligned to the column- and row-axes of the first image. This can be achieved by placing
the calibration plate in the first image approximately aligned with the image borders. Otherwise, the
distances between the above mentioned points make no sense and the upper left corner and the size of
the rectified image must be determined in a manner that is adapted for the configuration at hand.
With the shifted pose and the size of the rectified image, the rectification map for the first image can be
derived.
gen_image_to_world_plane_map (MapSingle1, CamParam1, PoseNewOrigin1, Width, \
Height, WidthRect, HeightRect, PixelSize, \
'bilinear')

9.3.2

Definition of the Rectification of the Second Image

The second image must be rectified such that it fits exactly to the right of the first rectified image. This
means that the upper left corner of the second rectified image must be identical with the upper right
corner of the first rectified image. Therefore, we need to know the coordinates of the upper right corner
of the first rectified image in the coordinate system that is defined by the calibration plate in the second
image.
First, we express the upper right corner of the first rectified image in the world coordinate system that
is defined by the calibration plate in the first image. It can be determined by a transformation from

Mosaicking I

HeightRect := int((LowerY - UpperY) / PixelSize)

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the origin into the upper left corner of the first rectified image (a translation in the example program)
followed by a translation along the upper border of the first rectified image. Together with the shift that
compensates the thickness of the calibration plate, this transformation is represented by the homogeneous
transformation matrix cp1 Hur1 (see figure 9.7), which can be defined in HDevelop by:
hom_mat3d_translate_local (HomMat3DIdentity, LeftX + PixelSize * WidthRect, \
UpperY, DiffHeight, cp1Hur1)

cp1

H ur1

Rectified image 1
Image 1

Rectified image 2
Image 2

Figure 9.7: Definition of the upper right corner of the first rectified image.

Then, we need the transformation between the two calibration plates of the calibration object. The
homogeneous transformation matrix cp1 Hcp2 describes how the world coordinate system defined by
the calibration plate in the first image is transformed into the world coordinate system defined by the
calibration plate in the second image (figure 9.8). This transformation must be known beforehand from
a precise measurement of the calibration object.
From these two transformations, it is easy to derive the transformation that transforms the world coordinate system of the second image such that its origin lies in the upper left corner of the second rectified
image. For this, the two transformations have to be combined appropriately (see figure 9.9):

cp2

Hul2

=
=

cp2

(

Hcp1 · cp1 Hur1

cp1

This can be implemented in HDevelop as follows:

Hcp2 )−1 · cp1 Hur1

(9.1)
(9.2)

9.3 Merging the Individual Images into One Larger Image

cp1

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H cp2

Rectified image 1
Image 1

Rectified image 2

Mosaicking I

Image 2

Figure 9.8: Transformation between the two world coordinate systems, each defined by the respective
calibration plate.

hom_mat3d_invert (cp1Hcp2, cp2Hcp1)
hom_mat3d_compose (cp2Hcp1, cp1Hur1, cp2Hul2)

With this, the pose of the calibration plate in the second image can be modified such that the origin of
the world coordinate system lies in the upper left corner of the second rectified image.
pose_to_hom_mat3d (Pose2, cam2Hcp2)
hom_mat3d_compose (cam2Hcp2, cp2Hul2, cam2Hul2)
hom_mat3d_to_pose (cam2Hul2, PoseNewOrigin2)

With the resulting new pose and the size of the rectified image, which can be the same as for the first
rectified image, the rectification map for the second image can be derived.
gen_image_to_world_plane_map (MapSingle2, CamParam2, PoseNewOrigin2, Width, \
Height, WidthRect, HeightRect, PixelSize, \
'bilinear')

9.3.3

Rectification of the Images

Once the rectification maps are created, every image pair from the two-camera setup can be rectified and
tiled very efficiently. The resulting mosaic image consists of the two rectified images and covers a part
as indicated in figure 9.10.

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Calibrated Mosaicking

cp1

cp2

Rectified image 1
Image 1

cp2

H ur1

H ul2

H cp1

Rectified image 2
Image 2

Figure 9.9: Definition of the upper left corner of the second rectified image.

Mosaic image
Image 1
Image 2

Figure 9.10: The position of the final mosaic image.

The rectification is carried out by the operator map_image.

9.3 Merging the Individual Images into One Larger Image

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map_image (Image1, MapSingle1, RectifiedImage1)
map_image (Image2, MapSingle2, RectifiedImage2)

Mosaicking I

This transforms the two images displayed in figure 9.11, into the two rectified images that are shown in
figure 9.12.

Figure 9.11: Two test images acquired with the two-camera setup.

Figure 9.12: Rectified images.

As a preparation for the tiling, the rectified images must be concatenated into one tuple, which then
contains both images.
concat_obj (RectifiedImage1, RectifiedImage2, Concat)

Then the two images can be tiled.
tile_images (Concat, Combined, 2, 'vertical')

The resulting mosaic image is displayed in figure 9.13.

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Figure 9.13: Mosaic image.

Uncalibrated Mosaicking

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Chapter 10

Uncalibrated Mosaicking

An example for such an application is given in figure 10.1. On the left side, six separate images are
displayed stacked upon each other. On the right side, the mosaic image generated from the six separate
images is shown. Note that the folds visible in the image do not result from the mosaicking. They are
due to some degradations on the PCB, which can be seen already in the separate images.
The mosaicking approach described in this section is designed for applications where it is not necessary
to achieve the high-precision mosaic images as described in chapter 9 on page 235. The advantages
compared to this approach are that no camera calibration is necessary and that the individual images can
be arranged automatically.
The example program solution_guide\3d_vision\mosaicking.hdev generates the mosaic image
displayed in figure 10.7 on page 253. First, the images are read from file and collected in one tuple.
gen_empty_obj (Images)
for J := 1 to 10 by 1
read_image (Image, ImgPath + ImgName + J$'02')
concat_obj (Images, Image, Images)
endfor

Then, the image pairs must be defined, i.e., which image should be mapped to which image.
From := [1,2,3,4,6,7,8,9,3]
To := [2,3,4,5,7,8,9,10,8]

Now, characteristic points must be extracted from the images, which are then used for the matching
between the image pairs. The resulting projective transformation matrices1 must be accumulated.
1 A projective transformation matrix describes a perspective projection. It consists of 3×3 values. If the last row contains the
values [0,0,1], it corresponds to a homogeneous transformation matrix of HALCON and therefore describes an affine transformation.

Mosaicking II

If you need an image of a large object, but the field of view of the camera does not allow to cover the
entire object with the desired resolution, you can use image mosaicking to generate a large image of the
entire object from a sequence of overlapping images of parts of the object.

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Uncalibrated Mosaicking

→

Figure 10.1: A first example for image mosaicking.

10.1 Rules for Taking Images for a Mosaic Image

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Num := |From|
ProjMatrices := []
for J := 0 to Num - 1 by 1
F := From[J]
T := To[J]
select_obj (Images, ImageF, F)
select_obj (Images, ImageT, T)
points_harris (ImageF, SigmaGrad, SigmaSmooth, Alpha, Threshold, \
RowFAll, ColFAll)
points_harris (ImageT, SigmaGrad, SigmaSmooth, Alpha, Threshold, \
RowTAll, ColTAll)
proj_match_points_ransac (ImageF, ImageT, RowFAll, ColFAll, RowTAll, \
ColTAll, 'sad', MaskSize, RowMove, ColMove, \
RowTolerance, ColTolerance, Rotation, \
MatchThreshold, 'gold_standard', \
DistanceThreshold, RandSeed, ProjMatrix, \
Points1, Points2)
ProjMatrices := [ProjMatrices,ProjMatrix]
endfor

gen_projective_mosaic (Images, MosaicImage, StartImage, From, To, \
ProjMatrices, StackingOrder, 'false', \
MosaicMatrices2D)

Note that image mosaicking is a tool for a quick and easy generation of large images from several
overlapping images. For this task, it is not necessary to calibrate the camera. If you need a high-precision
image mosaic, you should use the method described in chapter 9 on page 235.
In the following sections, the individual steps for the generation of a mosaic image are described.

10.1

Rules for Taking Images for a Mosaic Image

The following rules for the acquisition of the separate images should be considered:
• The images must overlap each other.
• The overlapping area of the images must be textured in order to allow the automatic matching
process to identify identical points in the images. The lack of texture in some overlapping areas
may be overcome by an appropriate definition of the image pairs (see section 10.2). If the whole
object shows little texture, the overlapping areas should be chosen larger.
• Overlapping images must have approximately the same scale. In general, the scale differences
should not exceed 5-10 %.
• The images should be radiometrically similar, at least in the overlapping areas, as no radiometric
adaption of the images is carried out. Otherwise, i.e., if the brightness differs heavily between
neighboring images, the seams between them will be clearly visible as can be seen in figure 10.2.

Mosaicking II

Finally, the image mosaic can be generated.

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Uncalibrated Mosaicking

→
Figure 10.2: A second example for image mosaicking.

The images are mapped onto a common image plane using a projective transformation. Therefore, to
generate a geometrically accurate image mosaic from images of non-flat objects, the separate images
must be acquired from approximately the same point of view, i.e., the camera can only be rotated around
its optical center (see figure 10.3).
When dealing with flat objects, it is possible to acquire the images from arbitrary positions and with
arbitrary orientations if the scale difference between the overlapping images is not too large (figure 10.4).
The lens distortions of the images are not compensated by the mosaicking process. Therefore, if lens
distortions are present in the images, they cannot be mosaicked with high accuracy, i.e., small distortions at the seams between neighboring images cannot be prevented (see figure 10.8 on page 254). To
eliminate this effect, the lens distortions can be compensated before starting the mosaicking process (see
section 3.4.2 on page 98).
If processing time is an issue, it is advisable to acquire the images in the same orientation, i.e., neither
the camera nor the object should be rotated around the optical axis, too much. Then, the rotation range
can be restricted for the matching process (see section 10.4 on page 256).

10.2

Definition of Overlapping Image Pairs

As shown in the introductory example, it is necessary to define the overlapping image pairs between
which the transformation is to be determined. The successive matching process will be carried out for
these image pairs only.

10.2 Definition of Overlapping Image Pairs

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Camera in three orientations
Common optical center

Object surface

Figure 10.3: Image acquisition for non-flat objects.

Mosaicking II

Camera in three positions

Object surface

Figure 10.4: Image acquisition for flat objects.

Figure 10.5 shows two configurations of separate images. For configuration (a), the definition of the
image pairs is simply (1,2) and (2,3), which can be defined in HDevelop as:
From := [1,2]
To := [2,3]

In any case, it is important to ensure that each image must be “connected” to all the other images. For
example, for configuration (b) of figure 10.5, it is not possible to define the image pairs as (1,2) and (3,4),

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Uncalibrated Mosaicking

1 2 3
(a)

1 2
3 4
(b)

Figure 10.5: Two configurations of overlapping images.

only, because images 1 and 2 would not be connected to images 3 and 4. In this case, it would, e.g., be
possible to define the three image pairs (1,2), (1,3), and (2,4).
From := [1,1,2]
To := [2,3,4]

Assuming there is no texture in the overlapping area of image two and four, the matching could be carried
out between images three and four instead.
From := [1,1,3]
To := [2,3,4]

If a larger number of separate images are mosaicked, or, e.g., an image configuration similar to the
one displayed in figure 10.6, where there are elongated rows of overlapping images, it is important to
thoroughly arrange the image pair configuration. Otherwise it is possible that some images do not fit
together precisely. This happens since the transformations between the images cannot be determined
with perfect accuracy because of very small errors in the point coordinates due to noise. These errors are
propagated from one image to the other.
Figure 10.7 shows such an image sequence of ten images of a BGA and the resulting mosaic image.
Figure 10.8 shows a cut-out of that mosaic image. It depicts the seam between image 5 and image 10 for
two image pair configurations, using the original images and the images where the lens distortions have
been eliminated, respectively. The position of the cut-out is indicated in figure 10.7 by a rectangle.
First, the matching has been carried out in the two image rows separately and the two rows are connected
via image pair 1 → 6.
From := [1,2,3,4,6,7,8,9,1]
To := [2,3,4,5,7,8,9,10,6]

In this configuration the two neighboring images 5 and 10 are connected along a relatively long path
(figure 10.9).

10.2 Definition of Overlapping Image Pairs

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1 2 3 4 5
6 7 8 9 10

→
Figure 10.7: Ten overlapping images and the resulting (rigid) mosaic image.

To improve the geometrical accuracy of the image mosaic, the connections between the two image rows
could be established by the image pair (3,8), as visualized in (figure 10.10)).
This can be achieved by defining the image pairs as follows.
From := [1,2,3,4,6,7,8,9,3]
To := [2,3,4,5,7,8,9,10,8]

As can be seen in figure 10.8, now the neighboring images fit better.

Mosaicking II

Figure 10.6: A configuration of ten overlapping images.

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Uncalibrated Mosaicking

with lens distortions

lens distortions eliminated

unfavorable configuration

good configuration
Figure 10.8: Seam between image 5 and image 10 for various configurations.

Figure 10.9: Unfavorable configuration of image pairs.

Recapitulating, there are three basic rules for the arrangement of the image pairs:
Take care that
1. each image is connected to all the other images.
2. the path along which neighboring images are connected is not too long.
3. the overlapping areas of image pairs are large enough and contain enough texture to ensure a
proper matching.
In principle, it is also possible to define more image pairs than required (number of images minus one).
However, then it cannot be controlled which pairs are actually used. Therefore, we do not recommend
this approach.

Figure 10.10: Good configuration of image pairs.

10.3 Detection of Characteristic Points

10.3

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Detection of Characteristic Points

HALCON provides you with various operators for the extraction of characteristic points (interest points).
The most important of these operators are
• points_foerstner
• points_harris and points_harris_binomial
• points_lepetit
• points_sojka
• saddle_points_sub_pix
All of these operators can determine the coordinates of interest points with subpixel accuracy.
In figure 10.11, a test image together with typical results of these interest operators is displayed.

The results of the operator points_harris are very reproducible, too. Admittedly, the points extracted by the operator points_harris are sometimes not meaningful to a human, e.g., they often
lie slightly beside a corner or an eye-catching image structure. Nevertheless, it is faster than the operator points_foerstner. The operator points_harris_binomial detects points of interest using the
binomial approximation of the points_harris operator. It is therefore faster than points_harris.
The operator points_lepetit extracts points of interest like corners or blob-like structures from the
image. This operator can especially be used for very fast interest point extraction. It is the fastest out of
the six operators presented in this section.
The operator points_sojka is specialized in the extraction of corner points.
The operator saddle_points_sub_pix is designed especially for the extraction of saddle points, i.e.,
points whose image intensity is minimal along one direction and maximal along a different direction.
The number of interest points influence the execution time and the result of the subsequent matching
process. The more interest points are used, the longer the matching takes. If too few points are used the
probability of an erroneous result increases.
In most cases, the default parameters of the interest operators need not be changed. Only if
too many or too few interest points are found adaptations of the parameters might be necessary. For a description of the parameters, please refer to the respective pages of the Reference Manual (points_foerstner, points_harris, points_harris_binomial, points_lepetit,
points_sojka, saddle_points_sub_pix).

Mosaicking II

The operator points_foerstner classifies the interest points into two categories: junction-like features
and area-like features. The results are very reproducible even in images taken from a different point of
view. Therefore, it is very well suited for the extraction of points for the subsequent matching. It is very
accurate but computationally the most expensive operator out of the interest operators presented in this
section.

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Uncalibrated Mosaicking

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 10.11: Comparison of typical results of interest operators. a) Test image; b) Förstner, junctions; c)
Förstner, area; d) Harris; e) Harris, binomial f) Lepetit; g) Sojka; h) Saddle points.

10.4

Matching of Characteristic Points in Overlapping Areas
and Determination of the Transformation between the
Images

The most demanding task during the generation of an image mosaic is the matching process. The operator proj_match_points_ransac is able to perform the matching even if the two images are shifted

10.4 Matching of Characteristic Points

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and rotated arbitrarily.
proj_match_points_ransac (ImageF, ImageT, RowFAll, ColFAll, RowTAll, \
ColTAll, 'sad', MaskSize, RowMove, ColMove, \
RowTolerance, ColTolerance, Rotation, \
MatchThreshold, 'gold_standard', \
DistanceThreshold, RandSeed, ProjMatrix, Points1, \
Points2)

The only requirement is that the images should have approximately the same scale. If information about
shift and rotation is available it can be used to restrict the search space, which speeds up the matching
process and makes it more robust.

If the images that should be mosaicked contain repetitive patterns, like the two images of a BGA shown
in figure 10.12a, it may happen that the matching does not work correctly. In the resulting erroneous
mosaic image, the separate images may not fit together or may be heavily distorted. To achieve a correct matching result for such images, it is important to provide initial values for the shift between the
images with the parameters RowMove and ColMove. In addition, the search space should be restricted
to an area that contains only one instance of the repetitive pattern, i.e., the values of the parameters
RowTolerance and ColTolerance should be chosen smaller than the distance between the instances
of the repetitive pattern. With this, it is possible to obtain proper mosaic images, even for objects like
BGAs (see figure 10.12b).

(b)
(a)
Figure 10.12: Separate images (a) and mosaic image (b) of a BGA.

For a detailed description of the other parameters, please refer to the Reference Manual
(proj_match_points_ransac).

Mosaicking II

In case the matching fails, ensure that there are enough characteristic points and that the search space
and the maximum rotation are defined appropriately.

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Uncalibrated Mosaicking

The results of the operator proj_match_points_ransac are the projective transformation matrix and
the two tuples Points1 and Points2 that contain the indices of the matched input points from the two
images.
The projective transformation matrices resulting from the matching between the image pairs must be
accumulated.
ProjMatrices := [ProjMatrices,ProjMatrix]

Alternatively, if it is known that the mapping between the images is a rigid 2D transformation, the
operator proj_match_points_ransac can be used to determine the point correspondences only, since
it returns the indices of the corresponding points in the tuples Points1 and Points2. With this, the
corresponding point coordinates can be selected.
RowF
ColF
RowT
ColT

:=
:=
:=
:=

subset(RowFAll,Points1)
subset(ColFAll,Points1)
subset(RowTAll,Points2)
subset(ColTAll,Points2)

Then, the rigid transformation between the image pair can be determined with the operator
vector_to_rigid. Note that we have to add 0.5 to the coordinates to make the extracted pixel positions fit the coordinate system that is used by the operator gen_projective_mosaic.
vector_to_rigid (RowF + 0.5, ColF + 0.5, RowT + 0.5, ColT + 0.5, HomMat2D)

Because gen_projective_mosaic expects a 3×3 transformation matrix, but vector_to_rigid returns a 2×3 matrix, we have to add the last row [0,0,1] to the transformation matrix before we can
accumulate it.
ProjMatrix := [HomMat2D,0,0,1]
ProjMatrices := [ProjMatrices,ProjMatrix]

Furthermore, the operator proj_match_points_ransac_guided is available.
Like
proj_match_points_ransac, it can be used to calculate the projective transformation matrix between two images by finding correspondences between points.
But in contrast to
proj_match_points_ransac, it is based on a known approximation of the projective transformation matrix. Thus, it can be used, for example, to speed up the matching of very large images
by implementing an image-pyramid-based projective matching algorithm. The HDevelop example
program hdevelop\Tools\Mosaicking\mosaicking_pyramid.hdev shows how to implement the
image-pyramid-based approach and compares the runtime for different numbers of pyramid levels.

10.5

Generation of the Mosaic Image

Once the transformations between the image pairs are known the mosaic image can be generated with
the operator gen_projective_mosaic.

10.6 Bundle Adjusted Mosaicking

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gen_projective_mosaic (Images, MosaicImage, StartImage, From, To, \
ProjMatrices, StackingOrder, 'false', \
MosaicMatrices2D)

It requires the images to be given in a tuple. All images are projected into the image plane of a so-called
start image. The start image can be defined by its position in the image tuple (starting with 1) with the
parameter StartImage.
Additionally, the image pairs must be specified together with the corresponding transformation matrices.
The order in which the images are added to the mosaic image can be specified with the parameter
StackingOrder. The first index in this array will end up at the bottom of the image stack while the
last one will be on top. If ’default’ is given instead of an array of integers, the canonical order (the order
in which the images are given) will be used.
If the domains of the images should be transformed as well, the parameter TransformRegion must be
set to ’true’.

10.6

Bundle Adjusted Mosaicking

It is also possible to generate the mosaic based on the matching results of all overlapping image pairs. The
transformation matrices between the images are then determined together within one bundle adjustment.
For this, the operators bundle_adjust_mosaic and gen_bundle_adjusted_mosaic are used.
The main advantage of the bundle adjusted mosaicking compared with the mosaicking based on single
image pairs is that the bundle adjustment determines the geometry of the mosaic as robustly as possible.
Typically, this leads to more accurate results. Another advantage is that there is no need to figure out
a good pair configuration, you simply pass the matching results of all overlapping image pairs. What
is more, it is possible to define the class of transformations that is used for the transformation between
the individual images. A disadvantage of the bundle adjusted mosaicking is that it takes more time to
perform the matching between all overlapping image pairs instead of just using a subset. Furthermore, if
the matching between two images was erroneous, sometimes the respective image pair is difficult to find
in the set of all image pairs.
With this, it is obvious that the bundle adjustment is worthwhile if there are multiple overlaps between
the images, i.e., if there are more than n − 1 overlapping image pairs, with n being the number of
images. Another reason for using the bundle adjusted mosaicking is the possibility to define the class of
transformations.
The example program solution_guide\3d_vision\bundle_adjusted_mosaicking.hdev shows
how to generate the bundle adjusted mosaic from the ten images of the BGA displayed in figure 10.7 on
page 253. The design of the program is very similar to that of the example program solution_guide\
3d_vision\mosaicking.hdev, which is described in the introduction of chapter 10 on page 247. The
main differences are that

Mosaicking II

The output parameter MosaicMatrices2D contains the projective 3×3 transformation matrices for
the mapping of the separate images into the mosaic image. These matrices can, e.g., be used to
transform features extracted from the separate images into the mosaic image by using the operators projective_trans_pixel, projective_trans_region, projective_trans_contour_xld,
or projective_trans_image.

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• the matching is carried out between all overlapping images,
• in addition to the projective transformation matrices also the coordinates of the corresponding
points must be accumulated, and
• the operator gen_projective_mosaic is replaced with the operators bundle_adjust_mosaic
and gen_bundle_adjusted_mosaic.
First, the matching is carried out between all overlapping image pairs, which can be defined as follows:
From := [1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5]
To := [6,7,2,6,7,8,3,7,8,9,4,8,9,10,5,9,10]

In addition to the accumulation of the projective transformation matrices, as described in section 10.4 on
page 256, also the coordinates of the corresponding points as well as the number of corresponding points
must be accumulated.
Rows1 := [Rows1,subset(RowFAll,Points1)]
Cols1 := [Cols1,subset(ColFAll,Points1)]
Rows2 := [Rows2,subset(RowTAll,Points2)]
Cols2 := [Cols2,subset(ColTAll,Points2)]
NumCorrespondences := [NumCorrespondences,|Points1|]

This data is needed by the operator bundle_adjust_mosaic, which determines the bundle adjusted
transformation matrices.
bundle_adjust_mosaic (10, StartImage, From, To, ProjMatrices, Rows1, Cols1, \
Rows2, Cols2, NumCorrespondences, Transformation, \
MosaicMatrices2D, Rows, Cols, Error)

The parameter Transformation defines the class of transformations that is used for the transformation
between the individual images. Possible values are ’projective’, ’affine’, ’similarity’, and ’rigid’. Thus, if
you know, e.g., that the camera looks perpendicular onto a planar object and that the camera movement
between the images is restricted to rotations and translations in the object plane, you can choose the
transformation class ’rigid’. If translations may also occur in the direction perpendicular to the object
plane, you must use ’similarity’ because this transformation class allows scale differences between the
images. If the camera looks tilted onto the object, the transformation class ’projective’ must be used,
which can be approximated by the transformation class ’affine’. Figure 10.13 shows cut-outs of the
resulting mosaic images. They depict the seam between image 5 and image 10. The mosaic images
have been created using the images where the lens distortions have been eliminated. The position of the
cut-out within the whole mosaic image is indicated by the rectangle in figure 10.7 on page 253.
Finally, with the transformation matrices MosaicMatrices2D, which are determined by
the operator bundle_adjust_mosaic, the mosaic can be generated with the operator
gen_bundle_adjusted_mosaic.
gen_bundle_adjusted_mosaic (Images, MosaicImage, MosaicMatrices2D, \
StackingOrder, TransformRegion, TransMat2D)

10.7 Spherical Mosaicking

(a)

(b)

(c)

(d)

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Figure 10.13: Seam between image 5 and image 10 for different classes of transformations: (a) projective,
(b) affine, (c) similarity, and (d) rigid.

10.7

Spherical Mosaicking

To create a spherical mosaic, you first perform the matching as described in the previous sections to determine the projective transformation matrices between the individual images. This information is the input
for the operator stationary_camera_self_calibration, which determines the internal camera parameters of the camera and the rotation matrices for each image. Based on this information, the operator
gen_spherical_mosaic then creates the mosaic image. Please have a look at the HDevelop example program Calibration\Self-Calibration\stationary_camera_self_calibration.hdev
for more information about how to use these operators.
As an alternative, you can map the images on the six sides of a cube using gen_cube_map_mosaic.
Cube maps are especially useful in computer graphics.

Mosaicking II

The methods described in the previous sections arranged the images on a plane. As the name suggests,
using spherical mosaicking you can arrange them on a sphere instead. Note that this method can only be
used if the camera is only rotated around its optical center or zoomed. If the camera movement includes
a translation or if the rotation is not performed exactly around the optical center, the resulting mosaic
image will not be accurate and can therefore not be used for high-accuracy applications.

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Rectification of Arbitrary Distortions

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Chapter 11

Rectification of Arbitrary
Distortions
For many applications like OCR or bar code reading, distorted images must be rectified prior to the
extraction of information. The distortions may be caused by the perspective projection and by the radial
lens distortions as well as by the decentering lens distortions, a non-flat object surface, or by any other
reason. In the first three cases, i.e., if the object surface is flat and the camera shows only radial or
decentering distortions, the rectification can be carried out very precisely as described in section 3.4.1
on page 91. For the remaining cases, a piecewise bilinear rectification can be carried out. In HALCON,
this kind of rectification is called grid rectification.

The main idea of the grid rectification is that the mapping for the rectification is determined from an
image of the object, where the object is covered by a known pattern.
First, this pattern, which is called rectification grid, must be created with the operator
create_rectification_grid.
create_rectification_grid (WidthOfGrid, NumSquares, 'rectification_grid.ps')

The resulting PostScript file must be printed. An example for such a rectification grid is shown in
figure 11.2a.
Now, the object must be wrapped with the rectification grid and an image of the wrapped object must be
taken (figure 11.2b).
From this image, the mapping that describes the transformation from the distorted image into the rectified
image can be derived. For this, first, the rectification grid must be extracted. Then, the rectification map
is derived from the distorted grid. This can be achieved by the following lines of code:

Rectification

The following example (hdevelop\Tools\Grid-Rectification\grid_rectification.hdev)
shows how the grid rectification can be used to rectify the image of a cylindrically shaped object (figure 11.1).

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Figure 11.1: Cylindrical object: a) Original image; b) rectified image.

Figure 11.2: a) Example of a rectification grid. b) Cylindrical object wrapped with the rectification grid.

find_rectification_grid (Image, GridRegion, MinContrast, Radius)
reduce_domain (Image, GridRegion, ImageReduced)
saddle_points_sub_pix (ImageReduced, 'facet', SigmaSaddlePoints, Threshold, \
Row, Col)
connect_grid_points (ImageReduced, ConnectingLines, Row, Col, \
SigmaConnectGridPoints, MaxDist)
gen_grid_rectification_map (ImageReduced, ConnectingLines, Map, Meshes, \
GridSpacing, 0, Row, Col, 'bilinear')

11.1 Basic Principle

C-265

Using the derived map, any image that shows the same distortions can be rectified such that the parts
that were covered by the rectification grid appear undistorted in the rectified image (figure 11.1b). This
mapping is performed by the operator map_image.
map_image (ImageReduced, Map, ImageMapped)

In the following section, the basic principle of the grid rectification is described. Then, some hints for
taking images of the rectification grid are given. In section 11.3 on page 268, the use of the involved
HALCON operators is described in more detail based on the above example application. Finally, it is
described briefly how to use self-defined grids for the generation of rectification maps.

11.1

Basic Principle

Figure 11.3: Rectification grid.

In the distorted image, the black and white fields do not appear as squares (figure 11.4a) because of the
non-planar object surface, the perspective distortions, and the lens distortions.
To determine the mapping for the rectification of the distorted image, the distorted rectification grid must
be extracted. For this, first, the corners of the black and white fields must be extracted with the operator
saddle_points_sub_pix (figure 11.4b). These corners must be connected along the borders of the
black and white fields with the operator connect_grid_points (figure 11.4c). Finally, the connecting
lines must be combined into meshes (figure 11.4d) with the operator gen_grid_rectification_map,
which also determines the mapping for the rectification of the distorted image.
If you want to use a self-defined grid, the grid points must be defined by yourself. Then, the operator
gen_arbitrary_distortion_map can be used to determine the mapping (see section 11.4 on page
271 for an example).

Rectification

The basic principle of the grid rectification is that a mapping from the distorted image into the rectified
image is determined from a distorted image of the rectification grid whose undistorted geometry is well
known: The black and white fields of the printed rectification grid are squares (figure 11.3).

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Figure 11.4: Distorted rectification grid: a) Image; b) extracted corners of the black and white fields; c)
lines that connect the corners; d) extracted rectification grid.

The mapping is determined such that the distorted rectification grid will be mapped into its original
undistorted geometry (figure 11.5). With this mapping, any image that shows the same distortions can
be rectified easily with the operator map_image. Note that within the meshes a bilinear interpolation
is carried out. Therefore, it is important to use a rectification grid with an appropriate grid size (see
section 11.2 for details).

11.2

Rules for Taking Images of the Rectification Grid

If you want to achieve accurate results, please follow the rules given in this section:
• The image must not be overexposed or underexposed: otherwise, the extraction of the corners of
the black and white fields of the rectification grid may fail.
• The contrast between the bright and the dark fields of the rectification grid should be as high as
possible.

11.2 Rules for Taking Images of the Rectification Grid

C-267

→
a)

Figure 11.5: Mapping of the distorted rectification grid (a) into the undistorted rectification grid (b).

• Ensure that the rectification grid is homogeneously illuminated.
• The images should contain as little noise as possible.
• The border length of the black and white fields should be at least 10 pixels.
In addition to these few rules for the taking of the images of the rectification grid, it is very important
to use a rectification grid with an appropriate grid size because the mapping is determined such that
within the meshes of the rectification grid a bilinear interpolation is applied. Because of this, non-linear
distortions within the meshes cannot be eliminated.

Rectification

The use of a rectification grid that is too coarse (figure 11.6a), i.e., whose grid size is too large, leads to
errors in the rectified image (figure 11.6b).

Figure 11.6: Cylindrical object covered with a very coarse rectification grid: a) Distorted image; b) rectified
image.

If it is necessary to fold the rectification grid, it should be folded along the borders of the black and white
fields. Otherwise, i.e., if the fold crosses these fields (figure 11.7a), the rectified image (figure 11.7b)
will contain distortions because of the bilinear interpolation within the meshes.

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Figure 11.7: Rectification grid folded across the borders of the black and white fields: a) Distorted image;
b) rectified image.

11.3

Machine Vision on Ruled Surfaces

In this section, the rectification of images of ruled surfaces is described in detail. Again,
the example of the cylindrically shaped object (hdevelop\Tools\Grid-Rectification\
grid_rectification.hdev) is used to explain the involved operators.
First, the operator create_rectification_grid is used to create a suitable rectification grid.
create_rectification_grid (WidthOfGrid, NumSquares, 'rectification_grid.ps')

The parameter WidthOfGrid defines the effectively usable size of the rectification grid in meters and
the parameter NumSquares sets the number of squares (black and white fields) per row. The rectification
grid is written to the PostScript file that is specified by the parameter GridFile.
To determine the mapping, an image of the rectification grid, wrapped around the object, must be taken
as described in section 11.2 on page 266. Figure 11.8a shows an image of a cylindrical object and
figure 11.8b shows the same object wrapped by the rectification grid.
Then, the rectification grid is searched in this image with the operator find_rectification_grid.
find_rectification_grid (Image, GridRegion, MinContrast, Radius)

The operator find_rectification_grid extracts image areas with a contrast of at least MinContrast
and fills up the holes in these areas. Note that in this case, contrast is defined as the gray value difference
of neighboring pixels in a slightly smoothed copy of the image (Gaussian smoothing with σ = 1.0).
Therefore, the value for the parameter MinContrast must be set significantly lower than the gray value
difference between the black and white fields of the rectification grid. Small areas of high contrast are
then eliminated by an opening with the radius Radius. The resulting region is used to restrict the search
space for the following steps with the operator reduce_domain (see figure 11.9a).

11.3 Machine Vision on Ruled Surfaces

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reduce_domain (Image, GridRegion, ImageReduced)

Figure 11.8: Cylindrical object: a) Without and b) with rectification grid.

The corners of the black and white fields appear as saddle points in the image. They can be extracted
with the operator saddle_points_sub_pix (see figure 11.9b).

The parameter Sigma controls the amount of Gaussian smoothing that is carried out before the actual
extraction of the saddle points. Which point is accepted as a saddle point is based on the value of the
parameter Threshold. If Threshold is set to higher values, fewer but more distinct saddle points are
returned than if Threshold is set to lower values. The filter method that is used for the extraction of the
saddle points can be selected by the parameter Filter. It can be set to ’facet’ or ’gauss’. The method
’facet’ is slightly faster. The method ’gauss’ is slightly more accurate but tends to be more sensitive to
noise.

Figure 11.9: Distorted rectification grid: a) Image reduced to the extracted area of the rectification grid; b)
extracted corners of the black and white fields; c) lines that connect the corners.

Rectification

saddle_points_sub_pix (ImageReduced, 'facet', SigmaSaddlePoints, Threshold, \
Row, Col)

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Rectification of Arbitrary Distortions

To generate a representation of the distorted rectification grid, the extracted saddle points must be connected along the borders of the black and white fields (figure 11.9c). This is done with the operator
connect_grid_points.
connect_grid_points (ImageReduced, ConnectingLines, Row, Col, \
SigmaConnectGridPoints, MaxDist)

Again, the parameter Sigma controls the amount of Gaussian smoothing that is carried out before the
extraction of the borders of the black and white fields. When a tuple of three values [sigma_min,
sigma_max, sigma_step] is passed instead of only one value, the operator connect_grid_points tests
every sigma within the given range from sigma_min to sigma_max with a step size of sigma_step and
chooses the sigma that causes the largest number of connecting lines. The same happens when a tuple
of only two values sigma_min and sigma_max is passed. However, in this case a fixed step size of 0.05
is used. The parameter MaxDist defines the maximum distance with which an edge may be linked to
the respectively closest saddle point. This helps to overcome the problem that edge detectors typically
return inaccurate results in the proximity of edge junctions. Figure 11.10 shows the connecting lines if
the parameter MaxDist has been selected inappropriately: In figure 11.10a, MaxDist has been selected
to small, whereas in figure 11.10b, it has been selected too large.

Figure 11.10: Connecting lines: Parameter MaxDist selected a) too small and b) too large.

Then, the rectification map is determined from the distorted grid with the operator
gen_grid_rectification_map.
gen_grid_rectification_map (ImageReduced, ConnectingLines, Map, Meshes, \
GridSpacing, 0, Row, Col, 'bilinear')

The parameter GridSpacing defines the size of the grid meshes in the rectified image. Each of the black
and white fields is projected onto a square of GridSpacing × GridSpacing pixels. The parameter
Rotation controls the orientation of the rectified image. The rectified image can be rotated by 0, 90,
180, or 270 degrees, or it is rotated such that the black circular mark is left of the white circular mark if
Rotation is set to ’auto’.
Using the derived rectification map, any image that shows the same distortions can be rectified very fast
with the operator map_image (see figure 11.11). Note that the objects must appears at exactly the same

11.4 Using Self-Defined Rectification Grids

C-271

position in the distorted images.
map_image (ImageReduced, Map, ImageMapped)

11.4

Using Self-Defined Rectification Grids

Up to now, we have used the predefined rectification grid together with the appropriate operators for its
segmentation. In this section, an alternative to this approach is presented. You can arbitrarily define
the rectification grid by yourself, but note that in this case you must also carry out the segmentation by
yourself.
This example shows how the grid rectification can be used to generate arbitrary distortion maps based on
self-defined grids.
The example application is a print inspection. It is assumed that some parts are missing and that smudges
are present. In addition, lines may be vertically shifted, e.g., due to an inaccurate paper transport, i.e.,
distortions in the vertical direction of the printed document may be present. These distortions should not
result in a rejection of the tested document. Therefore, it is not possible to simply compute the difference
image between a reference image and the image that must be checked.
Figure 11.12a shows the reference image and figure 11.12b the test image that must be checked.

Rectification

Figure 11.11: Rectified images: a) Rectification grid; b) object.

C-272

Rectification of Arbitrary Distortions

smudges

displaced lines
missing line

Figure 11.12: Images of one page of a document: a) Reference image; b) test image that must be
checked.

In a first step, the displacements between the lines in the reference document and the test document are
determined. With this, the rectification grid is defined. The resulting rectification map is applied to the
reference image to transform it into the geometry of the test image. Now, the difference image of the
mapped reference image and the test image can be computed.
The
HDevelop
example
program
solution_guide\3d_vision\
grid_rectification_arbitrary_distortion.hdev uses the component-based matching to
determine corresponding points in the reference image and the test image. First, the component model is
generated with the operator create_component_model. Then, the corresponding points are searched
in the test image with the operator find_component_model.
Based on the corresponding points of the reference and the test image (RowRef, ColRef, RowTest, and
ColTest), the coordinates of the grid points of the distorted grid are determined. In this example, the
row and column coordinates can be determined independently from each other because only the row
coordinates are distorted. Note that the upper left grid point of the undistorted grid is assumed to have
the coordinates (-0.5, -0.5). This means that the corresponding grid point of the distorted grid will be
mapped to the point (-0.5, -0.5). Because there are only vertical distortions in this example, the column
coordinates of the distorted grid are equidistant, starting at the value -0.5.

11.4 Using Self-Defined Rectification Grids

C-273

GridSpacing := 10
ColShift := mean(ColTest - ColRef)
RefGridColValues := []
for HelpCol := -0.5 to WidthTest + GridSpacing by GridSpacing
RefGridColValues := [RefGridColValues,HelpCol + ColShift]
endfor

The row coordinates of the distorted grid are determined by a linear interpolation between the above
determined pairs of corresponding row coordinates.
MinValue := 0
MaxValue := HeightTest + GridSpacing
sample_corresponding_values (RowTest, RowRef - 0.5, MinValue, MaxValue, \
GridSpacing, RefGridRowValues)

The interpolation is performed within the procedure which is part of the HDevelop example program
solution_guide\3d_vision\grid_rectification_arbitrary_distortion.hdev.
procedure sample_corresponding_values (Values, CorrespondingValues,
MinValue, MaxValue,
InterpolationInterval,
SampledCorrespondingValues):::

RefGridRow := []
RefGridCol := []
Ones := gen_tuple_const(|RefGridColValues|,1)
for r := 0 to |RefGridRowValues| - 1 by 1
RefGridRow := [RefGridRow,RefGridRowValues[r] * Ones]
RefGridCol := [RefGridCol,RefGridColValues]
endfor

The operator gen_arbitrary_distortion_map uses this distorted grid to derive the rectification map
that maps the reference image into the geometry of the test image1 .
gen_arbitrary_distortion_map (Map, GridSpacing, RefGridRow, RefGridCol, \
|RefGridColValues|, WidthRef, HeightRef, \
'bilinear')

With this rectification map, the reference image can be transformed into the geometry of the test image.
Note that the size of the mapped image depends on the number of grid cells and on the size of one grid
cell, which must be defined by the parameter GridSpacing. Possibly, the size of the mapped reference
image must be adapted to the size of the test image.
1 In this case, the reference image is mapped into the geometry of the test image to facilitate the marking of the differences in
the test image. Obviously, the rectification grid can also be defined such that the test image is mapped into the geometry of the
reference image.

Rectification

Now, the distorted grid is generated row by row.

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Rectification of Arbitrary Distortions

map_image (ImageRef, Map, ImageMapped)
crop_part (ImageMapped, ImagePart, 0, 0, WidthTest, HeightTest)

Finally, the test image can be subtracted from the mapped reference image.
sub_image (ImagePart, ImageTest, ImageSub, 1, 128)

Figure 11.13 shows the resulting difference image. In this case, missing parts appear dark while the
smudges appear bright.

Figure 11.13: Difference image: a) The entire image overlaid with a rectangle that indicates the position
of the cut-out. b) A cut-out.

The differences between the test image and the reference image can now be extracted easily from the
difference image with the operator threshold. If the difference image is not needed, e.g., for visualization purposes, the differences can be derived directly from the test image and the reference image with
the operator dyn_threshold.
Figure 11.14 shows the differences in a cut-out of the reference image (figure 11.14a) and of the test
image (figure 11.14b). The calibration marks near the left border of figure 11.14b indicate the vertical
position of the components that were used for the determination of the corresponding points. Vertical
shifts of the components with respect to the reference image are indicated by a vertical line of the respective length that is attached to the respective calibration mark. All other differences that could be detected
between the test image and the reference image are encircled.

Figure 11.14: Cut-out of the reference and the checked test image with the differences marked in the test
image: a) Reference image; b) checked test image.

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Rectification

11.4 Using Self-Defined Rectification Grids

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Rectification of Arbitrary Distortions

HDevelop Procedures Used in this Solution Guide

C-277

Appendix A

HDevelop Procedures Used in this
Solution Guide
gen_hom_mat3d_from_three_points

procedure gen_hom_mat3d_from_three_points (Origin, PointOnXAxis,
PointInXYPlane, HomMat3d):::
XAxis := [PointOnXAxis[0] - Origin[0],PointOnXAxis[1] - Origin[1], \
PointOnXAxis[2] - Origin[2]]
XAxisNorm := XAxis / sqrt(sum(XAxis * XAxis))
VectorInXYPlane := [PointInXYPlane[0] - Origin[0], \
PointInXYPlane[1] - Origin[1], \
PointInXYPlane[2] - Origin[2]]
cross_product (XAxisNorm, VectorInXYPlane, ZAxis)
ZAxisNorm := ZAxis / sqrt(sum(ZAxis * ZAxis))
cross_product (ZAxisNorm, XAxisNorm, YAxisNorm)
HomMat3d_WCS_to_RectCCS := [XAxisNorm[0],YAxisNorm[0],ZAxisNorm[0], \
Origin[0],XAxisNorm[1],YAxisNorm[1], \
ZAxisNorm[1],Origin[1],XAxisNorm[2], \
YAxisNorm[2],ZAxisNorm[2],Origin[2]]
hom_mat3d_invert (HomMat3d_WCS_to_RectCCS, HomMat3d)
return ()

This procedure uses the procedure
procedure cross_product (V1, V2,
CrossProduct := [V1[1] * V2[2] V1[2] * V2[0] V1[0] * V2[1] return ()

CrossProduct)
V1[2] * V2[1], \
V1[0] * V2[2], \
V1[1] * V2[0]]

Procedures

A.1

C-278

HDevelop Procedures Used in this Solution Guide

A.2

parameters_image_to_world_plane_centered

procedure parameters_image_to_world_plane_centered (CamParam, Pose,
CenterRow, CenterCol,
WidthMappedImage,
HeightMappedImage,
ScaleForCenteredImage,
PoseForCenteredImage):::
* Determine the scale for the mapping
* (here, the scale is determined such that in the
*
surroundings of the given point the image scale of the
*
mapped image is similar to the image scale of the original image)
Dist_ICS := 1
image_points_to_world_plane (CamParam, Pose, CenterRow, CenterCol, 1, \
CenterX, CenterY)
image_points_to_world_plane (CamParam, Pose, CenterRow + Dist_ICS, \
CenterCol, 1, BelowCenterX, BelowCenterY)
image_points_to_world_plane (CamParam, Pose, CenterRow, \
CenterCol + Dist_ICS, 1, RightOfCenterX, \
RightOfCenterY)
distance_pp (CenterY, CenterX, BelowCenterY, BelowCenterX, \
Dist_WCS_Vertical)
distance_pp (CenterY, CenterX, RightOfCenterY, RightOfCenterX, \
Dist_WCS_Horizontal)
ScaleVertical := Dist_WCS_Vertical / Dist_ICS
ScaleHorizontal := Dist_WCS_Horizontal / Dist_ICS
ScaleForCenteredImage := (ScaleVertical + ScaleHorizontal) / 2.0
* Determine the parameters for set_origin_pose such
* that the point given via get_mbutton will be in the center of the
* mapped image
DX := CenterX - ScaleForCenteredImage * WidthMappedImage / 2.0
DY := CenterY - ScaleForCenteredImage * HeightMappedImage / 2.0
DZ := 0
set_origin_pose (Pose, DX, DY, DZ, PoseForCenteredImage)
return ()

A.3 parameters_image_to_world_plane_entire

A.3

C-279

parameters_image_to_world_plane_entire

Procedures

procedure parameters_image_to_world_plane_entire (Image, CamParam, Pose,
WidthMappedImage,
HeightMappedImage,
ScaleForEntireImage,
PoseForEntireImage):::
* Transform the image border into the WCS (scale = 1)
full_domain (Image, ImageFull)
get_domain (ImageFull, Domain)
gen_contour_region_xld (Domain, ImageBorder, 'border')
contour_to_world_plane_xld (ImageBorder, ImageBorderWCS, CamParam, Pose, 1)
smallest_rectangle1_xld (ImageBorderWCS, MinY, MinX, MaxY, MaxX)
* Determine the scale of the mapping
ExtentX := MaxX - MinX
ExtentY := MaxY - MinY
ScaleX := ExtentX / WidthMappedImage
ScaleY := ExtentY / HeightMappedImage
ScaleForEntireImage := max([ScaleX,ScaleY])
* Shift the pose by the minimum X and Y coordinates
set_origin_pose (Pose, MinX, MinY, 0, PoseForEntireImage)
return ()

C-280

HDevelop Procedures Used in this Solution Guide

A.4

tilt_correction

procedure tilt_correction (DistanceImage, RegionDefiningReferencePlane,
DistanceImageCorrected):::
* Reduce the given region, which defines the reference plane
* to the domain of the distance image
get_domain (DistanceImage, Domain)
intersection (RegionDefiningReferencePlane, Domain, \
RegionDefiningReferencePlane)
* Determine the parameters of the reference plane
moments_gray_plane (RegionDefiningReferencePlane, DistanceImage, MRow, MCol, \
Alpha, Beta, Mean)
* Generate a distance image of the reference plane
get_image_pointer1 (DistanceImage, Pointer, Type, Width, Height)
area_center (RegionDefiningReferencePlane, Area, Row, Column)
gen_image_surface_first_order (ReferencePlaneDistance, Type, Alpha, Beta, \
Mean, Row, Column, Width, Height)
* Subtract the distance image of the reference plane
* from the distance image of the object
sub_image (DistanceImage, ReferencePlaneDistance, DistanceImageWithoutTilt, \
1, 0)
* Determine the scale factor for the reduction of the distance values
CosGamma := 1.0 / sqrt(Alpha * Alpha + Beta * Beta + 1)
* Reduce the distance values
scale_image (DistanceImageWithoutTilt, DistanceImageCorrected, CosGamma, 0)
return ()

A.5

calc_calplate_pose_movingcam

procedure calc_calplate_pose_movingcam (CalibObjInBasePose, ToolInCamPose,
ToolInBasePose, CalibObjInCamPose):::
* CalibObjInCamPose = cam_H_calplate
*
= cam_H_tool * tool_H_base * base_H_calplate
*
= ToolInCamPose * BaseInToolPose * CalibrationPose
pose_invert (ToolInBasePose, BaseInToolPose)
pose_compose (ToolInCamPose, BaseInToolPose, BaseInCamPose)
pose_compose (BaseInCamPose, CalibObjInBasePose, CalibObjInCamPose)
return ()

A.6 calc_calplate_pose_stationarycam

A.6

C-281

calc_calplate_pose_stationarycam

procedure calc_calplate_pose_stationarycam (CalObjInToolPose, BaseInCamPose,
ToolInBasePose,
CalObjInCamPose):::
* CalObjInCamPose = cam_H_calplate = cam_H_base * base_H_tool * \
* tool_H_calplate
*
= \
* BaseInCamPose*ToolInBasePose*CalObjInToolPose
pose_compose (BaseInCamPose, ToolInBasePose, ToolInCamPose)
pose_compose (ToolInCamPose, CalObjInToolPose, CalObjInCamPose)
return ()

define_reference_coord_system

procedure define_reference_coord_system (ImageName, CamParam, CalplateFile,
WindowHandle, PoseCamRef):::
read_image (RefImage, ImageName)
dev_display (RefImage)
caltab_points (CalplateFile, X, Y, Z)
* parameter settings for find_caltab and find_marks_and_pose
SizeGauss := 3
MarkThresh := 100
MinDiamMarks := 5
StartThresh := 128
DeltaThresh := 10
MinThresh := 18
Alpha := 0.9
MinContLength := 15
MaxDiamMarks := 100
find_caltab (RefImage, Caltab, CalplateFile, SizeGauss, MarkThresh, \
MinDiamMarks)
find_marks_and_pose (RefImage, Caltab, CalplateFile, CamParam, StartThresh, \
DeltaThresh, MinThresh, Alpha, MinContLength, \
MaxDiamMarks, RCoord, CCoord, PoseCamRef)
disp_3d_coord_system (WindowHandle, CamParam, PoseCamRef, 0.01)
return ()

Procedures

A.7

C-282

HDevelop Procedures Used in this Solution Guide

Index
2D projective matrix from RANSAC point
matching, 256
3D affine transformation of point, 89
3D alignment, 228
3D coordinates, 15
3D coordinates from binocular stereo disparity,
164
3D coordinates from multi-view stereo images,
172
3D coordinates with sheet of light, 192
3D distance from binocular stereo disparity, 164
3D homogeneous transformation matrix, 22
3D inspection, 10
3D measurement plane, 82
3D object model, 41
3D pose (position and orientation), 25
3D pose estimation, 105
3D pose of circle, 138
3D pose of rectangle, 138
3D reconstruction
guide, 11
3D reconstruction with binocular stereo, 154
3D reconstruction with multi-view stereo, 170
3D reconstruction with sheet of light (laser triangulation), 190
3D rotation, 18
3D transformation, 15
3D translation, 16
3D vision, 9
3D vision with single camera, 65
first example, 66
access 3D matching model (surface-based),
125, 131
accuracy of 3D measuring with single camera,
99
acquire image for grid rectification, 266
acquire images for camera calibration, 78

C-283

Index

acquire images for depth from focus, 208
acquire images for stereo camera calibration,
147
acquire images for uncalibrated mosaicking,
249
affine 3D transformation of points, 24
area scan camera model, 28
binocular stereo (uncalibrated), 166
bundle adjust mosaic images, 259
calibrate aberration for depth from focus, 211
calibrate camera before hand-eye calibration,
220
calibrate camera before or during hand-eye calibration, 217
calibrate camera for mosaicking, 236
calibrate camera parameters, 81
calibrate external camera parameters, 82
calibrate hand-eye system parameters, 225
calibrate internal camera parameters, 81
calibrate line scan camera parameters, 86
calibrate multiple cameras, 144
calibrate sheet-of-light setup using a special 3D
calibration object, 186
calibrate sheet-of-light system parameters, 182
calibrated camera setup model, 168
calibrated external camera parameters, 82
calibrated internal camera parameters, 81
camera calibration, 68
camera calibration results, 81
camera calibration troubleshooting, 85
camera coordinate system (3D), 28
camera model (3D), 27
camera scale factor, 34
change lens distortion, 90
change radial distortion of points, 90
change radial distortion of XLD contours, 90,
98

C-284

Index

check success of camera calibration, 81
connect points of rectification grid, 265
convert 3D pose into 3D homogeneous matrix,
26, 89, 101
convert 3D pose type, 26
correlation-based stereo, 149
create 3D homogeneous identity matrix, 24
create 3D matching model (deformable surfacebased), 129
create 3D matching model (shape-based), 112
create 3D matching model (surface-based), 124
create 3D object model from 3D coordinates,
124
create 3D pose, 25
create calibration plate, 77
create camera calibration data model, 69
create camera setup model, 170
create data model for hand-eye calibration, 221
create mosaic image, 258, 259
create rectification grid, 268
create sheet-of-light model, 191
create spherical mosaic, 261
create XLD contour of region, 89
delete observations from camera calibration
data model, 85
depth from focus, 199
example, 209
destroy 3D matching model (shape-based), 116
destroy 3D object model, 114
destroy camera calibration data model, 85
destroy sheet-of-light model, 193
determine z translation for SCARA robots, 227
disparity image with correlation-based stereo,
156
disparity image with multi-scanline stereo, 160
disparity image with multigrid stereo, 159
division model of lens distortion, 31
external camera parameters, 28
extract points for uncalibrated mosaicking, 255
find 3D matching model (deformable surfacebased), 131
find 3D matching model (shape-based), 116
find 3D matching model (surface-based), 126
find calibration plate, 80
find calibration plate marks and 3D pose, 80

focal length, 30
fundamental matrix from RANSAC point
matching, 166
get 3D object pose, 10
get measurement range for depth from focus,
203
grid rectification, 263
background information, 265
hand-eye calibration, 218
hand-eye calibration with moving camera or
stationary camera, 217
hand_eye calibration with articulated robot or
with SCARA robot, 215
hand_eye calibration with camera or 3D sensor,
216
image center point, 34
image plane, 28
image plane coordinate system, 28
internal camera parameters, 28
lens distortion models
guide, 70
lens distortion of camera parameters, 90, 98
lens distortion of image, 98
line scan camera model, 37
mapping for grid rectification, 265
mapping from image coordinates to 3D coordinates, 91
mapping to change lens distortion, 98
mapping to rectify arbitrary distortion, 265
match points for uncalibrated mosaicking, 256
measure volume with depth from focus, 210
mosaicking (image stitching) calibrated, 235
mosaicking (image stitching) uncalibrated, 247
multi-scanline stereo, 149
multigrid stereo, 149
multiply 3D homogeneous matrix, 26
obtain calibration plate, 74
parallel projection, 30
perspective projection, 30
pinhole camera, 27
polynomial model of lens distortion, 31
pose estimation for 3D alignment, 229

Index

read 3D model (shape-based), 115
read 3D object model, 112, 124, 128
read 3D pose, 26
reconstruct 3D distance image with correlationbased stereo, 161
reconstruct 3D distance image with multiscanline stereo, 164
reconstruct 3D distance image with multigrid
stereo, 163
reconstruct 3D distance with focus images, 209
reconstruct 3D information with sheet of light,
191
reconstruct 3D information with stereo (binocular), 154
reconstruct 3D information with stereo (multiview), 170
reconstruct 3D point from lines of sight, 165
reconstruct uncalibrated 3D information via
sheet of light, 194
rectification, 90
rectify image of ruled surface, 268
rectify image with user-specific rectification
grid, 271
rectify image(s), 91
rectify image(s) for stereo, 151
rectify images for mosaicking, 243
relative camera pose from RANSAC point
matching, 167
remove artifacts of sheet-of-light results, 195

resolution of stereo vision, 144
rigid 3D transformation, 22
robot vision, 13, 215
Scheimpflug principle, 35, 144, 182
select illumination for depth from focus, 204
self-calibrate projective camera parameters, 261
set 3D coordinate system of camera setup
model, 168
set calibration object for camera calibration, 74
set camera calibration parameters, 81
set image pairs of stereo model, 171
set initial camera parameters for camera calibration, 69
set mosaicking image pairs, 250
set observed points for camera calibration, 78
set origin of 3D pose, 26
set poses of calibration object for hand-eye calibration, 223
set poses of the robot tool for hand-eye calibration, 223
set sheet-of-light model parameter, 191
set up calibrated mosaicking, 236
set up camera for depth from focus, 202
set up depth from focus application, 202
set up sheet-of-light system, 179
set up stereo camera system, 143
sheet of light (laser triangulation), 177
sheet-of-light result, 192
solve depth from focus problems, 210
special applications for depth from focus, 212
speed up 3D matching (shape-based), 116
speed up depth from focus, 201
standard lens for depth from focus, 213
stereo (binocular), 148
stereo (multi-view), 167
stereo vision
background information, 141
overview, 139
suitable objects for depth from focus, 205
supported configurations for hand-eye calibration, 215
telecentric camera, 27
tile images, 245
tilt lenses, 32, 35, 144, 182
transform 3D coordinates into pixel coordinates
(projection), 89

Index

pose estimation from 3D matching (deformable
surface-based), 127
pose estimation from 3D matching (shapebased), 111
pose estimation from 3D matching (surfacebased), 123
pose estimation from matching (descriptorbased), 137
pose estimation from matching (perspective deformable), 136
pose estimation from point correspondences,
106
pose estimation from primitives fitting, 132
prepare 3D object model, 112
prepare the calibration input data, 221
problem handling for 3D matching (shapebased), 119

C-285

C-286

Index

transform 3D point into image coordinates, 30
transform 3D point into pixel coordinates (projection), 89
transform 3D shape model into pixel coordinates, 116
transform image coordinates into 3D coordinates, 86
transform image into 3D coordinates, 91
transform image plane coordinates into pixel
coordinates, 34
transform pixel coordinates into 3D coordinates, 87, 93, 101
transform region into 3D coordinates, 89
transform XLD contour into 3D coordinates, 89
transformation into/from world coordinates, 27
translate 3D homogeneous matrix, 24
translate 3D homogeneous matrix around local
axes, 24
translate 3D pose, 93
use 3D camera for sheet-of-light measuring,
196
use hand-eye system parameters, 227
user-specific calibration object, 77
world coordinate system (3D), 27, 28
write 3D pose, 26

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Author                          : MVTec Software GmbH
Title                           : 3D Vision
Subject                         : HALCON 12.0.2
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3D Vision Halcon 12.0 Solution Guide Iii C 2019 04 29 09 58 30

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