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Wavelet Toolbox™ User’s Guide R2012b Michel Misiti Yves Misiti Georges Oppenheim Jean-Michel Poggi How to Contact MathWorks Web Newsgroup www.mathworks.com/contact_TS.html Technical Support www.mathworks.com comp.soft-sys.matlab suggest@mathworks.com bugs@mathworks.com doc@mathworks.com service@mathworks.com info@mathworks.com Product enhancement suggestions Bug reports Documentation error reports Order status, license renewals, passcodes Sales, pricing, and general information 508-647-7000 (Phone) 508-647-7001 (Fax) The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 For contact information about worldwide offices, see the MathWorks Web site. Wavelet Toolbox™ User’s Guide © COPYRIGHT 1997–2012 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks, Inc. FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentation by, for, or through the federal government of the United States. By accepting delivery of the Program or Documentation, the government hereby agrees that this software or documentation qualifies as commercial computer software or commercial computer software documentation as such terms are used or defined in FAR 12.212, DFARS Part 227.72, and DFARS 252.227-7014. Accordingly, the terms and conditions of this Agreement and only those rights specified in this Agreement, shall pertain to and govern the use, modification, reproduction, release, performance, display, and disclosure of the Program and Documentation by the federal government (or other entity acquiring for or through the federal government) and shall supersede any conflicting contractual terms or conditions. If this License fails to meet the government’s needs or is inconsistent in any respect with federal procurement law, the government agrees to return the Program and Documentation, unused, to The MathWorks, Inc. Trademarks MATLAB and Simulink are registered trademarks of The MathWorks, Inc. See www.mathworks.com/trademarks for a list of additional trademarks. Other product or brand names may be trademarks or registered trademarks of their respective holders. Patents MathWorks products are protected by one or more U.S. patents. Please see www.mathworks.com/patents for more information. Revision History March 1997 September 2000 June 2001 July 2002 June 2004 July 2004 October 2004 March 2005 June 2005 September 2005 March 2006 September 2006 March 2007 September 2007 October 2007 March 2008 October 2008 March 2009 September 2009 March 2010 September 2010 April 2011 September 2011 March 2012 September 2012 First printing Second printing Online only Online only Online only Third printing Online only Online only Fourth printing Online only Online only Online only Online only Online only Fifth printing Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only New for Version 1.0 Revised for Version 2.0 (Release 12) Revised for Version 2.1 (Release 12.1) Revised for Version 2.2 (Release 13) Revised for Version 3.0 (Release 14) Revised for Version 3.0 Revised for Version 3.0.1 (Release 14SP1) Revised for Version 3.0.2 (Release 14SP2) Minor revision for Version 3.0.2 Minor revision for Version 3.0.3 (Release R14SP3) Minor revision for Version 3.0.4 (Release 2006a) Revised for Version 3.1 (Release 2006b) Revised for Version 4.0 (Release 2007a) Revised for Version 4.1 (Release 2007b) Revised for Version 4.1 Revised for Version 4.2 (Release 2008a) Revised for Version 4.3 (Release 2008b) Revised for Version 4.4 (Release 2009a) Minor revision for Version 4.4.1 (Release 2009b) Revised for Version 4.5 (Release 2010a) Revised for Version 4.6 (Release 2010b) Revised for Version 4.7 (Release 2011a) Revised for Version 4.8 (Release 2011b) Revised for Version 4.9 (Release 2012a) Revised for Version 4.10 (Release 2012b) Contents Acknowledgments Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters 1 Wavelet Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daubechies Wavelets: dbN . . . . . . . . . . . . . . . . . . . . . . . . . . Symlet Wavelets: symN . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coiflet Wavelets: coifN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biorthogonal Wavelet Pairs: biorNr.Nd . . . . . . . . . . . . . . . Meyer Wavelet: meyr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gaussian Derivatives Family: gaus . . . . . . . . . . . . . . . . . . . Mexican Hat Wavelet: mexh . . . . . . . . . . . . . . . . . . . . . . . . Morlet Wavelet: morl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional Real Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wavelet Families and Associated Properties — I . . . . . . . . Wavelet Families and Associated Properties — II . . . . . . . 1-2 1-7 1-8 1-9 1-10 1-12 1-14 1-15 1-16 1-17 1-18 1-22 1-23 Adding Your Own Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . Preparing to Add a New Wavelet Family . . . . . . . . . . . . . . Adding a New Wavelet Family . . . . . . . . . . . . . . . . . . . . . . . After Adding a New Wavelet Family . . . . . . . . . . . . . . . . . . 1-26 1-26 1-32 1-40 Lifting Method for Constructing Wavelets . . . . . . . . . . . Lifting Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polyphase Representation . . . . . . . . . . . . . . . . . . . . . . . . . . Split, Predict, and Update . . . . . . . . . . . . . . . . . . . . . . . . . . Haar Wavelet Via Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . Bior2.2 Wavelet Via Lifting . . . . . . . . . . . . . . . . . . . . . . . . . Lifting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Primal Lifting from Haar . . . . . . . . . . . . . . . . . . . . . . . . . . . Integer-to-Integer Wavelet Transform . . . . . . . . . . . . . . . . . 1-41 1-42 1-44 1-46 1-46 1-48 1-49 1-51 1-52 v Continuous Wavelet Analysis 2 1-D Continuous Wavelet Analysis . . . . . . . . . . . . . . . . . . . Command Line Continuous Wavelet Analysis . . . . . . . . . . Continuous Analysis Using the Graphical Interface . . . . . Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Complex Continuous Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Continuous Analysis Using the Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Continuous Analysis Using the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 2-4 2-8 2-18 2-21 2-22 2-24 2-29 DFT-Based Continuous Wavelet Analysis — Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CWT of Sum of Disjoint Sinusoids . . . . . . . . . . . . . . . . . . . . Approximate Scale-Frequency Conversions . . . . . . . . . . . . Signal Reconstruction from CWT Coefficients . . . . . . . . . . Signal Approximation with Modified CWT Coefficients . . . 2-30 2-30 2-33 2-37 2-38 Interactive DFT-Based Continuous Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manual Selection of CWT Coefficients . . . . . . . . . . . . . . . . . 2-41 2-46 Discrete Wavelet Analysis 3 1-D Decimated Wavelet Transforms . . . . . . . . . . . . . . . . . One-Dimensional Analysis Using the Command Line . . . . One-Dimensional Analysis Using the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Contents 3-2 3-4 3-13 3-28 Fast Wavelet Transform (FWT) Algorithm . . . . . . . . . . . . Filters Used to Calculate the DWT and IDWT . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Why Does Such an Algorithm Exist? . . . . . . . . . . . . . . . . . . One-Dimensional Wavelet Capabilities . . . . . . . . . . . . . . . . Two-Dimensional Wavelet Capabilities . . . . . . . . . . . . . . . . 3-37 3-37 3-40 3-45 3-49 3-50 Border Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal Extensions: Zero-Padding, Symmetrization, and Smooth Padding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-52 Discrete Stationary Wavelet Transform (SWT) . . . . . . . ε -Decimated DWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to Calculate the ε -Decimated DWT: SWT . . . . . . . . . Inverse Discrete Stationary Wavelet Transform (ISWT) . . More About SWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-62 3-62 3-63 3-67 3-68 One-Dimensional Discrete Stationary Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Analysis Using the Command Line . . . . Interactive 1-D Stationary Wavelet Transform Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Importing and Exporting from the GUI . . . . . . . . . . . . . . . . 3-52 3-69 3-70 3-80 3-84 One-Dimensional Multisignal Analysis . . . . . . . . . . . . . . . 3-86 One-Dimensional Multisignal Analysis — Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-87 Interactive One-Dimensional Multisignal Analysis . . . . . . 3-95 Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-129 Two-Dimensional Discrete Wavelet Analysis . . . . . . . . . Two-Dimensional Analysis — Command Line . . . . . . . . . . Interactive Two-Dimensional Wavelet Analysis . . . . . . . . . Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-137 3-138 3-147 3-156 Two-Dimensional Discrete Stationary Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-166 Two-Dimensional Analysis Using the Command Line . . . . 3-166 vii Interactive 2-D Stationary Wavelet Transform Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-175 Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-179 Three-Dimensional Discrete Wavelet Analysis . . . . . . . . Performing Three-Dimensional Analysis Using the Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performing Three-Dimensional Analysis Using the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-181 3-181 3-182 3-189 Wavelet Packets 4 viii Contents About Wavelet Packet Analysis . . . . . . . . . . . . . . . . . . . . . 4-2 One-Dimensional Wavelet Packet Analysis . . . . . . . . . . . Compressing a Signal Using Wavelet Packets . . . . . . . . . . De-Noising a Signal Using Wavelet Packets . . . . . . . . . . . . 4-7 4-11 4-14 Two-Dimensional Wavelet Packet Analysis . . . . . . . . . . . Compressing an Image Using Wavelet Packets . . . . . . . . . 4-15 4-18 Importing and Exporting from Graphical Tools . . . . . . Saving Information to Disk . . . . . . . . . . . . . . . . . . . . . . . . . Loading Information into the Graphical Tools . . . . . . . . . . 4-22 4-22 4-26 Wavelet Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Wavelets to Wavelet Packets . . . . . . . . . . . . . . . . . . . Wavelet Packets in Action: An Introduction . . . . . . . . . . . . Building Wavelet Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . Wavelet Packet Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Organizing the Wavelet Packets . . . . . . . . . . . . . . . . . . . . . Choosing the Optimal Decomposition . . . . . . . . . . . . . . . . . Some Interesting Subtrees . . . . . . . . . . . . . . . . . . . . . . . . . . Wavelet Packets 2-D Decomposition Structure . . . . . . . . . . 4-30 4-30 4-31 4-35 4-38 4-40 4-41 4-46 4-50 Wavelet Packets for Compression and De-Noising . . . . . . . 4-51 Introduction to Object-Oriented Features ........... 4-52 Objects in the Wavelet Toolbox Software ............ 4-53 Examples Using Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . plot and wpviewcf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . drawtree and readtree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change Terminal Node Coefficients . . . . . . . . . . . . . . . . . . Thresholding Wavelet Packets . . . . . . . . . . . . . . . . . . . . . . . 4-55 4-55 4-59 4-61 4-63 Description of Objects in the Wavelet Toolbox Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . WTBO Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NTREE Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DTREE Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . WPTREE Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-67 4-67 4-68 4-69 4-71 Advanced Use of Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . Building a Wavelet Tree Object (WTREE) . . . . . . . . . . . . . Building a Right Wavelet Tree Object (RWVTREE) . . . . . . Building a Wavelet Tree Object (WVTREE) . . . . . . . . . . . . Building a Wavelet Tree Object (EDWTTREE) . . . . . . . . . . 4-74 4-74 4-75 4-77 4-79 Denoising, Nonparametric Function Estimation, and Compression 5 Denoising and Nonparametric Function Estimation . . Threshold Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . Soft or Hard Thresholding . . . . . . . . . . . . . . . . . . . . . . . . . . Dealing with Unscaled Noise and Nonwhite Noise . . . . . . . Denoising in Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extension to Image De-Noising . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Wavelet Variance Adaptive Thresholding ............................................... 5-2 5-4 5-7 5-8 5-10 5-12 5-14 ix One-Dimensional Adaptive Thresholding of Wavelet Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Interactive Local Thresholding . . . . . . . Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Contents 5-17 5-17 5-25 Multivariate Wavelet Denoising . . . . . . . . . . . . . . . . . . . . . Multivariate Wavelet Denoising — Command Line . . . . . . Interactive Multivariate Wavelet Denoising . . . . . . . . . . . Importing and Exporting from the GUI . . . . . . . . . . . . . . . . 5-27 5-27 5-34 5-38 Multiscale Principal Components Analysis . . . . . . . . . . . Multiscale Principal Components Analysis — Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interactive Multiscale Principal Components Analysis . . . Importing and Exporting from the GUI . . . . . . . . . . . . . . . . 5-41 Data Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compression Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-51 5-53 True Compression for Images . . . . . . . . . . . . . . . . . . . . . . . Effects of Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . True Compression Methods . . . . . . . . . . . . . . . . . . . . . . . . . Quantitative and Perceptual Quality Measures . . . . . . . . . More Information on True Compression . . . . . . . . . . . . . . . 5-55 5-55 5-58 5-59 5-60 Two-Dimensional True Compression . . . . . . . . . . . . . . . . Two-Dimensional True Compression — Command Line . . Interactive Two-Dimensional True Compression . . . . . . . . Importing and Exporting from the GUI . . . . . . . . . . . . . . . . 5-61 5-61 5-69 5-79 One-Dimensional Wavelet Regression Estimation . . . . Regression for Equally-Spaced Observations . . . . . . . . . . . Regression for Randomly-Spaced Observations . . . . . . . . . Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-80 5-80 5-84 5-41 5-45 5-49 5-85 Matching Pursuit 6 Sparse Representation in Redundant Dictionaries . . . Redundant Dictionaries and Sparsity . . . . . . . . . . . . . . . . . Nonlinear Approximation in Dictionaries . . . . . . . . . . . . . . 6-2 6-2 6-3 Matching Pursuit Algorithms . . . . . . . . . . . . . . . . . . . . . . . Basic Matching Pursuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonal Matching Pursuit . . . . . . . . . . . . . . . . . . . . . . . Weak Orthogonal Matching Pursuit . . . . . . . . . . . . . . . . . . 6-4 6-4 6-7 6-8 Matching Pursuit — Command Line . . . . . . . . . . . . . . . . . Creating Dictionaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matching Pursuit With Dictionaries . . . . . . . . . . . . . . . . . . Matching Pursuit — Electricity Consumption Data . . . . . . 6-9 6-9 6-11 6-12 Matching Pursuit — Interactive Analysis . . . . . . . . . . . . Matching Pursuit 1-D Interactive Tool . . . . . . . . . . . . . . . . Interactive Matching Pursuit of Electricity Consumption Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-22 6-22 6-38 Generating MATLAB Code from Wavelet Toolbox GUI 7 Generating MATLAB Code for 1-D Decimated Wavelet Denoising and Compression . . . . . . . . . . . . . . . . . . . . . . Wavelet 1-D Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generating MATLAB Code for 2-D Decimated Wavelet Denoising and Compression . . . . . . . . . . . . . . . . . . . . . . 2-D Decimated Discrete Wavelet Transform Denoising . . . 2-D Decimated Discrete Wavelet Transform Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2 7-2 7-13 7-13 7-17 xi Generating MATLAB Code for 1-D Stationary Wavelet Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-D Stationary Wavelet Transform Denoising . . . . . . . . . . 7-20 7-20 Generating MATLAB Code for 2-D Stationary Wavelet Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-D Stationary Wavelet Transform Denoising . . . . . . . . . . 7-27 7-27 Generating MATLAB Code for 1-D Wavelet Packet Denoising and Compression . . . . . . . . . . . . . . . . . . . . . . 1-D Wavelet Packet Denoising . . . . . . . . . . . . . . . . . . . . . . . 7-31 7-31 Generating MATLAB Code for 2-D Wavelet Packet Denoising and Compression . . . . . . . . . . . . . . . . . . . . . . 2-D Wavelet Packet Compression . . . . . . . . . . . . . . . . . . . . 7-35 7-35 GUI Reference A xii Contents General Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Color Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connection of Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using the Mouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Controlling the Colormap . . . . . . . . . . . . . . . . . . . . . . . . . . . Using Menus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using the View Axes Button . . . . . . . . . . . . . . . . . . . . . . . . . A-2 A-2 A-3 A-5 A-7 A-9 A-10 Continuous Wavelet Tool Features . . . . . . . . . . . . . . . . . . A-12 Wavelet 2-D Tool Features . . . . . . . . . . . . . . . . . . . . . . . . . . A-13 Wavelet Packet Tool Features (1-D and 2-D) . . . . . . . . . . Node Action Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . A-14 A-14 Wavelet Display Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-16 Wavelet Packet Display Tool . . . . . . . . . . . . . . . . . . . . . . . . A-17 Index xiii xiv Contents _ Acknowledgments The authors wish to express their gratitude to all the colleagues who directly or indirectly contributed to the making of the Wavelet Toolbox™ software. Specifically • To Pierre-Gilles Lemarié-Rieusset (Evry) and Yves Meyer (ENS Cachan) for their help with wavelet questions • To Lucien Birgé (Paris 6), Pascal Massart (Paris 11), and Marc Lavielle (Paris 5) for their help with statistical questions • To David Donoho (Stanford) and to Anestis Antoniadis (Grenoble), who give generously so many valuable ideas Other colleagues and friends who have helped us enormously are Patrice Abry (ENS Lyon), Samir Akkouche (Ecole Centrale de Lyon), Mark Asch (Paris 11), Patrice Assouad (Paris 11), Roger Astier (Paris 11), Jean Coursol (Paris 11), Didier Dacunha-Castelle (Paris 11), Claude Deniau (Marseille), Patrick Flandrin (Ecole Normale de Lyon), Eric Galin (Ecole Centrale de Lyon), Christine Graffigne (Paris 5), Anatoli Juditsky (Grenoble), Gérard Kerkyacharian (Paris 10), Gérard Malgouyres (Paris 11), Olivier Nowak (Ecole Centrale de Lyon), Dominique Picard (Paris 7), and Franck Tarpin-Bernard (Ecole Centrale de Lyon). One of our first opportunities to apply the ideas of wavelets connected with signal analysis and its modeling occurred in collaboration with the team “Analysis and Forecast of the Electrical Consumption” of Electricité de France (Clamart-Paris) directed first by Jean-Pierre Desbrosses, and then by Hervé Laffaye, and which included Xavier Brossat, Yves Deville, and Marie-Madeleine Martin. And finally, apologies to those we may have omitted. xv Acknowledgments About the Authors Michel Misiti, Georges Oppenheim, and Jean-Michel Poggi are mathematics professors at Ecole Centrale de Lyon, University of Marne-La-Vallée and Paris 5 University. Yves Misiti is a research engineer specializing in Computer Sciences at Paris 11 University. The authors are members of the “Laboratoire de Mathématique” at Orsay-Paris 11 University France. Their fields of interest are statistical signal processing, stochastic processes, adaptive control, and wavelets. The authors’ group has published numerous theoretical papers and carried out applications in close collaboration with industrial teams. For instance: • Robustness of the piloting law for a civilian space launcher for which an expert system was developed • Forecasting of the electricity consumption by nonlinear methods • Forecasting of air pollution Notes by Yves Meyer The history of wavelets is not very old, at most 10 to 15 years. The field experienced a fast and impressive start, characterized by a close-knit international community of researchers who freely circulated scientific information and were driven by the researchers’ youthful enthusiasm. Even as the commercial rewards promised to be significant, the ideas were shared, the trials were pooled together, and the successes were shared by the community. There are lots of successes for the community to share. Why? Probably because the time is ripe. Fourier techniques were liberated by the appearance of windowed Fourier methods that operate locally on a time-frequency approach. In another direction, Burt-Adelson’s pyramidal algorithms, the quadrature mirror filters, and filter banks and subband coding are available. The mathematics underlying those algorithms existed earlier, but new computing techniques enabled researchers to try out new ideas rapidly. The numerical image and signal processing areas are blooming. The wavelets bring their own strong benefits to that environment: a local outlook, a multiscaled outlook, cooperation between scales, and a time-scale analysis. They demonstrate that sines and cosines are not the only useful xvi Acknowledgments functions and that other bases made of weird functions serve to look at new foreign signals, as strange as most fractals or some transient signals. Recently, wavelets were determined to be the best way to compress a huge library of fingerprints. This is not only a milestone that highlights the practical value of wavelets, but it has also proven to be an instructive process for the researchers involved in the project. Our initial intuition generally was that the proper way to tackle this problem of interweaving lines and textures was to use wavelet packets, a flexible technique endowed with quite a subtle sharpness of analysis and a substantial compression capability. However, it was a biorthogonal wavelet that emerged victorious and at this time represents the best method in terms of cost as well as speed. Our intuitions led one way, but implementing the methods settled the issue by pointing us in the right direction. For wavelets, the period of growth and intuition is becoming a time of consolidation and implementation. In this context, a toolbox is not only possible, but valuable. It provides a working environment that permits experimentation and enables implementation. Since the field still grows, it has to be vast and open. The Wavelet Toolbox product addresses this need, offering an array of tools that can be organized according to several criteria: • Synthesis and analysis tools • Wavelet and wavelet packets approaches • Signal and image processing • Discrete and continuous analyses • Orthogonal and redundant approaches • Coding, de-noising and compression approaches What can we anticipate for the future, at least in the short term? It is difficult to make an accurate forecast. Nonetheless, it is reasonable to think that the pace of development and experimentation will carry on in many different fields. Numerical analysis constantly uses new bases of functions to encode its operators or to simplify its calculations to solve partial differential equations. The analysis and synthesis of complex transient signals touches musical xvii Acknowledgments instruments by studying the striking up, when the bow meets the cello string. The analysis and synthesis of multifractal signals, whose regularity (or rather irregularity) varies with time, localizes information of interest at its geographic location. Compression is a booming field, and coding and de-noising are promising. For each of these areas, the Wavelet Toolbox software provides a way to introduce, learn, and apply the methods, regardless of the user’s experience. It includes a command-line mode and a graphical user interface mode, each very capable and complementing to the other. The user interfaces help the novice to get started and the expert to implement trials. The command line provides an open environment for experimentation and addition to the graphical interface. In the journey to the heart of a signal’s meaning, the toolbox gives the traveler both guidance and freedom: going from one point to the other, wandering from a tree structure to a superimposed mode, jumping from low to high scale, and skipping a breakdown point to spot a quadratic chirp. The time-scale graphs of continuous analysis are often breathtaking and more often than not enlightening as to the structure of the signal. Here are the tools, waiting to be used. Yves Meyer Professor, Ecole Normale Supérieure de Cachan and Institut de France Notes by Ingrid Daubechies Wavelet transforms, in their different guises, have come to be accepted as a set of tools useful for various applications. Wavelet transforms are good to have at one’s fingertips, along with many other mostly more traditional tools. Wavelet Toolbox software is a great way to work with wavelets. The toolbox, together with the power of MATLAB® software, really allows one to write complex and powerful applications, in a very short amount of time. The Graphic User Interface is both user-friendly and intuitive. It provides an excellent interface to explore the various aspects and applications of wavelets; it takes away the tedium of typing and remembering the various function calls. xviii Acknowledgments Ingrid C. Daubechies Professor, Princeton University, Department of Mathematics and Program in Applied and Computational Mathematics xix Preface xx 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters • “Wavelet Families” on page 1-2 • “Adding Your Own Wavelets” on page 1-26 • “Lifting Method for Constructing Wavelets” on page 1-41 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters Wavelet Families The Wavelet Toolbox software includes a large number of wavelets that you can use for both continuous and discrete analysis. For discrete analysis, examples include orthogonal wavelets (Daubechies’ extremal phase and least asymmetric wavelets) and B-spline biorthogonal wavelets. For continuous analysis, the Wavelet Toolbox software includes Morlet, Meyer, derivative of Gaussian, and Paul wavelets. The choice of wavelet is dictated by the signal or image characteristics and the nature of the application. If you understand the properties of the analysis and synthesis wavelet, you can choose a wavelet that is optimized for your application. Wavelet families vary in terms of several important properties. Examples include: • Support of the wavelet in time and frequency and rate of decay. • Symmetry or antisymmetry of the wavelet. The accompanying perfect reconstruction filters have linear phase. • Number of vanishing moments. Wavelets with increasing numbers of vanishing moments result in sparse representations for a large class of signals and images. • Regularity of the wavelet. Smoother wavelets provide sharper frequency resolution. Additionally, iterative algorithms for wavelet construction converge faster. • Existence of a scaling function, φ. For continuous analysis, the Wavelet Toolbox software provides a Fourier-transform based analysis for select analysis and synthesis wavelets. See cwtft and icwtft for details. For wavelets whose Fourier transforms satisfy certain constraints, you can define a single integral inverse. This allows you to reconstruct a time and scale-localized approximation to your input signal. See “Inverse Continuous Wavelet Transform” for a basic theoretical motivation. Signal Reconstruction from Continuous Wavelet Transform Coefficients illustrates the use of the 1-2 Wavelet Families inverse continuous wavelet transform (CWT) for simulated and real-world signals. Also, see the function reference pages for icwtft and icwtlin. Entering waveinfo at the command line displays a survey of the main properties of available wavelet families. For a specific wavelet family, use waveinfo with the wavelet family short name. You can find the wavelet family short names listed in the following table and on the reference page for waveinfo. Wavelet Family Short Name Wavelet Family Name 'haar' Haar wavelet 'db' Daubechies wavelets 'sym' Symlets 'coif' Coiflets 'bior' Biorthogonal wavelets 'rbio' Reverse biorthogonal wavelets 'meyr' Meyer wavelet 'dmey' Discrete approximation of Meyer wavelet 'gaus' Gaussian wavelets 'mexh' Mexican hat wavelet 'morl' Morlet wavelet 'cgau' Complex Gaussian wavelets 'shan' Shannon wavelets 'fbsp' Frequency B-Spline wavelets 'cmor' Complex Morlet wavelets To display detailed information about the Daubechies’ least asymmetric orthogonal wavelets, enter: waveinfo('sym') To compute the wavelet and scaling function (if available), use wavefun. 1-3 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters The Morlet wavelet is suitable for continuous analysis. There is no scaling function associated with the Morlet wavelet. To compute the Morlet wavelet, you can enter: [psi,xval] = wavefun('morl',10); plot(xval,psi); title('Morlet Wavelet'); For wavelets associated with a multiresolution analysis, you can compute both the scaling function and wavelet. The following code returns the scaling function and wavelet for the Daubechies’ extremal phase wavelet with 4 vanishing moments. [phi,psi,xval] = wavefun('db4',10); subplot(211); 1-4 Wavelet Families plot(xval,phi); title('db4 Scaling Function'); subplot(212); plot(xval,psi); title('db4 Wavelet'); In discrete wavelet analysis, the analysis and synthesis filters are of more interest than the associated scaling function and wavelet. You can use wfilters to obtain the analysis and synthesis filters. Obtain the decomposition (analysis) and reconstruction (synthesis) filters for the B-spline biorthogonal wavelet. Specify 3 vanishing moments in the 1-5 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters synthesis wavelet and 5 vanishing moments in the analysis wavelet. Plot the filters’ impulse responses. [LoD,HiD,LoR,HiR] = wfilters('bior3.5'); subplot(221); stem(LoD); title('Lowpass Analysis Filter'); subplot(222); stem(HiD); title('Highpass Analysis Filter'); subplot(223); stem(LoR); title('Lowpass Synthesis Filter'); subplot(224); stem(HiR); title('Highpass Synthesis Filter'); 1-6 Wavelet Families Daubechies Wavelets: dbN The dbN wavelets are the Daubechies’ extremal phase wavelets. N refers to the number of vanishing moments. These filters are also referred to in the literature by the number of filter taps, which is 2N. More about this family can be found in [Dau92] page 195. Enter waveinfo('db') at the MATLAB command prompt to obtain a survey of the main properties of this family. 1-7 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters Daubechies Wavelets db4 on the Left and db8 on the Right The db1 wavelet is also known as the Haar wavelet. The Haar wavelet is the only orthogonal wavelet with linear phase. Using waveinfo('haar'), you can obtain a survey of the main properties of this wavelet. Symlet Wavelets: symN The symN wavelets are also known as Daubechies’ least-asymmetric wavelets. The symlets are more symmetric than the extremal phase wavelets. In symN, N is the number of vanishing moments. These filters are also referred to in the literature by the number of filter taps, which is 2N. More about symlets can be found in [Dau92], pages 198, 254-257. Enter waveinfo('sym') at the MATLAB command prompt to obtain a survey of the main properties of this family. 1-8 Wavelet Families Symlets sym4 on the Left and sym8 on the Right Coiflet Wavelets: coifN Coiflet scaling functions also exhibit vanishing moments. In coifN, N is the number of vanishing moments for both the wavelet and scaling functions. These filters are also referred to in the literature by the number of filter taps, which is 2N.. For the coiflet construction, see [Dau92] pages 258–259. Enter waveinfo('coif') at the MATLAB command prompt to obtain a survey of the main properties of this family. Coiflets coif3 on the Left and coif5 on the Right 1-9 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters If s is a sufficiently regular continuous time signal, for large j the coefficient s, φ− j,k is approximated by 2− j / 2 s(2− j k) . If s is a polynomial of degree d, d ≤ N – 1, the approximation becomes an equality. This property is used, connected with sampling problems, when calculating the difference between an expansion over the φj,l of a given signal and its sampled version. Biorthogonal Wavelet Pairs: biorNr.Nd While the Haar wavelet is the only orthogonal wavelet with linear phase, you can design biorthogonal wavelets with linear phase. Biorthogonal wavelets feature a pair of scaling functions and associated scaling filters — one for analysis and one for synthesis. There is also a pair of wavelets and associated wavelet filters — one for analysis and one for synthesis. The analysis and synthesis wavelets can have different numbers of vanishing moments and regularity properties. You can use the wavelet with the greater number of vanishing moments for analysis resulting in a sparse representation, while you use the smoother wavelet for reconstruction. See [Dau92] pages 259, 262–85 and [Coh92] for more details on the construction of biorthogonal wavelet bases. Enter waveinfo('bior') at the command line to obtain a survey of the main properties of this family. The following code returns the B-spline biorthogonal reconstruction and decomposition filters with 3 and 5 vanishing moments and plots the impulse responses. The impulse responses of the lowpass filters are symmetric with respect to the midpoint. The impulse responses of the highpass filters are antisymmetric with respect to the midpoint. [LoD,HiD,LoR,HiR] = wfilters('bior3.5'); subplot(221); stem(LoD); 1-10 Wavelet Families title('Lowpass Analysis Filter'); subplot(222); stem(HiD); title('Highpass Analysis Filter'); subplot(223); stem(LoR); title('Lowpass Synthesis Filter'); subplot(224); stem(HiR); title('Highpass Synthesis Filter'); 1-11 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters Reverse Biorthogonal Wavelet Pairs: rbioNr.Nd This family is obtained from the biorthogonal wavelet pairs previously described. You can obtain a survey of the main properties of this family by typing waveinfo('rbio') from the MATLAB command line. Reverse Biorthogonal Wavelet rbio1.5 Meyer Wavelet: meyr Both ψ and φ are defined in the frequency domain, starting with an auxiliary function ν (see [Dau92] pages 117, 119, 137, 152). By typing waveinfo('meyr') at the MATLAB command prompt, you can obtain a survey of the main properties of this wavelet. 1-12 Wavelet Families Meyer Wavelet The Meyer wavelet and scaling function are defined in the frequency domain: • Wavelet function ⎛π ⎛ 3 ⎞⎞ ψˆ (ω ) = (2π )−1 / 2 eiω / 2 sin ⎜ ν ⎜ ω − 1 ⎟ ⎟ if ⎠⎠ ⎝ 2 ⎝ 2π 2π 4π ≤ ω ≤ 3 3 ⎛π ⎛ 3 ⎞⎞ ψˆ (ω ) = (2π )−1 / 2 eiω / 2 cos ⎜ ν ⎜ ω − 1 ⎟ ⎟ if ⎠⎠ ⎝ 2 ⎝ 4π 4π 8π ≤ ω ≤ 3 3 ⎡ 2π 8π ⎤ ω ∉⎢ , ⎥ ⎣ 3 3 ⎦ and ψˆ (ω ) = 0 if where ν (a) = a4 ( 35 − 84 a + 70 a2 − 20 a3 ) a ∈ [0, 1] • Scaling function φˆ(ω ) = (2π )−1 / 2 ω ≤ if 2π 3 ⎛π ⎛ 3 ⎞⎞ φˆ(ω ) = (2π )−1 / 2 cos ⎜ ν ⎜ ω − 1⎟ ⎟ ⎠⎠ ⎝ 2 ⎝ 2π φˆ(ω ) = 0 if ω > if 2π 4π ≤ ω ≤ 3 3 4π 3 1-13 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters By changing the auxiliary function, you get a family of different wavelets. For the required properties of the auxiliary function ν (see“References” for more information). This wavelet ensures orthogonal analysis. The function ψ does not have finite support, but ψ decreases to 0 when faster than any inverse polynomial ∀n ∈ Ν, ∃Cn such that ψ ( x) ≤ Cn (1 + x 2 ) , −n This property holds also for the derivatives 2 ∀k ∈ N , ∀n ∈ N , ∃Ck,n , such that ψ (k) x ≤ Ck,n (1 + x ) − n The wavelet is infinitely differentiable. Note Although the Meyer wavelet is not compactly supported, there exists a good approximation leading to FIR filters that you can use in the DWT. Enter waveinfo('dmey') at the MATLAB command prompt to obtain a survey of the main properties of this pseudo-wavelet. Gaussian Derivatives Family: gaus This family is built starting from the Gaussian function f ( x) = C p e− x by taking the pth derivative of f. 2 The integer p is the parameter of this family and in the previous formula, Cp is such that f ( p) 2 = 1 where f (p) is the pth derivative of f. You can obtain a survey of the main properties of this family by typing waveinfo('gaus') from the MATLAB command line. 1-14 Wavelet Families Gaussian Derivative Wavelet gaus8 You can use even-order derivative of Gaussian wavelets in the Fourier-transform based CWT See cwtft for details. Mexican Hat Wavelet: mexh This wavelet is proportional to the second derivative function of the Gaussian probability density function. The wavelet is a special case of a larger family of derivative of Gaussian (DOG) wavelets. There is no scaling function associated with this wavelet. Enter waveinfo('mexh') at the MATLAB command prompt to obtain a survey of the main properties of this wavelet. You can compute the wavelet with wavefun. [psi,xval] = wavefun('mexh',10); plot(xval,psi); title('Mexican Hat Wavelet'); 1-15 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters You can use the Mexican hat wavelet in the Fourier-transform based CWT. See cwtft for details. Morlet Wavelet: morl Both real-valued and complex-valued versions of this wavelet exist. Enter waveinfo('morl') at the MATLAB command line to obtain the properties of the real-valued Morlet wavelet. The real-valued Morlet wavelet is defined as: ψ ( x) = Ce− x cos(5 x) 2 1-16 Wavelet Families The constant C is used for normalization in view of reconstruction. [psi,xval] = wavefun('morl',10); plot(xval,psi); title('Real-valued Morlet Wavelet'); The Morlet wavelet does not technically satisfy the admissibility condition.. Additional Real Wavelets Some other real wavelets are available in the toolbox. 1-17 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters Complex Wavelets The toolbox also provides a number of complex-valued wavelets for continuous wavelet analysis. Complex-valued wavelets provide phase information and are therefore very important in the time-frequency analysis of nonstationary signals. Complex Gaussian Wavelets: cgau This family is built starting from the complex Gaussian function f ( x) = C p e−ix e− x by taking the pth derivative of f. The integer p is the parameter of this family and in the previous formula, Cp is such that 2 f ( p) 2 = 1 where f (p) is the pth derivative of f. You can obtain a survey of the main properties of this family by typing waveinfo('cgau') from the MATLAB command line. Complex Gaussian Wavelet cgau8 1-18 Wavelet Families Complex Morlet Wavelets: cmor See [Teo98] pages 62–65. A complex Morlet wavelet is defined by ψ ( x) = 1 π fb x2 2iπ fc x f b e e depending on two parameters: • fb is a bandwidth parameter. • fc is a wavelet center frequency. You can obtain a survey of the main properties of this family by typing waveinfo('cmor') from the MATLAB command line. Complex Morlet Wavelet morl 1.5-1 Complex Frequency B-Spline Wavelets: fbsp See [Teo98] pages 62–65. 1-19 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters A complex frequency B-spline wavelet is defined by ψ ( x) = ⎛ ⎛ f x ⎞⎞ fb ⎜ sinc ⎜ b ⎟ ⎟ ⎝ m ⎠⎠ ⎝ m e2iπ fc x depending on three parameters: • m is an integer order parameter (m ≥ 1). • fb is a bandwidth parameter. • fc is a wavelet center frequency. You can obtain a survey of the main properties of this family by typing waveinfo('fbsp') from the MATLAB command line. Complex Frequency B-Spline Wavelet fbsp 2-0.5-1 Complex Shannon Wavelets: shan See [Teo98] pages 62–65. 1-20 Wavelet Families This family is obtained from the frequency B-spline wavelets by setting m to 1. A complex Shannon wavelet is defined by ψ ( x) = fb sinc ( fb x ) e2iπ fc x depending on two parameters: • fb is a bandwidth parameter. • fc is a wavelet center frequency. You can obtain a survey of the main properties of this family by typing waveinfo('shan') from the MATLAB command line. Complex Shannon Wavelet shan 0.5-1 1-21 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters Wavelet Families and Associated Properties — I Property morl mexh meyr Crude ■ ■ Infinitely regular ■ ■ haar dbN symN coifN biorNr.Nd ■ ■ ■ ■ ■ ■ ■ Arbitrary regularity ■ Compactly supported orthogonal ■ Compactly supported biothogonal ■ Symmetry ■ ■ ■ ■ ■ ■ Asymmetry Near symmetry ■ Arbitrary number of vanishing moments ■ ■ ■ ■ ■ ■ Vanishing moments for φ Existence of φ ■ ■ ■ ■ ■ Orthogonal analysis ■ ■ ■ ■ ■ Biorthogonal analysis ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Discrete transform ■ ■ ■ ■ ■ Fast algorithm ■ ■ ■ ■ ■ Exact reconstruction ≈ ■ FIR filters Continuous transform Explicit expression ■ ■ ■ ■ ■ ■ ■ For splines Crude wavelet — A wavelet is said to be crude when satisfying only the admissibility condition. Regularity 1-22 Wavelet Families Orthogonal Biorthogonal — See “Biorthogonal Wavelet Pairs: biorNr.Nd” on page 1-10. Vanishing moments Exact reconstruction — See “Reconstruction Filters” in the Wavelet Toolbox Getting Started Guide. Continuous — See “Continuous Wavelet Transform” in the Wavelet Toolbox Getting Started Guide. Discrete — See “Critically-Sampled Discrete Wavelet Transform” in the Wavelet Toolbox Getting Started Guide. FIR filters — See “Filters Used to Calculate the DWT and IDWT” on page 3-37. Wavelet Families and Associated Properties — II Property rbioNr.Nd gaus dmey cgau cmor fbsp shan Crude ■ ■ ■ ■ ■ Infinitely regular ■ ■ ■ ■ ■ ■ ■ ■ ■ Arbitrary regularity ■ Compactly supported orthogonal Compactly supported biothogonal ■ Symmetry ■ ■ ■ Asymmetry Near symmetry Arbitrary number of vanishing moments ■ 1-23 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters Property rbioNr.Nd gaus dmey cgau cmor fbsp shan ■ ≈ ■ ■ ■ ■ ■ ■ ■ ■ Complex valued ■ ■ ■ ■ Complex continuous transform ■ ■ ■ ■ Vanishing moments for φ ■ Existence of φ Orthogonal analysis Biorthogonal analysis ■ Exact reconstruction ■ FIR filters ■ Continuous transform ■ Discrete transform ■ ■ Fast algorithm ■ ■ Explicit expression For splines ■ ■ ■ ■ FIR-based approximation Crude wavelet Regularity Orthogonal Biorthogonal — See “Biorthogonal Wavelet Pairs: biorNr.Nd” on page 1-10. Vanishing moments Exact reconstruction — See “Reconstruction Filters” in the Wavelet Toolbox Getting Started Guide. Continuous — See “Continuous Wavelet Transform” in the Wavelet Toolbox Getting Started Guide. 1-24 Wavelet Families Discrete — See “Critically-Sampled Discrete Wavelet Transform” in the Wavelet Toolbox Getting Started Guide. FIR filters — See “Filters Used to Calculate the DWT and IDWT” on page 3-37. 1-25 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters Adding Your Own Wavelets This section shows you how to add your own wavelet families to the toolbox. In this section... “Preparing to Add a New Wavelet Family” on page 1-26 “Adding a New Wavelet Family” on page 1-32 “After Adding a New Wavelet Family” on page 1-40 Preparing to Add a New Wavelet Family Wavelet Toolbox software contains a large number of the most commonly-used wavelet families. Additionally, using wavemngr, you can add new wavelets to the existing ones to implement your favorite wavelet or try out one of your own design. The toolbox allows you to define new wavelets for use with both the command line functions and the graphical interface tools. Caution The toolbox does not check that your wavelet meets all the mathematical requisites to constitute a valid wavelet. wavemngr affords extensive wavelet management. However, this section focuses only on the addition of a wavelet family. For more complete information, see the wavemngr reference page. To add a new wavelet, you must 1 Choose the full name of the wavelet family (fn). 2 Choose the short name of the wavelet family (fsn). 3 Determine the wavelet type (wt). 4 Define the orders of wavelets within the given family (nums). 5 Build a MAT-file or a MATLAB file (file). 6 For wavelets without FIR filters: Define the effective support. 1-26 Adding Your Own Wavelets These steps are described below. Choose the Wavelet Family Full Name The full name of the wavelet family, fn, must be a string. Do not use predefined wavelet family names. To see the predefined wavelet family names, enter: wavemngr('read') The predefined wavelet family names are the displayed in the first column of the output. Predefined wavelet family names are Haar, Daubechies, Symlets, Coiflets, BiorSplines, ReverseBior, Meyer, DMeyer, Gaussian, Mexican_hat, Morlet, Complex Gaussian, Shannon, Frequency B-Spline, and Complex Morlet. Choose the Wavelet Family Short Name The short name of the wavelet family, fsn, must be a string of four characters or less. Do not use predefined wavelet family short names. To see the predefined wavelet family short names, enter: wavemngr('read') The predefined wavelet family short names are the displayed in the second column of the output.. Determine the Wavelet Type We distinguish five types of wavelets: 1 Orthogonal wavelets with FIR filters These wavelets can be defined through the scaling filter h. The scaling filter is a lowpass filter. For orthogonal wavelets, the same scaling filter is used for decomposition (analysis) and reconstruction (synthesis). Predefined families of such wavelets include Haar, Daubechies, Coiflets, and Symlets. 2 Biorthogonal wavelets with FIR filters 1-27 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters These wavelets can be defined through the two scaling filters hr and hd, for reconstruction and decomposition respectively. The BiorSplines wavelet family is a predefined family of this type. 3 Orthogonal wavelets without FIR filters, but with a scaling function These wavelets can be defined through the definition of the wavelet function and the scaling function. The Meyer wavelet family is a predefined family of this type. 4 Wavelets without FIR filters and without a scaling function These wavelets can be defined through the definition of the wavelet function. Predefined families of such wavelets include Morlet and Mexican_hat. 5 Complex wavelets without FIR filters and without a scaling function These wavelets can be defined through the definition of the wavelet function. Predefined families of such wavelets include Complex Gaussian and Shannon. Define the Orders of Wavelets Within the Given Family If a family contains many wavelets, the short name and the order are appended to form the wavelet name. Argument nums is a string containing the orders separated with blanks. This argument is not used for wavelet families that only have a single wavelet (Haar, Meyer, and Morlet for example). For example, for the first extremal-phase Daubechies wavelets, fsn = 'db' nums = '1 2 3' yields the three wavelets db1, db2, and db3. For the first B-spline biorthogonal wavelets, fsn = 'bior' nums = '1.1 1.3 1.5 2.2' yields the four wavelets bior1.1, bior1.3, bior1.5, and bior2.2. 1-28 Adding Your Own Wavelets You can display this information for the predefined wavelets with wavemngr('read',1) Build a MAT-File or Code File wavemngr requires a file argument, which is a string containing a MATLAB function or MAT file name. If a family contains many wavelets, a MATLAB code file (with a .m extension) must be defined and must be of a specific form that depends on the wavelet type. The specific file formats are described in the remainder of this section. If a family contains a single wavelet, then a MAT-file can be defined for wavelets of type 1. It must have the wavelet family short name (fsn) argument as its name and must contain a single variable whose name is fsn and whose value is the scaling filter. An code file can also be defined as discussed below. Note If no file extension is specified, a .m extension is used as default. Type 1 (Orthogonal with FIR Filter). The syntax of the first line in the MATLAB function is function w = file(wname) where the input argument wname is a string containing the wavelet name, and the output argument w is the corresponding scaling filter. The filter w must be of even length. If the scaling filter is not of even length, the filter is zero-padded by the toolbox. For predefined wavelets, the sum of the scaling filter coefficients is 1. This follows the convention followed by Daubechies. m0 ( ) 1 2 hn e jn 1-29 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters When you access these coefficients using wfilters, the coefficients are scaled by the square root of 2. The toolbox normalizes your filter so that the resulting sum is 1. Examples of such files for predefined wavelets are dbwavf.m for Daubechies, coifwavf.m for coiflets, and symwavf.m for symlets. Type 2 (Biorthogonal with FIR Filter). The syntax of the first line in the MATLAB function is function [wr,wd] = file(wname) where the input argument wname is a string containing the wavelet name and the output arguments wr and wd are the corresponding reconstruction and decomposition scaling filters, respectively. The filters wr and wd must be of the same even length. In general, initial biorthogonal filters do not meet these requirements, so they are zero-padded by the toolbox. For predefined wavelets, the sum of the scaling filter coefficients is 1. This follows the convention followed by Daubechies. m0 ( ) 1 2 hn e jn When you access these coefficients using wfilters, the coefficients are scaled by the square root of 2. The toolbox normalizes your filter so that the resulting sum is 1. The file biorwavf.m (for BiorSplines) is an example of a file for a type 2 predefined wavelet family. Type 3 (Orthogonal with Scale Function). The syntax of the first line in the MATLAB function is function [phi,psi,t] = file(lb,ub,n,wname) 1-30 Adding Your Own Wavelets which returns values of the scaling function phi and the wavelet function psi on t, a linearly-spaced n-point grid of the interval [lb ub]. The argument wname is optional (see Note below). The file meyer.m is an example of a file for a type 3 predefined wavelet family. Type 4 or Type 5 (No FIR Filter; No Scale Function). The syntax of the first line in the MATLAB function is function [psi,t] = file(lb,ub,n,wname) or function [psi,t] = file(lb,ub,n,wname, additional arguments) which returns values of the wavelet function psi ont a linearly-spaced n-point grid of the interval [lb ub]. The argument wname is optional (see Note below). Examples of type 4 files for predefined wavelet families are mexihat.m (for Mexican_hat) and morlet.m (for Morlet). Examples of type 5 files for predefined wavelet families are shanwavf.m (for Shannon) and cmorwavf.m (for Complex Morlet). Note For the types 3, 4, and 5, the wname argument is optional unless the new wavelet family contains more than one wavelet and if you plan to use this new family in the GUI mode. For the types 4 and 5, a complete example of using the additional arguments can be found on the fbspwavf reference page. Define the Effective Support This definition is required only for wavelets of types 3, 4, and 5, since they are not compactly supported. 1-31 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters Defining the effective support means specifying an upper and lower bound. The following table includes the lower and upper bounds for a few of the toolbox wavelets. Family Lower Bound (lb) Upper Bound (ub) Meyer –8 8 Mexican_hat –5 5 Morlet –4 4 Adding a New Wavelet Family To add a new wavelet, usewavemngr in one of two forms: wavemngr('add',fn,fsn,wt,nums,file) or wavemngr('add',fn,fsn,wt,nums,file,b). Here are a few examples to illustrate how you would use wavemngr to add some of the predefined wavelet families. New wavelet family names and short names are used for illustration purposes. Type 1 wavelet — Add wavelets ndb1,ndb2,ndb3,ndb4,ndb5. wavemngr('add','Ndaubechies','ndb',1,'1 2 3 4 5','dbwavf'); Type 1 wavelet — Add wavelets ndaub1,ndaub2,.... dbwavf calls dbaux to compute the Daubechies extremal phase wavelets with more than 10 vanishing moments. The computation in dbaux becomes unstable when the number of vanishing moments becomes large. Therefore dbaux errors when you specify the number of vanishing moments greater than 45. wavemngr('add','Ndaubwav','ndaub',1,'1 2 3 4 5 **','dbwavf'); Type 2 wavelet — Add wavelets nbio1.1, nbio1.3. wavemngr('add','Nbiorwavf','nbio',2,'1.1 1.3','biorwavf'); Type 3 wavelet — Add the wavelet Nmeyer with effective support [-8,8]. 1-32 Adding Your Own Wavelets wavemngr('add','Nmeyer','nmey',3,'','meyer',[-8,8]); Type 4 wavelet — Add the wavelet Nmorlet with effective support [-4,4]. wavemngr('add','Nmorlet','nmor',4,'','morlet',[-4,4]); You can delete the wavelets you have created with wavemngr('del',familyShortName). For example: wavemngr('del','nmey'); Example 1 Let us take the example of Binlets proposed by Strang and Nguyen in pages 216-217 of the book Wavelets and Filter Banks (see [StrN96] in “References”). Note The files used in this example can be found in the wavedemo folder. The full family name is Binlets. The short name of the wavelet family is binl. The wavelet type is 2 (Biorthogonal with FIR filters). The order of the wavelet within the family is 7.9 (we just use one in this example). The file used to generate the filters is binlwavf.m Then to add the new wavelet, type % Add new family of biorthogonal wavelets. wavemngr('add','Binlets','binl',2,'7.9','binlwavf') % List wavelets families. wavemngr('read') ans = 1-33 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters =================================== Haar haar Daubechies db Symlets sym Coiflets coif BiorSplines bior ReverseBior rbio Meyer meyr DMeyer dmey Gaussian gaus Mexican_hat mexh Morlet morl Complex Gaussian cgau Shannon shan Frequency B-Spline fbsp Complex Morlet cmor Binlets binl =================================== If you want to get online information on this new family, you can build an associated help file which would look like the following: function binlinfo %BINLINFO Information on biorthogonal wavelets (binlets). % % Biorthogonal Wavelets (Binlets) % % Family Binlets % Short name binl % Order Nr,Nd Nr = 7 , Nd = 9 % % Orthogonal no % Biorthogonal yes % Compact support yes % DWT possible % CWT possible % % binl Nr.Nd ld lr % effective length effective length % of LoF_D of HiF_D 1-34 Adding Your Own Wavelets % binl 7.9 7 9 The associated file to generate the filters (binlwavf.m) is function [Rf,Df] = binlwavf(wname) %BINLWAVF Biorthogonal wavelet filters (Binlets). % [RF,DF] = BINLWAVF(W) returns two scaling filters % associated with the biorthogonal wavelet specified % by the string W. % W = 'binlNr.Nd' where possible values for Nr and Nd are: Nr = 7 Nd = 9 % The output arguments are filters: % RF is the reconstruction filter % DF is the decomposition filter % Check arguments. if errargn('binlwavf',nargin,[0 1],nargout,[0:2]), error('*'); end % suppress the following line for extension Nr = 7; Nd = 9; % for possible extension % more wavelets in 'Binlets' family %---------------------------------if nargin==0 Nr = 7; Nd = 9; elseif isempty(wname) Nr = 7; Nd = 9; else if ischar(wname) lw = length(wname); ab = abs(wname); ind = find(ab==46 | 47info.m (for example, see dbinfo.m or morlinfo.m). This file will be associated automatically with the Wavelet Family button in the Wavelet Display option of the graphical tools. 1-40 Lifting Method for Constructing Wavelets Lifting Method for Constructing Wavelets The so-called first generation wavelets and scaling functions are dyadic dilations and translates of a single function. Fourier methods play a key role in the design of these wavelets. However, the requirement that the wavelet basis consist of translates and dilates of a single function imposes some constraints that limit the utility of the multiresolution idea at the core of wavelet analysis. The utility of wavelet methods is extended by the design of second generation wavelets via lifting. Typical settings where translation and dilation of a single function cannot be used include: • Designing wavelets on bounded domains — This includes the construction of wavelets on an interval, or bounded domain in a higher-dimensional Euclidean space. • Weighted wavelets — In certain applications, such as the solution of partial differential equations, wavelets biorthogonal with respect to a weighted inner product are needed. • Irregularly-spaced data — In many real-world applications, the sampling interval between data samples is not equal. Designing new first generation wavelets requires expertise in Fourier analysis. The lifting method proposed by Sweldens (see [Swe98] in “References”) removes the necessity of expertise in Fourier analysis and allows you to generate an infinite number of discrete biorthogonal wavelets starting from an initial one. In addition to generation of first generation wavelets with lifting, the lifting method also enables you to design second generation wavelets, which cannot be designed using Fourier-based methods. With lifting, you can design wavelets that address the shortcomings of the first generation wavelets. The following section introduces the theory behind lifting, presents the lifting functions of Wavelet Toolbox software and gives two short examples: • “Lifting Background” on page 1-42 • “Lifting Functions” on page 1-49 1-41 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters For more information on lifting, see [Swe98], [Mal98], [StrN96], and [MisMOP03] in “References”. Lifting Background The DWT implemented by a filter bank is defined by four filters as described in “Fast Wavelet Transform (FWT) Algorithm” on page 3-37. Two main properties of interest are • The perfect reconstruction property • The link with “true” wavelets (how to generate, starting from the filters, orthogonal or biorthogonal bases of the space of the functions of finite energy) To illustrate the perfect reconstruction property, the following filter bank contains two decomposition filters and two synthesis filters. The decomposition and synthesis filters may constitute a pair of biorthogonal bases or an orthogonal basis. The capital letters denote the Z-transforms of the filters.. ~ H 2 2 H X X 2 ~ G 2 G This leads to the following two conditions for a perfect reconstruction (PR) filter bank: ~ ~ H ( z) H ( z) G( z)G( z) 2 z L1 and ~ ~ H ( z) H ( z) G( z)G( z) 0 1-42 Lifting Method for Constructing Wavelets The first condition is usually (incorrectly) called the perfect reconstruction condition and the second is the anti-aliasing condition. The z–L+1 term implies that perfect reconstruction is achieved up to a delay of one sample less than the filter length, L. This results if the analysis filters are shifted to be causal. Lifting designs perfect reconstruction filter banks by beginning from the basic nature of the wavelet transform. Wavelet transforms build sparse representations by exploiting the correlation inherent in most real world data. For example, plot the example of electricity consumption over a 3-day period. load leleccum; plot(leleccum) grid on; axis tight; 1-43 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters The data do not exhibit arbitrary changes from sample to sample. Neighboring samples exhibit correlation. A relatively low (high) value at index (sample) n is associated with a relatively low (high) value at index n-1 and n+1. This implies that if you have only the odd or even samples from the data, you can predict the even or odd samples. How accurate your prediction is obviously depends on the nature of the correlation between adjacent samples and how closely your predictor approximates that correlation. Polyphase Representation The polyphase representation of a signal is an important concept in lifting. You can view each signal as consisting of phases, which consist of taking every N-th sample beginning with some index. For example, if you index a time series from n=0 and take every other sample starting at n=0, you extract 1-44 Lifting Method for Constructing Wavelets the even samples. If you take every other sample starting from n=1, you extract the odd samples. These are the even and odd polyphase components of the data. Because your increment between samples is 2, there are only two phases. If you increased your increment to 4, you can extract 4 phases. For lifting, it is sufficient to concentrate on the even and odd polyphase components. The following diagram illustrates this operation for an input signal. 2 Xe 2 Xo X Z where Z denotes the unit advance operator and the downward arrow with the number 2 represents downsampling by two. In the language of lifting, the operation of separating an input signal into even and odd components is known as the split operation, or the lazy wavelet. To understand lifting mathematically, it is necessary to understand the z-domain representation of the even and odd polyphase components. The z-transform of the even polyphase component is X 0 ( z) x(2n) z n n The z-transform of the odd polyphase component is X1 ( z) x(2n 1) z n n You can write the z-transform of the input signal as the sum of dilated versions of the z-transforms of the polyphase components. X ( z) x(2n) z2n x(2n 1) z2n1 X 0 ( z2 ) z1 X1 ( z2 ) n n 1-45 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters Split, Predict, and Update A single lifting step can be described by the following three basic operations: • Split — the signal into disjoint components. A common way to do this is to extract the even and odd polyphase components explained in “Polyphase Representation” on page 1-44. This is also known as the lazy wavelet. • Predict — the odd polyphase component based on a linear combination of samples of the even polyphase component. The samples of the odd polyphase component are replaced by the difference between the odd polyphase component and the predicted value. The predict operation is also referred to as the dual lifting step. • Update — the even polyphase component based on a linear combination of difference samples obtained from the predict step. The update step is also referred to as the primal lifting step. In practice, a normalization is incorporated for both the primal and dual liftings. The following diagram illustrates one lifting step. Xe X + P Split Xo - A U D Haar Wavelet Via Lifting Using the operations defined in “Split, Predict, and Update” on page 1-46, you can implement the Haar wavelet via lifting. • Split — Divide the signal into even and odd polyphase components. • Predict — Replace x(2n+1) with d(n)=x(2n+1)–x(2n). The predict operator is simply x(2n). 1-46 Lifting Method for Constructing Wavelets • Update — Replace x(2n) with x(2n)+d(n)/2. This is equal to (x(2n)+x(2n+1))/2. The dual lifting in the z domain can be written in the following matrix form 0 X 0 ( z) 1 P ( z) 1 X1 ( z) with P(z)=1. The primal lifting can be written in the z domain in the following matrix form 0 X 0 ( z) 1 S( z) 1 0 1 P ( z ) 1 X1 ( z) with S(z)=1/2. Finally, the primal and dual normalization can be incorporated as follows. 2 0 0 0 X 0 ( z) 1 S( z) 1 1 1 P ( z) 1 X1 ( z) 0 2 To construct this lifting step in MATLAB, enter: LiftHaar = liftwave('haar'); displs(LiftHaar) The following is displayed in the MATLAB command window. LiftHaar = {... 'd' [ -1.00000000] 'p' [ 0.50000000] [ 1.41421356] [ 0.70710678] }; [0] [0] [] 'd' denotes the dual lifting. Note that for convenience, the negative sign is incorporated into the dual lifting step in the Wavelet Toolbox software. 'p' denotes the primal lifting and [ 1.41421356] [ 0.70710678] are the primal and dual normalization constants. LiftHaar{1,3} and LiftHaar{2,3} give 1-47 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters the highest degree of the Laurent polynomials, which describe the dual and primal liftings. In this case, both are zero because the dual and primal liftings are both described by scalars. Bior2.2 Wavelet Via Lifting This examples presents the lifting scheme for the bior2.2 biorthogonal scaling and wavelet filters. In the Haar lifting scheme, the dual lifting (predict operator) differenced the odd and even samples. In this example, define a new predict operator that computes the average of the two neighboring even samples. Subtract the average from the intervening odd sample. 1 d(n) x(2n 1) [ x(2n) x(2n 2)] 2 In the z-domain you can write the dual lifting step as 1 0 X 0 ( z) 1 (1 z) 1 X1 ( z) 2 To obtain the primal lifting, or update, examine the primal lifting in “Haar Wavelet Via Lifting” on page 1-46. The update is defined in such a way that the sum of the approximation coefficients is proportional to the mean of the input data vector. 1 x(n) 2 a(n) n n To obtain the same result in this lifting step, define the update as 1 0 1-48 1 1 1 0 ( z 1) X 0 ( z) 4 1 (1 z) 1 X1 ( z) 1 2 Lifting Method for Constructing Wavelets To obtain this lifting scheme at the command line, enter: liftwave('bior2.2') Lifting Functions The lifting functions of the toolbox are organized into five groups: • “Lifting Schemes ” on page 1-49 • “Biorthogonal Quadruplets of Filters and Lifting Schemes” on page 1-49 • “Usual Biorthogonal Quadruplets” on page 1-50 • “Lifting Wavelet Transform (LWT)” on page 1-50 • “Laurent Polynomials and Matrices” on page 1-51 Lifting Schemes Function Name Description lsinfo Information about lifting schemes displs Display a lifting scheme addlift Add primal or dual elementary lifting steps to a lifting scheme wavenames Wavelets with lifting schemes Biorthogonal Quadruplets of Filters and Lifting Schemes These functions connect lifting schemes to biorthogonal quadruplets of filters and associated scaling and wavelet function pairs. Function Name Description liftfilt Apply elementary lifting steps on quadruplet of filters filt2ls Transform a quadruplet of filters to a lifting scheme 1-49 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters Function Name Description ls2filt Transform a lifting scheme to a quadruplet of filters bswfun Compute and plot biorthogonal “scaling and wavelet” functions Usual Biorthogonal Quadruplets These functions provide some basic lifting schemes associated with some usual orthogonal or biorthogonal (“true”) wavelets and the “lazy” one. These schemes can be used to initialize a lifting procedure. Function Name Description wavenames Provides usual wavelet names available for LWT liftwave Provides lifting scheme associated with a usual wavelet wave2lp Provides Laurent polynomials associated with a usual wavelet Lifting Wavelet Transform (LWT) These functions contain the direct and inverse lifting wavelet transform (LWT) files for both 1-D and 2-D signals. LWT reduces to the polyphase version of the DWT algorithm with zero-padding extension mode and without extra-coefficients. 1-50 Function Name Description lwt 1-D lifting wavelet transform ilwt Inverse 1-D lifting wavelet transform lwtcoef Extract or reconstruct 1-D LWT wavelet coefficients lwt2 2-D lifting wavelet transform Lifting Method for Constructing Wavelets Function Name Description ilwt2 Inverse 2-D lifting wavelet transform lwtcoef2 Extract or reconstruct 2-D LWT wavelet coefficients Laurent Polynomials and Matrices These functions permit an entry to representation and calculus of Laurent polynomials and matrices. Function Name Description laurpoly Constructor for the class of Laurent polynomials laurmat Constructor for the class of Laurent matrices The lifting folder and the two object folders @laurpoly and @laurmat contain many other files. Primal Lifting from Haar These two simple examples illustrate the basic lifting capabilities of Wavelet Toolbox software. A primal lifting starting from Haar wavelet. Start from the Haar wavelet and get the corresponding lifting scheme. lshaar = liftwave('haar'); displs(lshaar); Add a primal ELS to the lifting scheme. els = {'p',[-0.125 0.125],0}; lsnew = addlift(lshaar,els); displs(lsnew); Transform the lifting scheme to biorthogonal filters quadruplet and plot the resulting scaling function and wavelet. 1-51 1 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters [LoD,HiD,LoR,HiR] = ls2filt(lsnew); bswfun(LoD,HiD,LoR,HiR,'plot'); Integer-to-Integer Wavelet Transform In several applications it is desirable to have a wavelet transform that maps integer inputs to integer scaling and wavelet coefficients. You can accomplish easily using lifting. Start with the Haar transform for an integer to integer wavelet transform and apply a primal lifting step. lshaar = liftwave('haar','int2int'); els = {'p',[-0.125 0.125],0}; 1-52 Lifting Method for Constructing Wavelets lsnewint = addlift(lshaar,els); Obtain the integer-to-integer wavelet transform of a 1-D signal and invert the transform to demonstrate perfect reconstruction. x = 1:8; [cA,cD] = lwt(x,lsnewint); xnew = ilwt(cA,cD,lsnewint) 1-53 1 1-54 Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters 2 Continuous Wavelet Analysis • “1-D Continuous Wavelet Analysis” on page 2-2 • “One-Dimensional Complex Continuous Wavelet Analysis” on page 2-21 • “DFT-Based Continuous Wavelet Analysis — Command Line” on page 2-30 • “Interactive DFT-Based Continuous Wavelet Analysis” on page 2-41 2 Continuous Wavelet Analysis 1-D Continuous Wavelet Analysis The Wavelet Toolbox software enables you to perform a continuous wavelet analysis of your univariate or bivariate 1-D input signals. You can perform continuous wavelet analyses at the command line or with graphical user interfaces accessible through wavemenu. Key features include: • Continuous wavelet transform (CWT) of a 1-D input signal using real-valued and complex-valued wavelets. The Wavelet Toolbox software features CWT algorithms based on the correlation of the signal with an analyzing wavelet, cwt, and based on the discrete Fourier transform of the input signal and analyzing wavelet, cwtft. • Inverse CWT of 1–D input signal. For select analyzing wavelets, you can invert the CWT to reconstruct a time and scale-localized approximation to your input signal. See icwtft and icwtlin for details. • Wavelet cross spectrum and coherence. You can use wcoher to compute the wavelet cross spectrum and coherence between two time series. The wavelet cross spectrum and coherence can reveal localized similarities between two time series in time and scale. See Wavelet Coherence for examples. • Pattern-adapted wavelets for signal analysis. A strength of wavelet analysis is the ability to design wavelets that mimic the structures you wish to detect. Using pat2cwav and wavemngr you can add custom wavelets optimized to detect specified patterns in your data. See Pattern Adapted Wavelets for Signal Detection for examples. In this section, you’ll learn how to • Load a signal • Perform a continuous wavelet transform of a signal • Produce a plot of the coefficients • Produce a plot of coefficients at a given scale • Produce a plot of local maxima of coefficients across scales • Select the displayed plots 2-2 1-D Continuous Wavelet Analysis • Switch from scale to pseudo-frequency information • Zoom in on detail • Display coefficients in normal or absolute mode • Choose the scales at which analysis is performed Since you can perform analyses either from the command line or using the graphical interface tools, this section has subsections covering each method. The final subsection discusses how to exchange signal and coefficient information between the disk and the graphical tools. 2-3 2 Continuous Wavelet Analysis Command Line Continuous Wavelet Analysis This example involves a noisy sinusoidal signal. 1 Load a signal. From the MATLAB prompt, type load noissin; You now have the signal noissin in your workspace: whos 2-4 1-D Continuous Wavelet Analysis Name Size Bytes Class noissin 1x1000 8000 double array 2 Perform a Continuous Wavelet Transform. Use the cwt command. Type c = cwt(noissin,1:48,'db4'); The arguments to cwt specify the signal to be analyzed, the scales of the analysis, and the wavelet to be used. The returned argument c contains the coefficients at various scales. In this case, c is a 48-by-1000 matrix with each row corresponding to a single scale. 3 Plot the coefficients. The cwt command accepts a fourth argument. This is a flag that, when present, causes cwt to produce a plot of the absolute values of the continuous wavelet transform coefficients. The cwt command can accept more arguments to define the different characteristics of the produced plot. For more information, see the cwt reference page. c = cwt(noissin,1:48,'db4','plot'); A plot appears. 2-5 2 Continuous Wavelet Analysis Of course, coefficient plots generated from the command line can be manipulated using ordinary MATLAB graphics commands. 4 Choose scales for the analysis. The second argument to cwt gives you fine control over the scale levels on which the continuous analysis is performed. In the previous example, we used all scales from 1 to 48, but you can construct any scale vector subject to these constraints: • All scales must be real positive numbers. • The scale increment must be positive. 2-6 1-D Continuous Wavelet Analysis • The highest scale cannot exceed a maximum value depending on the signal. Let’s repeat the analysis using every other scale from 2 to 128. Type c = cwt(noissin,2:2:128,'db4','plot'); A new plot appears: This plot gives a clearer picture of what’s happening with the signal, highlighting the periodicity. 2-7 2 Continuous Wavelet Analysis Continuous Analysis Using the Graphical Interface We now use the Continuous Wavelet 1-D tool to analyze the same noisy sinusoidal signal we examined earlier using the command line interface in “Command Line Continuous Wavelet Analysis” on page 2-4. 1 Start the Continuous Wavelet 1-D Tool. From the MATLAB prompt, type wavemenu The Wavelet Toolbox Main Menu appears. Click the Continuous Wavelet 1-D menu item. The continuous wavelet analysis tool for one-dimensional signal data appears. 2-8 1-D Continuous Wavelet Analysis 2 Load a signal. Choose the File > Load Signal menu option. When the Load Signal dialog box appears, select the MAT-file noissin.mat, which should reside in the MATLAB folder toolbox/wavelet/wavedemo. Click the OK button. The noisy sinusoidal signal is loaded into the Continuous Wavelet 1-D tool. The default value for the sampling period is equal to 1 (second). 3 Perform a Continuous Wavelet Transform. 2-9 2 Continuous Wavelet Analysis To start our analysis, let’s perform an analysis using the db4 wavelet at scales 1 through 48, just as we did using command line functions in the previous section. In the upper right portion of the Continuous Wavelet 1-D tool, select the db4 wavelet and scales 1–48. 4 Click the Analyze button. After a pause for computation, the tool displays the coefficients plot, the coefficients line plot corresponding to the scale a = 24, and the local maxima plot, which displays the chaining across scales (from a = 48 down to a = 1) of the coefficients local maxima. 2-10 1-D Continuous Wavelet Analysis 5 View Wavelet Coefficients Line. Select another scale a = 40 by clicking in the coefficients plot with the right mouse button. See step 9 to know, more precisely, how to select the desired scale. Click the New Coefficients Line button. The tool updates the plot. 6 View Maxima Line. Click the Refresh Maxima Line button. The local maxima plot displays the chaining across scales of the coefficients local maxima from a = 40 down to a = 1. 2-11 2 Continuous Wavelet Analysis Hold down the right mouse button over the coefficients plot. The position of the mouse is given by the Info frame (located at the bottom of the screen) in terms of location (X) and scale (Sca). 2-12 1-D Continuous Wavelet Analysis 7 Switch from scale to Pseudo-Frequency Information. Using the option button on the right part of the screen, select Frequencies instead of Scales. Again hold down the right mouse button over the coefficients plot, the position of the mouse is given in terms of location (X) and frequency (Frq) in Hertz. 2-13 2 Continuous Wavelet Analysis This facility allows you to interpret scale in terms of an associated pseudo-frequency, which depends on the wavelet and the sampling period.. 8 Deselect the last two plots using the check boxes in the Selected Axes frame. 2-14 1-D Continuous Wavelet Analysis 9 Zoom in on detail. Drag a rubber band box (by holding down the left mouse button) over the portion of the signal you want to magnify. 2-15 2 Continuous Wavelet Analysis 10 Click the X+ button (located at the bottom of the screen) to zoom horizontally only. The Continuous Wavelet 1-D tool enlarges the displayed signal and coefficients plot (for more information on zooming, see “Connection of Plots” on page A-3 in the Wavelet Toolbox User’s Guide). 2-16 1-D Continuous Wavelet Analysis As with the command line analysis on the preceding pages, you can change the scales or the analyzing wavelet and repeat the analysis. To do this, just edit the necessary fields and click the Analyze button. 11 View normal or absolute coefficients. The Continuous Wavelet 1-D tool allows you to plot either the absolute values of the wavelet coefficients, or the coefficients themselves. 2-17 2 Continuous Wavelet Analysis More generally, the coefficients coloration can be done in several different ways. For more details on the Coloration Mode, see “Controlling the Coloration Mode” on page A-8. Choose either one of the absolute modes or normal modes from the Coloration Mode menu. In normal modes, the colors are scaled between the minimum and maximum of the coefficients. In absolute modes, the colors are scaled between zero and the maximum absolute value of the coefficients. Importing and Exporting Information from the Graphical Interface The Continuous Wavelet 1-D graphical interface tool lets you import information from and export information to disk. You can • Load signals from disk into the Continuous Wavelet 1-D tool. • Save wavelet coefficients from the Continuous Wavelet 1-D tool to disk. Loading Signals into the Continuous Wavelet 1-D Tool To load a signal you’ve constructed in your MATLAB workspace into the Continuous Wavelet 1-D tool, save the signal in a MAT-file (with extension mat or other). For instance, suppose you’ve designed a signal called warma and want to analyze it in the Continuous Wavelet 1-D tool. save warma warma The workspace variable warma must be a vector. sizwarma = size(warma) sizwarma = 1 2-18 1000 1-D Continuous Wavelet Analysis To load this signal into the Continuous Wavelet 1-D tool, use the menu option File > Load Signal. A dialog box appears that lets you select the appropriate MAT-file to be loaded. Note The first one-dimensional variable encountered in the file is considered the signal. Variables are inspected in alphabetical order. Saving Wavelet Coefficients The Continuous Wavelet 1-D tool lets you save wavelet coefficients to disk. The toolbox creates a MAT-file in the current folder with the extension wc1 and a name you give it. To save the continuous wavelet coefficients from the present analysis, use the menu option File > Save > Coefficients. A dialog box appears that lets you specify a folder and filename for storing the coefficients. Consider the example analysis: File > Example Analysis > with haar at scales [1:1:64] → Cantor curve. After saving the continuous wavelet coefficients to the file cantor.wc1, load the variables into your workspace: load cantor.wc1 -mat whos Name Size Bytes Class coeff 64x2188 1120256 double array scales 1x64 512 double array wname 1x4 8 char array Variables coefs and scales contain the continuous wavelet coefficients and the associated scales. More precisely, in the above example, coefs is a 2-19 2 Continuous Wavelet Analysis 64-by-2188 matrix, one row for each scale; and scales is the 1-by-64 vector 1:64. Variable wname contains the wavelet name. 2-20 One-Dimensional Complex Continuous Wavelet Analysis One-Dimensional Complex Continuous Wavelet Analysis This section takes you through the features of complex continuous wavelet analysis using the Wavelet Toolbox software and focuses on the differences between the real and complex continuous analysis. You can refer to the section “Command Line Continuous Wavelet Analysis” on page 2-4 if you want to learn how to • Zoom in on detail • Display coefficients in normal or absolute mode • Choose the scales at which the analysis is performed • Switch from scale to pseudo-frequency information • Exchange signal and coefficient information between the disk and the graphical tools Wavelet Toolbox software requires only one function for complex continuous wavelet analysis of a real valued signal: cwt. You’ll find full information about this function in its reference page. In this section, you’ll learn how to • Load a signal • Perform a complex continuous wavelet transform of a signal • Produce plots of the coefficients Since you can perform analyses either from the command line or using the graphical interface tools, this section has subsections covering each method. 2-21 2 Continuous Wavelet Analysis Complex Continuous Analysis Using the Command Line This example involves a cusp signal. 1 Load a signal. From the MATLAB prompt, type load cuspamax; You now have the signal cuspamax in your workspace: whos 2-22 One-Dimensional Complex Continuous Wavelet Analysis Name Size Bytes Class caption 1x71 142 char array cuspamax 1x1024 8192 double array caption caption = x = linspace(0,1,1024); y = exp(-128*((x-0.3).^2))-3*(abs(x-0.7).^0.4); caption is a string that contains the signal definition. 2 Perform a Continuous Wavelet Transform. Use the cwt command. Type c = cwt(cuspamax,1:2:64,'cgau4'); The arguments to cwt specify the signal to be analyzed, the scales of the analysis, and the wavelet to be used. The returned argument c contains the coefficients at various scales. In this case, c is a complex 32-by-1024 matrix, each row of which corresponds to a single scale. 3 Plot the coefficients. The cwt command accepts a fourth argument. This is a flag that, when present, causes cwt to produce four plots related to the complex continuous wavelet transform coefficients: • Real and imaginary parts • Modulus and angle The cwt command can accept more arguments to define the different characteristics of the produced plots. For more information, see the cwt reference page. Type c = cwt(cuspamax,1:2:64,'cgau4','plot'); 2-23 2 Continuous Wavelet Analysis A plot appears: Of course, coefficient plots generated from the command line can be manipulated using ordinary MATLAB graphics commands. Complex Continuous Analysis Using the Graphical Interface We now use the Complex Continuous Wavelet 1-D tool to analyze the same cusp signal we examined using the command line interface in the previous section. 1 Start the Complex Continuous Wavelet 1-D Tool. From the MATLAB prompt, type wavemenu The Wavelet Toolbox Main Menu appears. 2-24 One-Dimensional Complex Continuous Wavelet Analysis Click the Complex Continuous Wavelet 1-D menu item. The continuous wavelet analysis tool for one-dimensional signal data appears. 2-25 2 Continuous Wavelet Analysis 2 Load a signal. Choose the File > Load Signal menu option. When the Load Signal dialog box appears, select the MAT-file cuspamax.mat, which should reside in the MATLAB folder toolbox/wavelet/wavedemo. Click the OK button. The cusp signal is loaded into the Complex Continuous Wavelet 1-D tool. The default value for the sampling period is equal to 1 (second). 3 Perform a Complex Continuous Wavelet Transform To start our analysis, let’s perform an analysis using the cgau4 wavelet at scales 1 through 64 in steps of 2, just as we did using command-line functions in “One-Dimensional Complex Continuous Wavelet Analysis” on page 2-21. In the upper-right portion of the Complex Continuous Wavelet 1-D tool, select the cgau4 wavelet and scales 1–64 in steps of 2. 2-26 One-Dimensional Complex Continuous Wavelet Analysis Click the Analyze button. After a pause for computation, the tool displays the usual plots associated to the modulus of the coefficients on the left side, and the angle of the coefficients on the right side. 2-27 2 Continuous Wavelet Analysis Each side has exactly the same representation that we found in “Continuous Analysis Using the Graphical Interface” on page 2-8. Select the plots related to the modulus of the coefficients using the Modulus option button in the Selected Axes frame. 2-28 One-Dimensional Complex Continuous Wavelet Analysis The figure now looks like the one in the real Continuous Wavelet 1-D tool. Importing and Exporting Information from the Graphical Interface To know how to import and export information from the Complex Continuous Wavelet Graphical Interface, see the corresponding paragraph in “Command Line Continuous Wavelet Analysis” on page 2-4. The only difference is that the variable coefs is a complex matrix (see “Saving Wavelet Coefficients” on page 2-19). 2-29 2 Continuous Wavelet Analysis DFT-Based Continuous Wavelet Analysis — Command Line In this section... “CWT of Sum of Disjoint Sinusoids” on page 2-30 “Approximate Scale-Frequency Conversions” on page 2-33 “Signal Reconstruction from CWT Coefficients” on page 2-37 “Signal Approximation with Modified CWT Coefficients” on page 2-38 To implement a DFT-based continuous wavelet analysis in the MATLAB command window, use cwtft and icwtft. For the mathematical basis of the DFT-based continuous wavelet analysis and synthesis see: • “DFT-Based Continuous Wavelet Transform” • “Inverse Continuous Wavelet Transform” CWT of Sum of Disjoint Sinusoids The signal is a sum of two disjoint sinusoids. The sampling frequency is 1023 Hz. The total signal duration is 1 second. The frequencies of the two sine waves are 4 and 8 Hz. The 4-Hz sine wave has support on the initial 1/2 second of the 1-second interval. The 8-Hz sine wave has support on the final 1/2 second. N = 1024; t = linspace(0,1,N); dt =1/(N-1); Y = sin(8*pi*t).*(t<=0.5) + sin(16*pi*t).*(t>0.5); Obtain the continuous wavelet transform (CWT) using the default analytic Morlet wavelet, and plot the results. sig = {Y,dt}; cwtS1 = cwtft(sig,'plot'); 2-30 DFT-Based Continuous Wavelet Analysis — Command Line The figure shows a plot of the original signal. The CWT moduli, real and imaginary parts of the CWT coefficients, and the CWT coefficient arguments (phase angles) also appear as plots. You can display the reconstructed signal by enabling the radio button at the bottom-left corner of the plot. Enabling the radio button superimposes the reconstructed signal on the original signal in the top-left corner of the figure. The relative maximum and quadratic (L2 norm) errors appear under the plot. 2-31 2 Continuous Wavelet Analysis You can customize your continuous wavelet analysis by providing additional inputs to cwtft. In the following example, specify the analyzing wavelet as the Paul wavelet of order 8. Specify the initial scale, the spacing between scales, and the number of scales. By default, the scale vector is logarithmic to the base 2. % smallest scale, spacing between scales, number of scales dt = 1/1023; s0 = 2*dt; ds = 0.5; NbSc = 20; % scale vector is % scales = s0*2.^((0:NbSc-1)*ds); wname = 'paul'; SIG = {Y,dt}; % Create SCA input as cell array SCA = {s0,ds,NbSc}; % Specify wavelet and parameters WAV = {wname,8}; 2-32 DFT-Based Continuous Wavelet Analysis — Command Line % Compute and plot the CWT cwtS2 = cwtft(SIG,'scales',SCA,'wavelet',WAV,'plot'); The oscillations at 4 and 8 Hz are clearly visible as alternating positive and negative real and imaginary parts. The 4-Hz oscillation occurs at a longer scale than the 8-Hz oscillation. In the plot of the CWT moduli, you see the transition from the 4-Hz (longer scale) sinusoid to the 8-Hz sinusoid (shorter scale) around 0.5 seconds. Approximate Scale-Frequency Conversions There is not a direct correspondence between Fourier wavelength and scale. However, you can find conversion factors for select wavelets that yield an 2-33 2 Continuous Wavelet Analysis approximate scale-frequency correspondence. You can find these factors for wavelets supported by cwtft listed on the reference page. This example shows you how to change the scale axis to an approximate frequency axis for analysis. Use the sum of disjoint sinusoids as the input signal. Set the initial scale to 6*dt, the scale increment to 0.15, and the number of scales to 50. Use the Fourier factor for the Morlet wavelet to convert the scale vector to an approximate frequency vector in hertz. Plot the result. figure; s0 = 6*dt; ds = 0.15; NbSc = 50; wname = 'morl'; SCA = {s0,ds,NbSc}; cwtsig = cwtft({Y,dt},'scales',SCA,'wavelet',wname); MorletFourierFactor = 4*pi/(6+sqrt(2+6^2)); Scales = cwtsig.scales.*MorletFourierFactor; Freq = 1./Scales; imagesc(t,[],abs(cwtsig.cfs)); indices = get(gca,'ytick'); set(gca,'yticklabel',Freq(indices)); xlabel('Time'); ylabel('Hz'); title('Time-Frequency Analysis with CWT'); 2-34 DFT-Based Continuous Wavelet Analysis — Command Line You can see the signal contains significant energy at approximately 4 Hz over the first 1/2 second. In the final 1/2 second interval, the predominant signal energy transitions higher in frequency to approximately 8 Hz. Repeat the above example using the Paul analyzing wavelet with order, m, equal to 8. Use a contour plot of the real part of the CWT to visualize the sine waves at 4 and 8-Hz. The real part exhibits oscillations in the sign of the wavelet coefficients at those frequencies. s0 = 6*dt; ds = 0.15; NbSc = 50; m = 8; % scale vector is % scales = s0*2.^((0:NbSc-1)*ds); wname = 'paul'; 2-35 2 Continuous Wavelet Analysis SIG = {Y,dt}; % Create SCA input as cell array SCA = {s0,ds,NbSc}; % Specify wavelet and parameters WAV = {wname,m}; cwtPaul = cwtft(SIG,'scales',SCA,'wavelet',WAV); scales = cwtPaul.scales; PaulFourierFactor = 4*pi/(2*m+1); Freq = 1./(PaulFourierFactor.*scales); contour(t,Freq,real(cwtPaul.cfs)); xlabel('Time'); ylabel('Hz'); colorbar; title('Real Part of CWT using Paul Wavelet (m=8)'); axis([0 1 1 15]); grid on; 2-36 DFT-Based Continuous Wavelet Analysis — Command Line Signal Reconstruction from CWT Coefficients You can use the critically sampled (decimated) and oversampled (nondecimated) discrete wavelet transforms (DWT) to achieve perfect reconstruction of the input signal from the wavelet coefficients. To obtain a time and scale-dependent approximation to a signal, you can use a possibly-modified subset of the decimated or undecimated DWT coefficients. The inversion of the CWT is not as straightforward. The simplest CWT inversion utilizes the single integral formula due to Morlet, which employs a Dirac delta function as the synthesizing wavelet. See “Inverse Continuous Wavelet Transform” for a brief mathematical motivation. icwtft and icwtlin both implement the single integral formula. Because of necessary approximations in the implementation of the single integral inverse CWT, you cannot expect to obtain perfect reconstruction. However, you can use the inverse CWT to obtain useful position and scale-dependent approximations to the input signal. Implement the inverse CWT with logarithmically-spaced scales. N = 1024; t = linspace(0,1,N); dt =1/(N-1); Y = sin(8*pi*t).*(t<=0.5) + sin(16*pi*t).*(t>0.5); dt = 1/1023; s0 = 2*dt; ds = 0.5; NbSc = 20; % scale vector is % scales = s0*2.^((0:NbSc-1)*ds); wname = 'paul'; SIG = {Y,dt}; % Create SCA input as cell array SCA = {s0,ds,NbSc}; % Specify wavelet and parameters WAV = {wname,8}; cwtS2 = cwtft(SIG,'scales',SCA,'wavelet',WAV); YR1 = icwtft(cwtS2,'plot','signal',SIG); norm(Y-YR1,2) Enable the radio button in the left corner of the figure to plot the reconstructed signal. 2-37 2 Continuous Wavelet Analysis Signal Approximation with Modified CWT Coefficients Obtain the CWT of a noisy Doppler (frequency-modulated) signal using the analytic Morlet wavelet. Reconstruct an approximation by selecting a subset of the CWT coefficients. By eliminating the smallest scales, you obtain a lowpass approximation to the signal. The lowpass approximation produces a smooth approximation to the lower-frequency features of the noisy Doppler signal. The high-frequency (small scale) features at the beginning of the signal are lost. % Define the signal load noisdopp; Y = noisdopp; N = length(Y); % Define parameters before analysis dt = 1; 2-38 DFT-Based Continuous Wavelet Analysis — Command Line s0 = wname SIG = SCA = WAV = 2*dt; ds = 0.4875; = 'morl'; {Y,dt}; {s0,ds,NbSc}; {wname,[]}; NbSc = 20; % Compute CWT analysis cwtS4 = cwtft(SIG,'scales',SCA,'wavelet',WAV); % Thresholding step building the new structure cwtS5 = cwtS4; newCFS = zeros(size(cwtS4.cfs)); newCFS(11:end,:) = cwtS4.cfs(11:end,:); cwtS5.cfs = newCFS; % Reconstruction from the modified structure YRDen = icwtft(cwtS5,'signal',SIG); plot(Y,'k-.'); hold on; plot(YRDen,'r','linewidth',3); axis tight; legend('Original Signal', 'Selective inverse CWT'); title('Signal approximation based on a subset of CWT coefficients'); 2-39 2 2-40 Continuous Wavelet Analysis Interactive DFT-Based Continuous Wavelet Analysis Interactive DFT-Based Continuous Wavelet Analysis You can use the Continuous Wavelet 1-D (Using FFT) tool to analyze the same signals examined in “DFT-Based Continuous Wavelet Analysis — Command Line” on page 2-30. 1 At the MATLAB command prompt, enter wavemenu 2 Click the Continuous Wavelet 1-D (Using FFT) menu item. 3 Choose the File > Load Data option. When the Pick a file dialog appears, select noisdopp.mat from the toolbox/wavelet/wavedemo folder. 2-41 2 Continuous Wavelet Analysis 4 Using the menu default parameters, click Analyze. 2-42 Interactive DFT-Based Continuous Wavelet Analysis 5 Reconstruct the signal based on all the default dyadic scales. Click Scales Selection. Select all scales by clicking All. Click Synthesize. 2-43 2 Continuous Wavelet Analysis In the top left, the synthesized signal plot is superimposed on the original signal. The relative maximum and L2 errors are displayed under the plot. The single integral CWT inversion does not produce perfect reconstruction, but the relative errors using the default logarithmically–spaced scales are small. 6 Obtain a signal approximation from selected scales. Click None in the Selection of Scales panel to undo the scale selection. Then, select only scale indices greater than 10 and reconstruct an approximation to the original signal. Hold the Ctrl key while selecting scale indices 11–21. The scale indices correspond to the following physical scales. dt = 1; s0 = 2*dt; ds = 0.4875; 2-44 Interactive DFT-Based Continuous Wavelet Analysis nb = 21; physical_scales = s0*pow.^((0:nb-1)*ds); 7 Click Synthesize. The reconstructed signal from scale indices 11–21 is a lowpass approximation to the noisy Doppler signal. 8 Analyze using linear scales. In the Scales drop-down menu in the upper right, select Linear default and click Analyze. 2-45 2 Continuous Wavelet Analysis Note The other options under Scales include Dyadic default and Manual. If you select Manual, a Define Scales button appears. Click Define Scales to set the parameters for your scale vector. Manual Selection of CWT Coefficients Select coefficients manually by graphically selecting the CWT coefficients. Reconstruct the signal from the selected coefficients. Click Manual Selection of Coefficients. The Select the Coefficients Manually panel appears with a single box containing all the CWT coefficient moduli. 2-46 Interactive DFT-Based Continuous Wavelet Analysis You can change the CWT coefficient view to Angle, Real, or Imaginary. To select a subset of coefficients, draw a box by left-clicking and dragging the mouse. When you release the mouse button, a semi-transparent box with a green border is superimposed on the plot. You can place multiple boxes on the same plot. To synthesize a signal based on the selected coefficients, click Synthesize. 2-47 2 Continuous Wavelet Analysis To select, unselect, or delete a box, right-click in the box. A context menu appears that allows you to select, unselect, or delete the box. After you select the coefficients within the box, the border of the box displays in green. When the coefficients within the box are not selected, the border of the box displays in red. You can move a box by clicking the left mouse button inside the box while simultaneously pressing the Shift key. The border of the box changes to yellow, and you can drag the box to the desired location. You must keep the Shift key pressed while you are moving the box. Quit the manual selection mode by clicking the Close button. In the Show synthesized signals from panel on the right, you can turn the plot of your synthesized signal on and off by checking and unchecking Manual selection. 2-48 Interactive DFT-Based Continuous Wavelet Analysis Using the File > Save > Synthesized signal menu, you can save the available synthesized signals. Using the File > Save > Decomposition menu, you can save the wavelet analysis as a MAT file. 2-49 2 2-50 Continuous Wavelet Analysis 3 Discrete Wavelet Analysis • “1-D Decimated Wavelet Transforms” on page 3-2 • “Fast Wavelet Transform (FWT) Algorithm” on page 3-37 • “Border Effects” on page 3-52 • “Discrete Stationary Wavelet Transform (SWT)” on page 3-62 • “One-Dimensional Discrete Stationary Wavelet Analysis” on page 3-69 • “One-Dimensional Multisignal Analysis” on page 3-86 • “Two-Dimensional Discrete Wavelet Analysis” on page 3-137 • “Two-Dimensional Discrete Stationary Wavelet Analysis” on page 3-166 • “Three-Dimensional Discrete Wavelet Analysis” on page 3-181 3 Discrete Wavelet Analysis 1-D Decimated Wavelet Transforms This section takes you through the features of one-dimensional critically-sampled wavelet analysis using the Wavelet Toolbox software. The toolbox provides these functions for one-dimensional signal analysis. For more information, see the reference pages. Analysis-Decomposition Functions Function Name Purpose dwt Single-level decomposition wavedec Decomposition wmaxlev Maximum wavelet decomposition level Synthesis-Reconstruction Functions Function Name Purpose idwt Single-level reconstruction waverec Full reconstruction wrcoef Selective reconstruction upcoef Single reconstruction Decomposition Structure Utilities 3-2 Function Name Purpose detcoef Extraction of detail coefficients appcoef Extraction of approximation coefficients upwlev Recomposition of decomposition structure 1-D Decimated Wavelet Transforms De-noising and Compression Function Name Purpose ddencmp Provide default values for de-noising and compression wbmpen Penalized threshold for wavelet 1-D or 2-D de-noising wdcbm Thresholds for wavelet 1-D using Birgé-Massart strategy wdencmp Wavelet de-noising and compression wden Automatic wavelet de-noising wthrmngr Threshold settings manager In this section, you’ll learn how to • Load a signal • Perform a single-level wavelet decomposition of a signal • Construct approximations and details from the coefficients • Display the approximation and detail • Regenerate a signal by inverse wavelet transform • Perform a multilevel wavelet decomposition of a signal • Extract approximation and detail coefficients • Reconstruct the level 3 approximation • Reconstruct the level 1, 2, and 3 details • Display the results of a multilevel decomposition • Reconstruct the original signal from the level 3 decomposition • Remove noise from a signal • Refine an analysis • Compress a signal • Show a signal’s statistics and histograms 3-3 3 Discrete Wavelet Analysis Since you can perform analyses either from the command line or using the graphical interface tools, this section has subsections covering each method. The final subsection discusses how to exchange signal and coefficient information between the disk and the graphical tools. One-Dimensional Analysis Using the Command Line This example involves a real-world signal — electrical consumption measured over the course of 3 days. This signal is particularly interesting because of noise introduced when a defect developed in the monitoring equipment as the measurements were being made. Wavelet analysis effectively removes the noise. 1 Load the signal and select a portion for wavelet analysis. load leleccum; s = leleccum(1:3920); l_s = length(s); 2 Perform a single-level wavelet decomposition of a signal. Perform a single-level decomposition of the signal using the db1 wavelet. [cA1,cD1] = dwt(s,'db1'); This generates the coefficients of the level 1 approximation (cA1) and detail (cD1). 3 Construct approximations and details from the coefficients. To construct the level 1 approximation and detail (A1 and D1) from the coefficients cA1 and cD1, type A1 = upcoef('a',cA1,'db1',1,l_s); D1 = upcoef('d',cD1,'db1',1,l_s); or A1 = idwt(cA1,[],'db1',l_s); D1 = idwt([],cD1,'db1',l_s); 3-4 1-D Decimated Wavelet Transforms 4 Display the approximation and detail. To display the results of the level-one decomposition, type subplot(1,2,1); plot(A1); title('Approximation A1') subplot(1,2,2); plot(D1); title('Detail D1') 5 Regenerate a signal by using the Inverse Wavelet Transform. To find the inverse transform, enter A0 = idwt(cA1,cD1,'db1',l_s); err = max(abs(s-A0)) 6 Perform a multilevel wavelet decomposition of a signal. 3-5 3 Discrete Wavelet Analysis To perform a level 3 decomposition of the signal (again using the db1 wavelet), type [C,L] = wavedec(s,3,'db1'); The coefficients of all the components of a third-level decomposition (that is, the third-level approximation and the first three levels of detail) are returned concatenated into one vector, C. Vector L gives the lengths of each component. 7 Extract approximation and detail coefficients. To extract the level 3 approximation coefficients from C, type cA3 = appcoef(C,L,'db1',3); To extract the levels 3, 2, and 1 detail coefficients from C, type cD3 = detcoef(C,L,3); cD2 = detcoef(C,L,2); cD1 = detcoef(C,L,1); or [cD1,cD2,cD3] = detcoef(C,L,[1,2,3]); 8 Reconstruct the Level 3 approximation and the Level 1, 2, and 3 details. To reconstruct the level 3 approximation from C, type A3 = wrcoef('a',C,L,'db1',3); To reconstruct the details at levels 1, 2, and 3, from C, type 3-6 1-D Decimated Wavelet Transforms D1 = wrcoef('d',C,L,'db1',1); D2 = wrcoef('d',C,L,'db1',2); D3 = wrcoef('d',C,L,'db1',3); 9 Display the results of a multilevel decomposition. To display the results of the level 3 decomposition, type subplot(2,2,1); plot(A3); title('Approximation A3') subplot(2,2,2); plot(D1); title('Detail D1') subplot(2,2,3); plot(D2); title('Detail D2') subplot(2,2,4); plot(D3); title('Detail D3') 3-7 3 Discrete Wavelet Analysis 10 Reconstruct the original signal from the Level 3 decomposition. To reconstruct the original signal from the wavelet decomposition structure, type A0 = waverec(C,L,'db1'); err = max(abs(s-A0)) 11 Crude de-noising of a signal. Using wavelets to remove noise from a signal requires identifying which component or components contain the noise, and then reconstructing the signal without those components. 3-8 1-D Decimated Wavelet Transforms In this example, we note that successive approximations become less and less noisy as more and more high-frequency information is filtered out of the signal. The level 3 approximation, A3, is quite clean as a comparison between it and the original signal. To compare the approximation to the original signal, type subplot(2,1,1);plot(s);title('Original'); axis off subplot(2,1,2);plot(A3);title('Level 3 Approximation'); axis off Of course, in discarding all the high-frequency information, we’ve also lost many of the original signal’s sharpest features. 3-9 3 Discrete Wavelet Analysis Optimal de-noising requires a more subtle approach called thresholding. This involves discarding only the portion of the details that exceeds a certain limit. 12 Remove noise by thresholding. Let’s look again at the details of our level 3 analysis. To display the details D1, D2, and D3, type subplot(3,1,1); plot(D1); title('Detail Level 1'); axis off subplot(3,1,2); plot(D2); title('Detail Level 2'); axis off subplot(3,1,3); plot(D3); title('Detail Level 3'); axis off Most of the noise occurs in the latter part of the signal, where the details show their greatest activity. What if we limited the strength of the details by restricting their maximum values? This would have the effect of cutting 3-10 1-D Decimated Wavelet Transforms back the noise while leaving the details unaffected through most of their durations. But there’s a better way. Note that cD1, cD2, and cD3 are just MATLAB vectors, so we could directly manipulate each vector, setting each element to some fraction of the vectors’ peak or average value. Then we could reconstruct new detail signals D1, D2, and D3 from the thresholded coefficients. To denoise the signal, use the ddencmp command to calculate the default parameters and the wdencmp command to perform the actual de-noising, type [thr,sorh,keepapp] = ddencmp('den','wv',s); clean = wdencmp('gbl',C,L,'db1',3,thr,sorh,keepapp); Note that wdencmp uses the results of the decomposition (C and L) that we calculated in step 6. We also specify that we used the db1 wavelet to perform the original analysis, and we specify the global thresholding option 'gbl'. See ddencmp and wdencmp in the reference pages for more information about the use of these commands. To display both the original and denoised signals, type subplot(2,1,1); plot(s(2000:3920)); title('Original') subplot(2,1,2); plot(clean(2000:3920)); title('denoised') 3-11 3 Discrete Wavelet Analysis We’ve plotted here only the noisy latter part of the signal. Notice how we’ve removed the noise without compromising the sharp detail of the original signal. This is a strength of wavelet analysis. While using command line functions to remove the noise from a signal can be cumbersome, the software’s graphical interface tools include an easy-to-use de-noising feature that includes automatic thresholding. More information on the de-noising process can be found in the following sections: • Remove noise from a signal • “Denoising and Nonparametric Function Estimation” on page 5-2 in the Wavelet Toolbox User’s Guide 3-12 1-D Decimated Wavelet Transforms • “One-Dimensional Adaptive Thresholding of Wavelet Coefficients” on page 5-17 • “One-Dimensional Wavelet Variance Adaptive Thresholding ” on page 5-14 in the Wavelet Toolbox User’s Guide One-Dimensional Analysis Using the Graphical Interface In this section, we explore the same electrical consumption signal as in the previous section, but we use the graphical interface tools to analyze the signal. 1 Start the 1-D Wavelet Analysis Tool. From the MATLAB prompt, type wavemenu 3-13 3 Discrete Wavelet Analysis The Wavelet Toolbox Main Menu appears. Click the Wavelet 1-D menu item. The discrete wavelet analysis tool for one-dimensional signal data appears. 2 Load a signal. From the File menu, choose the Load > Signal option. 3-14 1-D Decimated Wavelet Transforms When the Load Signal dialog box appears, select the MAT-file leleccum.mat, which is in the MATLAB folder toolbox/wavelet/wavedemo. Click the OK button. The electrical consumption signal is loaded into the Wavelet 1-D tool. 3 Perform a single-level wavelet decomposition. To start our analysis, let’s perform a single-level decomposition using the db1 wavelet, just as we did using the command-line functions in “One-Dimensional Analysis Using the Command Line” on page 3-4. In the upper right portion of the Wavelet 1-D tool, select the db1 wavelet and single-level decomposition. 3-15 3 Discrete Wavelet Analysis Click the Analyze button. After a pause for computation, the tool displays the decomposition. 4 Zoom in on relevant detail. One advantage of using the graphical interface tools is that you can zoom in easily on any part of the signal and examine it in greater detail. Drag a rubber band box (by holding down the left mouse button) over the portion of the signal you want to magnify. Here, we’ve selected the noisy part of the original signal. 3-16 1-D Decimated Wavelet Transforms Click the X+ button (located at the bottom of the screen) to zoom horizontally. The Wavelet 1-D tool zooms all the displayed signals. 3-17 3 Discrete Wavelet Analysis The other zoom controls do more or less what you’d expect them to. The X- button, for example, zooms out horizontally. The history function keeps track of all your views of the signal. Return to a previous zoom level by clicking the left arrow button. 5 Perform a multilevel decomposition. Again, we’ll use the graphical tools to emulate what we did in the previous section using command line functions. To perform a level 3 decomposition of the signal using the db1 wavelet: Select 3 from the Level menu at the upper right, and then click the Analyze button again. After the decomposition is performed, you’ll see a new analysis appear in the Wavelet 1-D tool. Selecting Different Views of the Decomposition The Display mode menu (middle right) lets you choose different views of the wavelet decomposition. 3-18 1-D Decimated Wavelet Transforms The default display mode is called “Full Decomposition Mode.” Other alternatives include: • “Separate Mode,” which shows the details and the approximations in separate columns. • “Superimpose Mode,” which shows the details on a single plot superimposed in different colors. The approximations are plotted similarly. • “Tree Mode,” which shows the decomposition tree, the original signal, and one additional component of your choice. Click on the decomposition tree to select the signal component you’d like to view. • “Show and Scroll Mode,” which displays three windows. The first shows the original signal superimposed on an approximation you select. The second window shows a detail you select. The third window shows the wavelet coefficients. • “Show and Scroll Mode (Stem Cfs)” is very similar to the “Show and Scroll Mode” except that it displays, in the third window, the wavelet coefficients as stem plots instead of colored blocks. You can change the default display mode on a per-session basis. Select the desired mode from the View > Default Display Mode submenu. Note The Compression and De-noising windows opened from the Wavelet 1-D tool will inherit the current coefficient visualization attribute (stems or colored blocks). Depending on which display mode you select, you may have access to additional display options through the More Display Options button. 3-19 3 Discrete Wavelet Analysis These options include the ability to suppress the display of various components, and to choose whether or not to display the original signal along with the details and approximations. 6 Remove noise from a signal. The graphical interface tools feature a de-noising option with a predefined thresholding strategy. This makes it very easy to remove noise from a signal. Bring up the de-noising tool: click the denoise button, located in the middle right of the window, underneath the Analyze button. The Wavelet 1-D De-noising window appears. While a number of options are available for fine-tuning the de-noising algorithm, we’ll accept the defaults of soft fixed form thresholding and unscaled white noise. Continue by clicking the denoise button. The denoised signal appears superimposed on the original. The tool also plots the wavelet coefficients of both signals. 3-20 1-D Decimated Wavelet Transforms Zoom in on the plot of the original and denoised signals for a closer look. Drag a rubber band box around the pertinent area, and then click the XY+ button. The denoise window magnifies your view. By default, the original signal is shown in red, and the denoised signal in yellow. 3-21 3 Discrete Wavelet Analysis Dismiss the Wavelet 1-D De-noising window: click the Close button. You cannot have the denoise and Compression windows open simultaneously, so close the Wavelet 1-D De-noising window to continue. When the Update Synthesized Signal dialog box appears, click No. If you click Yes, the Synthesized Signal is then available in the Wavelet 1-D main window. 7 Refine the analysis. The graphical tools make it easy to refine an analysis any time you want to. Up to now, we’ve looked at a level 3 analysis using db1. Let’s refine our analysis of the electrical consumption signal using the db3 wavelet at level 5. Select 5 from the Level menu at the upper right, and select the db3 from the Wavelet menu. Click the Analyze button. 8 Compress the signal. The graphical interface tools feature a compression option with automatic or manual thresholding. 3-22 1-D Decimated Wavelet Transforms Bring up the Compression window: click the Compress button, located in the middle right of the window, underneath the Analyze button. The Compression window appears. While you always have the option of choosing by level thresholding, here we’ll take advantage of the global thresholding feature for quick and easy compression. Note If you want to experiment with manual thresholding, choose the By Level thresholding option from the menu located at the top right of the Wavelet 1-D Compression window. The sliders located below this menu then control the level-dependent thresholds, indicated by yellow dotted lines running horizontally through the graphs on the left of the window. The yellow dotted lines can also be dragged directly using the left mouse button. Click the Compress button, located at the center right. After a pause for computation, the electrical consumption signal is redisplayed in red with the compressed version superimposed in yellow. Below, we’ve zoomed in to get a closer look at the noisy part of the signal. 3-23 3 Discrete Wavelet Analysis You can see that the compression process removed most of the noise, but preserved 99.99% of the energy of the signal. 9 Show the residuals. From the Wavelet 1-D Compression tool, click the Residuals button. The More on Residuals for Wavelet 1-D Compression window appears. 3-24 1-D Decimated Wavelet Transforms Displayed statistics include measures of tendency (mean, mode, median) and dispersion (range, standard deviation). In addition, the tool provides frequency-distribution diagrams (histograms and cumulative histograms), as well as time-series diagrams: autocorrelation function and spectrum. The same feature exists for the Wavelet 1-D De-noising tool. Dismiss the Wavelet 1-D Compression window: click the Close button. When the Update Synthesized Signal dialog box appears, click No. 10 Show statistics. You can view a variety of statistics about your signal and its components. From the Wavelet 1-D tool, click the Statistics button. 3-25 3 Discrete Wavelet Analysis The Wavelet 1-D Statistics window appears displaying by default statistics on the original signal. Select the synthesized signal or signal component whose statistics you want to examine. Click the appropriate option button, and then click the Show Statistics button. Here, we’ve chosen to examine the synthesized signal using 100 bins instead of 30, which is the default: 3-26 1-D Decimated Wavelet Transforms Displayed statistics include measures of tendency (mean, mode, median) and dispersion (range, standard deviation). In addition, the tool provides frequency-distribution diagrams (histograms and cumulative histograms). You can plot these histograms separately using the Histograms button from the Wavelets 1-D window. Click the Approximation option button. A menu appears from which you choose the level of the approximation you want to examine. 3-27 3 Discrete Wavelet Analysis Select Level 1 and again click the Show Statistics button. Statistics appear for the level 1 approximation. Importing and Exporting Information from the Graphical Interface The Wavelet 1-D graphical interface tool lets you import information from and export information to disk and the MATLAB workspace. Saving Information to Disk You can save synthesized signals, coefficients, and decompositions from the Wavelet 1-D tool to the disk, where the information can be manipulated and later reimported into the graphical tool. 3-28 1-D Decimated Wavelet Transforms Saving Synthesized Signals. You can process a signal in the Wavelet 1-D tool and then save the processed signal to a MAT-file (with extension mat or other). For example, load the example analysis: File > Example Analysis > Basic Signals > with db3 at level 5 → Sum of sines, and perform a compression or de-noising operation on the original signal. When you close the De-noising or Compression window, update the synthesized signal by clicking Yes in the dialog box. Then, from the Wavelet 1-D tool, select the File > Save > Signal menu option. A dialog box appears allowing you to select a folder and filename for the MAT-file. For this example, choose the name synthsig. To load the signal into your workspace, simply type load synthsig; When the synthesized signal is obtained using any thresholding method except a global one, the saved structure is whos Name Size Bytes Class synthsig 1x1000 8000 double array thrParams 1x5 580 cell array wname 1x3 6 char array The synthesized signal is given by the variable synthsig. In addition, the parameters of the de-noising or compression process are given by the wavelet name (wname) and the level dependent thresholds contained in the thrParams variable, which is a cell array of length 5 (same as the level of the decomposition). For i from 1 to 5, thrParams{i} contains the lower and upper bounds of the thresholding interval and the threshold value (since interval dependent 3-29 3 Discrete Wavelet Analysis thresholds are allowed, see “One-Dimensional Adaptive Thresholding of Wavelet Coefficients” on page 5-17). For example, for level 1, thrParams{1} ans = 1.0e+03 * 0.0010 1.0000 0.0014 When the synthesized signal is obtained using a global thresholding method, the saved structure is Name Size Bytes Class synthsig 1x1000 8000 double array valTHR 1x1 8 double array wname 1x3 6 char array where the variable valTHR contains the global threshold: valTHR valTHR = 1.2922 Saving Discrete Wavelet Transform Coefficients. The Wavelet 1-D tool lets you save the coefficients of a discrete wavelet transform (DWT) to disk. The toolbox creates a MAT-file in the current folder with a name you choose. To save the DWT coefficients from the present analysis, use the menu option File > Save > Coefficients. A dialog box appears that lets you specify a folder and filename for storing the coefficients. Consider the example analysis: 3-30 1-D Decimated Wavelet Transforms File > Example Analysis > Basic Signals > with db1 at level 5 → Cantor curve. After saving the wavelet coefficients to the file cantor.mat, load the variables into your workspace: load cantor whos Name Size Bytes Class coefs 1x2190 17520 double array longs 1x7 56 double array thrParams 0x0 0 double array wname 1x3 6 char array Variable coefs contains the discrete wavelet coefficients. More precisely, in the above example coefs is a 1-by-2190 vector of concatenated coefficients, and longs is a vector giving the lengths of each component of coefs. Variable wname contains the wavelet name and thrParams is empty since the synthesized signal does not exist. Saving Decompositions. The Wavelet 1-D tool lets you save the entire set of data from a discrete wavelet analysis to disk. The toolbox creates a MAT-file in the current folder with a name you choose, followed by the extension wa1 (wavelet analysis 1-D). Open the Wavelet 1-D tool and load the example analysis: File > Example Analysis > Basic Signals > with db3 at level 5 → Sum of sines To save the data from this analysis, use the menu option File > Save > Decomposition. A dialog box appears that lets you specify a folder and filename for storing the decomposition data. Type the name wdecex1d. 3-31 3 Discrete Wavelet Analysis After saving the decomposition data to the file wdecex1d.wa1, load the variables into your workspace: load wdecex1d.wa1 -mat whos Name Size Bytes Class coefs 1x1023 8184 double array data_name 1x6 12 char array longs 1x7 56 double array thrParams 0x0 0 double array wave_name 1x3 6 char array Note Save options are also available when performing de-noising or compression inside the Wavelet 1-D tool. In the Wavelet 1-D De-noising window, you can save denoised signal and decomposition. The same holds true for the Wavelet 1-D Compression window. This way, you can save many different trials from inside the De-noising and Compression windows without going back to the main Wavelet 1-D window during a fine-tuning process. Note When saving a synthesized signal, a decomposition or coefficients to a MAT-file, the mat file extension is not necessary. You can save approximations individually for each level or save them all at once. Export to Workspace The Wavelet 1-D tool allows you to export your 1-D wavelet analysis to the MATLAB workspace in a number of formats. For example, load the example analysis for the freqbrk signal. 3-32 1-D Decimated Wavelet Transforms After the wavelet 1-D analysis loads, select File > Export to Workspace. You have the option to • Export Signal — This option exports the synthesized signal vector. • Export Coefficients — This option exports the vector of wavelet and scaling coefficients, the bookkeeping vector, and the analyzing wavelet in a structure array. The wavelet and scaling coefficient and bookkeeping vectors are identical to the output of wavedec. • Export Decomposition — This option is identical to Export Coefficients except that it also contains the name of the analyzed signal. • Export All Approximations — This option exports a L-by-N matrix where L is the value of Level and N is the length of the input signal. Each row of the matrix is the projection onto the approximation space at the corresponding level. For example, the first row of the matrix is the projection onto the approximation space at level 1. • Export All Details — This option exports a L-by-N matrix where L is the value of Level and N is the length of the input signal. Each row of the matrix is the projection onto the detail (wavelet) space at the corresponding 3-33 3 Discrete Wavelet Analysis level. For example, the first row of the matrix is the projection onto the detail space at level 1. Loading Information into the Wavelet 1-D Tool You can load signals, coefficients, or decompositions into the graphical interface. The information you load may have been previously exported from the graphical interface, and then manipulated in the workspace, or it may have been information you generated initially from the command line. In either case, you must observe the strict file formats and data structures used by the Wavelet 1-D tool, or else errors will result when you try to load information. Loading Signals. To load a signal you’ve constructed in your MATLAB workspace into the Wavelet 1-D tool, save the signal in a MAT-file (with extension mat or other). For instance, suppose you’ve designed a signal called warma and want to analyze it in the Wavelet 1-D tool. save warma warma The workspace variable warma must be a vector. sizwarma = size(warma) sizwarma = 1 1000 3-34 1-D Decimated Wavelet Transforms To load this signal into the Wavelet 1-D tool, use the menu option File > Load > Signal. A dialog box appears that lets you select the appropriate MAT-file to be loaded. Note The first one-dimensional variable encountered in the file is considered the signal. Variables are inspected in alphabetical order. Loading Discrete Wavelet Transform Coefficients. To load discrete wavelet transform coefficients into the Wavelet 1-D tool, you must first save the appropriate data in a MAT-file, which must contain at least the two variables coefs and longs. Variable coefs must be a vector of DWT coefficients (concatenated for the various levels), and variable longs a vector specifying the length of each component of coefs, as well as the length of the original signal. After constructing or editing the appropriate data in your workspace, type save myfile coefs longs Use the File > Load > Coefficients menu option from the Wavelet 1-D tool to load the data into the graphical tool. A dialog box appears, allowing you to choose the folder and file in which your data reside. 3-35 3 Discrete Wavelet Analysis Loading Decompositions. To load discrete wavelet transform decomposition data into the Wavelet 1-D graphical interface, you must first save the appropriate data in a MAT-file (with extension wa1 or other). The MAT-file contains the following variables. Variable Status Description coefs Required Vector of concatenated DWT coefficients longs Required Vector specifying lengths of components of coefs and of the original signal wave_name Required String specifying name of wavelet used for decomposition (e.g., db3) data_name Optional String specifying name of decomposition After constructing or editing the appropriate data in your workspace, type save myfile coefs longs wave_name Use the File > Load > Decomposition menu option from the Wavelet 1-D tool to load the decomposition data into the graphical tool. A dialog box appears, allowing you to choose the folder and file in which your data reside. Note When loading a signal, a decomposition or coefficients from a MAT-file, the extension of this file is free. The mat extension is not necessary. 3-36 Fast Wavelet Transform (FWT) Algorithm Fast Wavelet Transform (FWT) Algorithm In 1988, Mallat produced a fast wavelet decomposition and reconstruction algorithm [Mal89]. The Mallat algorithm for discrete wavelet transform (DWT) is, in fact, a classical scheme in the signal processing community, known as a two-channel subband coder using conjugate quadrature filters or quadrature mirror filters (QMFs). • The decomposition algorithm starts with signal s, next calculates the coordinates of A1 and D1, and then those of A2 and D2, and so on. • The reconstruction algorithm called the inverse discrete wavelet transform (IDWT) starts from the coordinates of AJ and DJ, next calculates the coordinates of AJ–1, and then using the coordinates of AJ–1 and DJ–1 calculates those of AJ–2, and so on. This section addresses the following topics: • “Filters Used to Calculate the DWT and IDWT” on page 3-37 • “Algorithms” on page 3-40 • “Why Does Such an Algorithm Exist?” on page 3-45 • “One-Dimensional Wavelet Capabilities” on page 3-49 • “Two-Dimensional Wavelet Capabilities” on page 3-50 Filters Used to Calculate the DWT and IDWT For an orthogonal wavelet, in the multiresolution framework, we start with the scaling function φ and the wavelet function ψ. One of the fundamental relations is the twin-scale relation (dilation equation or refinement equation): 1 ⎛ x⎞ φ ⎜ ⎟ = ∑ wnφ ( x − n) 2 ⎝ 2 ⎠ n∈Z All the filters used in DWT and IDWT are intimately related to the sequence (wn)n Z 3-37 3 Discrete Wavelet Analysis Clearly if φ is compactly supported, the sequence (wn) is finite and can be viewed as a filter. The filter W, which is called the scaling filter (nonnormalized), is • Finite Impulse Response (FIR) • Of length 2N • Of sum 1 1 • Of norm 2 • A low-pass filter For example, for the db3 scaling filter, load db3 db3 db3 = 0.2352 0.5706 0.3252 -0.0955 -0.0604 0.0249 sum(db3) ans = 1.0000 norm(db3) ans = 0.7071 From filter W, we define four FIR filters, of length 2N and of norm 1, organized as follows. Filters Low-Pass High-Pass Decomposition Lo_D Hi_D Reconstruction Lo_R Hi_R The four filters are computed using the following scheme. 3-38 Fast Wavelet Transform (FWT) Algorithm where qmf is such that Hi_R and Lo_R are quadrature mirror filters (i.e., Hi_R(k) = (–1) k Lo_R(2N + 1 – k)) for k = 1, 2, ..., 2N. Note that wrev flips the filter coefficients. So Hi_D and Lo_D are also quadrature mirror filters. The computation of these filters is performed using orthfilt. Next, we illustrate these properties with the db6 wavelet. Load the Daubechies’ extremal phase scaling filter and plot the coefficients. load db6; subplot(421); stem(db6,'markerfacecolor',[0 0 1]); title('Original scaling filter'); Use orthfilt to return the analysis (decomposition) and synethsis (reconstruction) filters. Obtain the discrete Fourier transforms (DFT) of the lowpass and highpass analysis filters. Plot the modulus of the DFT. LoDFT = fft(Lo_D,64); HiDFT = fft(Hi_D,64); freq = -pi+(2*pi)/64:(2*pi)/64:pi; subplot(427); plot(freq,fftshift(abs(LoDFT))); set(gca,'xlim',[-pi,pi]); xlabel('Radians/sample'); title('DFT Modulus - Lowpass Filter') subplot(428); plot(freq,fftshift(abs(HiDFT))); set(gca,'xlim',[-pi,pi]); xlabel('Radians/sample'); title('Highpass Filter'); 3-39 3 Discrete Wavelet Analysis Algorithms Given a signal s of length N, the DWT consists of log2N stages at most. Starting from s, the first step produces two sets of coefficients: approximation coefficients cA1, and detail coefficients cD1. These vectors are obtained by convolving s with the low-pass filter Lo_D for approximation, and with the high-pass filter Hi_D for detail, followed by dyadic decimation. More precisely, the first step is 3-40 Fast Wavelet Transform (FWT) Algorithm The length of each filter is equal to 2n. If N = length (s), the signals F and G are of length N + 2n – 1, and then the coefficients cA1 and cD1 are of length ⎛ ( N − 1) ⎞ floor ⎜ + n⎟ ⎝ 2 ⎠ The next step splits the approximation coefficients cA1 in two parts using the same scheme, replacing s by cA1 and producing cA2 and cD2, and so on. So the wavelet decomposition of the signal s analyzed at level j has the following structure: [cAj, cDj, ..., cD1]. 3-41 3 Discrete Wavelet Analysis This structure contains for J = 3 the terminal nodes of the following tree. • Conversely, starting from cAj and cDj , the IDWT reconstructs cAj–1, inverting the decomposition step by inserting zeros and convolving the results with the reconstruction filters. 3-42 Fast Wavelet Transform (FWT) Algorithm • For images, a similar algorithm is possible for two-dimensional wavelets and scaling functions obtained from one-dimensional wavelets by tensorial product. This kind of two-dimensional DWT leads to a decomposition of approximation coefficients at level j in four components: the approximation at level j + 1 and the details in three orientations (horizontal, vertical, and diagonal). The following charts describe the basic decomposition and reconstruction steps for images. 3-43 3 Discrete Wavelet Analysis So, for J = 2, the two-dimensional wavelet tree has the following form. 3-44 Fast Wavelet Transform (FWT) Algorithm Finally, let us mention that, for biorthogonal wavelets, the same algorithms hold but the decomposition filters on one hand and the reconstruction filters on the other hand are obtained from two distinct scaling functions associated with two multiresolution analyses in duality. In this case, the filters for decomposition and reconstruction are, in general, of different odd lengths. This situation occurs, for example, for “splines” biorthogonal wavelets used in the toolbox. By zero-padding, the four filters can be extended in such a way that they will have the same even length. Why Does Such an Algorithm Exist? The previous paragraph describes algorithms designed for finite-length signals or images. To understand the rationale, we must consider infinite-length signals. The methods for the extension of a given finite-length signal are described in “Border Effects” on page 3-52. Let us denote h = Lo_R and g = Hi_R and focus on the one-dimensional case. We first justify how to go from level j to level j+1, for the approximation vector. This is the main step of the decomposition algorithm for the computation of the approximations. The details are calculated in the same way using the filter g instead of filter h. Let (Ak(j))k Z be the coordinates of the vector Aj: A j = ∑ Ak( j)φ j,k k and Ak(j+1) the coordinates of the vector Aj+1: A j +1 = ∑ Ak( j +1)φ j +1,k k Ak(j+1) is calculated using the formula Ak( j +1) = ∑ hn−2k An( j) n 3-45 3 Discrete Wavelet Analysis This formula resembles a convolution formula. The computation is very simple. Let us define − n An( j) h (k) = h(−k), and Fk( j +1) = ∑ hk n The sequence F(j+1) is the filtered output of the sequence A(j) by the filter h . We obtain Ak(j+1) = F2k(j+1) We have to take the even index values of F. This is downsampling. The sequence A(j+1) is the downsampled version of the sequence F(j+1). The initialization is carried out using Ak(0) = s(k), where s(k) is the signal value at time k. There are several reasons for this surprising result, all of which are linked to the multiresolution situation and to a few of the properties of the functions φj,k and ψj,k. Let us now describe some of them. 1 The family (φ0,k , k ∈ Z) is formed of orthonormal functions. As a consequence for any j, the family (φ j,k , k ∈ Z) is orthonormal. 2 The double indexed family (ψ j,k , j ∈ Z, k ∈ Z) is orthonormal. 3 For any j, the (φ j ,k , k ∈ Z) are orthogonal to (ψ j′,k , j ′ ≤ j, k ∈ Z) . 3-46 Fast Wavelet Transform (FWT) Algorithm 4 Between two successive scales, we have a fundamental relation, called the twin-scale relation. Twin-Scale Relation for φ1,0 = φ ∑ hkφ0,k k∈Z φ j +1,0 = ∑ hkφ j,k k∈Z This relation introduces the algorithm’s h filter (hn = 2ωn ) . For more information, see “Filters Used to Calculate the DWT and IDWT” on page 3-37. 5 We check that: a The coordinate of φj+1,0 on φj,k is hk and does not depend on j. b The coordinate of φj+1,n on φj,k is equal to φ j +1,n , φ j,k = hk−2n . 6 These relations supply the ingredients for the algorithm. 7 Up to now we used the filter h. The high-pass filter g is used in the twin scales relation linking the ψ and φ functions. Between two successive scales, we have the following twin-scale fundamental relation. Twin-Scale Relation Between ψ 1,0 = ∑ gkφ0,k k∈Z ψ and φ ψ j +1,0 = ∑ gkφ j,k k∈Z 8 After the decomposition step, we justify now the reconstruction algorithm by building it. Let us simplify the notation. Starting from A1 and D1, let us study A0 = A1 + Dj1. The procedure is the same to calculate A = Aj+1 + Dj+1. Let us define αn, δn, α k0 by 3-47 3 Discrete Wavelet Analysis A1 = ∑ anφ1,n D1 = ∑ δ nψ 1,n n n A0 = ∑ ak0φ0,k k Let us assess the α k0 coordinates as ak0 = A0 , φ0,k = A1 + D1 , φ0,k = A1 , φ0,k + D1 , φ0,k = ∑ an φ1,n , φ0,k + ∑ δ n ψ 1,n , φ0,k n n = ∑ an hk−2n + ∑ δ n gk−2n n n We will focus our study on the first sum ∑ n δ n gk−2n ∑ n an hk−2n ; the second sum is handled in a similar manner. The calculations are easily organized if we note that (taking k = 0 in the previous formulas, makes things simpler) ∑ an h−2n = + a−1h2 +a0 h0 + a1h−2 + a2 h−4 + n = + a−1 h2 + 0h1 + a0 h0 + 0h−1 + a1 h−2 + 0h−3 + a2 h−4 + If we transform the (α n ) sequence into a new sequence (α n ) defined by ..., α–1, 0, α0, 0, α1, 0, α2, 0, ... that is precisely a 2n = an , a 2n + 1 = 0 Then ∑ an h−2n =∑ a n h−n n n and by extension 3-48 Fast Wavelet Transform (FWT) Algorithm ∑ an hk−2n =∑ a n hk−n n n Since ak0 = ∑ a n hk− n + ∑ δn gk− n n n the reconstruction steps are: 1 Replace the α and δ sequences by upsampled versions α˜ and δ inserting zeros. 2 Filter by h and g respectively. 3 Sum the obtained sequences. One-Dimensional Wavelet Capabilities Basic One-Dimensional Objects. Signal in original time Coefficients in scale-related time Objects Description s Original signal A k, 0 ≤ k ≤ j Approximation at level k D k, 1 ≤ k ≤ j Detail at level k cAk, 1 ≤ k ≤ j Approximation coefficients at level k cDk, 1 ≤ k ≤ j Detail coefficients at level k [cAj, cDj, ..., cD1] Wavelet decomposition at level j, j ≥ 1 Analysis-Decomposition Capabilities. 3-49 3 Discrete Wavelet Analysis Purpose Input Output File Single-level decomposition s cA1, cD1 dwt Single-level decomposition cAj cAj+1, cDj+1 dwt Decomposition s [cAj, cDj, ..., cD1] wavedec Synthesis-Reconstruction Capabilities. Purpose Input Output File Single-level reconstruction cA1, cD1 s or A0 idwt Single-level reconstruction cAj+1, cDj+1 cAj idwt Full reconstruction [cAj, cDj, ..., cD1] s or A0 waverec Selective reconstruction [cAj, cDj, ..., cD1] Al, Dm wrcoef Decomposition Structure Utilities. Purpose Input Output File Extraction of detail coefficients [cAj, cDj, ..., cD1] cDk, 1 ≤ k ≤ j detcoef Extraction of approximation coefficients [cAj, cDj, ..., cD1] cAk, 0≤ k ≤ j appcoef Recomposition of the decomposition structure [cAj, cDj, ..., cD1] [cAk, cDk, ..., cD1] 1 ≤ k ≤j upwlev To illustrate command-line mode for one-dimensional capabilities, see “One-Dimensional Analysis Using the Command Line” on page 3-4. . Two-Dimensional Wavelet Capabilities Basic Two-Dimensional Objects. 3-50 Fast Wavelet Transform (FWT) Algorithm Image in original resolution Coefficients in scale-related resolution Objects Description s Original image A0 Approximation at level 0 Ak , 1 ≤ k ≤ j Approximation at level k D k, 1 ≤ k ≤ j Details at level k cAk, 1 ≤ k ≤ j Approximation coefficients at level k cDk, 1 ≤ k ≤ j Detail coefficients at level k [cAj, cDj, ..., cD1] Wavelet decomposition at level j Dk stands for ⎡⎣ Dk(h) , Dk(v) , Dk( d) ⎤⎦ , the horizontal, vertical, and diagonal details at level k. The same holds for cDk, which stands for ⎡⎣ cDk(h) , cDk(v) , cDk( d) ⎤⎦ . The two-dimensional files are the same as those for the one-dimensional case, but with a 2 appended on the end of the command. For example, idwt becomes idwt2. For more information, see “One-Dimensional Wavelet Capabilities” on page 3-49. To illustrate command-line mode for two-dimensional capabilities, see “Two-Dimensional Analysis — Command Line” on page 3-138.. 3-51 3 Discrete Wavelet Analysis Border Effects Classically, the DWT is defined for sequences with length of some power of 2, and different ways of extending samples of other sizes are needed. Methods for extending the signal include zero-padding, smooth padding, periodic extension, and boundary value replication (symmetrization). The basic algorithm for the DWT is not limited to dyadic length and is based on a simple scheme: convolution and downsampling. As usual, when a convolution is performed on finite-length signals, border distortions arise. Signal Extensions: Zero-Padding, Symmetrization, and Smooth Padding To deal with border distortions, the border should be treated differently from the other parts of the signal. Various methods are available to deal with this problem, referred to as “wavelets on the interval” (see [CohDJV93] in “References”). These interesting constructions are effective in theory but are not entirely satisfactory from a practical viewpoint. Often it is preferable to use simple schemes based on signal extension on the boundaries. This involves the computation of a few extra coefficients at each stage of the decomposition process to get a perfect reconstruction. It should be noted that extension is needed at each stage of the decomposition process. Details on the rationale of these schemes are in Chapter 8 of the book Wavelets and Filter Banks, by Strang and Nguyen (see [StrN96] in “References”). The available signal extension modes are as follows (see dwtmode): • Zero-padding ('zpd'): This method is used in the version of the DWT given in the previous sections and assumes that the signal is zero outside the original support. The disadvantage of zero-padding is that discontinuities are artificially created at the border. 3-52 Border Effects • Symmetrization ('sym'): This method assumes that signals or images can be recovered outside their original support by symmetric boundary value replication. It is the default mode of the wavelet transform in the toolbox. Symmetrization has the disadvantage of artificially creating discontinuities of the first derivative at the border, but this method works well in general for images. • Smooth padding of order 1 ('spd'or 'sp1'): This method assumes that signals or images can be recovered outside their original support by a simple first-order derivative extrapolation: padding using a linear extension fit to the first two and last two values. Smooth padding works well in general for smooth signals. • Smooth padding of order 0 ('sp0'): This method assumes that signals or images can be recovered outside their original support by a simple constant extrapolation. For a signal extension this is the repetition of the first value on the left and last value on the right. • Periodic-padding (1) ('ppd'): This method assumes that signals or images can be recovered outside their original support by periodic extension. The disadvantage of periodic padding is that discontinuities are artificially created at the border. The DWT associated with these five modes is slightly redundant. But IDWT ensures a perfect reconstruction for any of the five previous modes whatever the extension mode used for DWT. • Periodic-padding (2) ('per'): If the signal length is odd, the signal is first extended by adding an extra-sample equal to the last value on the right. Then a minimal periodic extension is performed on each side. The same kind of rule exists for images. This extension mode is used for SWT (1-D & 2-D). This last mode produces the smallest length wavelet decomposition. But the extension mode used for IDWT must be the same to ensure a perfect reconstruction. 3-53 3 Discrete Wavelet Analysis Before looking at an illustrative example, let us compare some properties of the theoretical Discrete Wavelet Transform versus the actual DWT. The theoretical DWT is applied to signals that are defined on an infinite length time interval (Z). For an orthogonal wavelet, this transform has the following desirable properties: 1 Norm preservation Let cA and cD be the approximation and detail of the DWT coefficients of an infinite length signal X. Then the l2–norm is preserved: X 2 = cA 2 + cD 2 2 Orthogonality Let A and D be the reconstructed approximation and detail. Then, A and D are orthogonal and X 2 = A 2 + D 2 3 Perfect reconstruction X=A+D Since the DWT is applied to signals that are defined on a finite-length time interval, extension is needed for the decomposition, and truncation is necessary for reconstruction. To ensure the crucial property 3 (perfect reconstruction) for arbitrary choices of • The signal length • The wavelet • The extension mode the properties 1 and 2 can be lost. These properties hold true for an extended signal of length usually larger than the length of the original signal. So only the perfect reconstruction property is always preserved. Nevertheless if the 3-54 Border Effects DWT is performed using the periodic extension mode (’per’) and if the length of the signal is divisible by 2J, where J is the maximum level decomposition, the properties 1, 2, and 3 remain true. It is interesting to notice that if arbitrary extension is used, and decomposition performed using the convolution-downsampling scheme, perfect reconstruction is recovered using idwt or idwt2. This point is illustrated below. % Set initial signal and get filters. x = sin(0.3*[1:451]); w = 'db9'; [Lo_D,Hi_D,Lo_R,Hi_R] = wfilters(w); % In fact using a slightly redundant scheme, any signal % extension strategy works well. % For example use random padding. 3-55 3 Discrete Wavelet Analysis lx = length(x); lf = length(Lo_D); ex = [randn(1,lf) x randn(1,lf)]; axis([1 lx+2*lf -2 3]) subplot(211), plot(lf+1:lf+lx,x), title('Original signal') axis([1 lx+2*lf -2 3]) subplot(212), plot(ex), title('Extended signal') axis([1 lx+2*lf -2 3]) % Decomposition. la = floor((lx+lf-1)/2); ar = wkeep(dyaddown(conv(ex,Lo_D)),la); dr = wkeep(dyaddown(conv(ex,Hi_D)),la); % Reconstruction. xr = idwt(ar,dr,w,lx); 3-56 Border Effects % Check perfect reconstruction. err0 = max(abs(x-xr)) Now let us illustrate the differences between the first three methods both for 1-D and 2-D signals. Zero-Padding Using the GUI we will examine the effects of zero-padding. 1 From the MATLAB prompt, type dwtmode('zpd') 2 From the MATLAB prompt, type wavemenu. The Wavelet Toolbox Main Menu appears. 3 Click the Wavelet 1-D menu item.The discrete wavelet analysis tool for one-dimensional signal data appears. 4 From the File menu, choose the Example Analysis option and select Basic Signals > with db2 at level 5 > Two nearby discontinuities. 5 Select Display Mode: Show and Scroll. The detail coefficients clearly show the signal end effects. Symmetric Extension 6 From the MATLAB prompt, type dwtmode('sym') 3-57 3 Discrete Wavelet Analysis 7 Click the Wavelet 1-D menu item. The discrete wavelet analysis tool for one-dimensional signal data appears. 8 From the File menu, choose the Example Analysis option and select Basic Signals > with db2 at level 5 > Two nearby discontinuities. 9 From the MATLAB prompt, type dwtmode('spd') 10 Click the Wavelet 1-D menu item. The discrete wavelet analysis tool for one-dimensional signal data appears. 11 From the File menu, choose the Example Analysis option and select Basic Signals > with db2 at level 5 > Two nearby discontinuities. 12 Select Display Mode: Show and Scroll. The detail coefficients show the signal end effects are not present, and the discontinuities are well detected. Let us now consider an image example. Original Image 1 From the MATLAB prompt, type load geometry; % X contains the loaded image and % map contains the loaded colormap. nbcol = size(map,1); 3-58 Border Effects colormap(pink(nbcol)); image(wcodemat(X,nbcol)); Zero-Padding Now we set the extension mode to zero-padding and perform a decomposition of the image to level 3 using the sym4 wavelet. Then we reconstruct the approximation of level 3. 2 From the MATLAB prompt, type lev = 3; wname = 'sym4'; dwtmode('zpd') [c,s] = wavedec2(X,lev,wname); a = wrcoef2('a',c,s,wname,lev); image(wcodemat(a,nbcol)); 3-59 3 Discrete Wavelet Analysis Symmetric Extension Now we set the extension mode to symmetric extension and perform a decomposition of the image again to level 3 using the sym4 wavelet. Then we reconstruct the approximation of level 3. 3 From the MATLAB prompt, type dwtmode('sym') [c,s] = wavedec2(X,lev,wname); a = wrcoef2('a',c,s,wname,lev); image(wcodemat(a,nbcol)); Smooth Padding 3-60 Border Effects Now set the extension mode to smooth padding and perform a decomposition of the image again to level 3 using the sym4 wavelet. Then reconstruct the approximation of level 3. 4 From the MATLAB prompt, type dwtmode('spd') [c,s] = wavedec2(X,lev,wname); a = wrcoef2('a',c,s,wname,lev); image(wcodemat(a,nbcol)); 3-61 3 Discrete Wavelet Analysis Discrete Stationary Wavelet Transform (SWT) We know that the classical DWT suffers a drawback: the DWT is not a time-invariant transform. This means that, even with periodic signal extension, the DWT of a translated version of a signal X is not, in general, the translated version of the DWT of X. How to restore the translation invariance, which is a desirable property lost by the classical DWT? The idea is to average some slightly different DWT, called ε-decimated DWT, to define the stationary wavelet transform (SWT). This property is useful for several applications such as breakdown points detection. The main application of the SWT is de-noising. For more information on the rationale, see [CoiD95] in “References”. For examples, see “One-Dimensional Discrete Stationary Wavelet Analysis” on page 3-69and “Two-Dimensional Discrete Stationary Wavelet Analysis” on page 3-166. . The principle is to average several denoised signals. Each of them is obtained using the usual de-noising scheme (see “Denoising and Nonparametric Function Estimation” on page 5-2), but applied to the coefficients of an ε-decimated DWT. Note We define the SWT only for signals of length divisible by 2J, where J is the maximum decomposition level, and we use the DWT with periodic (per) extension. ε -Decimated DWT What is an ε-decimated DWT? There exist a lot of slightly different ways to handle the discrete wavelet transform. Let us recall that the DWT basic computational step is a convolution followed by a decimation. The decimation retains even indexed elements. But the decimation could be carried out by choosing odd indexed elements instead of even indexed elements. This choice concerns every step of the decomposition process, so at every level we chose odd or even. 3-62 Discrete Stationary Wavelet Transform (SWT) If we perform all the different possible decompositions of the original signal, we have 2J different decompositions, for a given maximum level J. Let us denote by εj = 1 or 0 the choice of odd or even indexed elements at step j. Every decomposition is labeled by a sequence of 0s and 1s: ε = ε1...,εJ. This transform is called the ε-decimated DWT. You can obtain the basis vectors of the ε-decimated DWT from those of the standard DWT by applying a shift and corresponds to a special choice of the origin of the basis functions. How to Calculate the ε -Decimated DWT: SWT It is possible to calculate all the ε-decimated DWT for a given signal of length N, by computing the approximation and detail coefficients for every possible sequence ε. Do this using iteratively, a slightly modified version of the basic step of the DWT of the form: [A,D] = dwt(X,wname,'mode','per','shift',e); The last two arguments specify the way to perform the decimation step. This is the classical one for e = 0, but for e = 1 the odd indexed elements are retained by the decimation. Of course, this is not a good way to calculate all the ε-decimated DWT, because many computations are performed many times. We shall now describe another way, which is the stationary wavelet transform (SWT). The SWT algorithm is very simple and is close to the DWT one. More precisely, for level 1, all the ε-decimated DWT (only two at this level) for a given signal can be obtained by convolving the signal with the appropriate filters as in the DWT case but without downsampling. Then the approximation and detail coefficients at level 1 are both of size N, which is the signal length. This can be visualized in the following figure. 3-63 3 Discrete Wavelet Analysis The general step j convolves the approximation coefficients at level j–1, with upsampled versions of the appropriate original filters, to produce the approximation and detail coefficients at level j. This can be visualized in the following figure. Next, we illustrate how to extract a given ε-decimated DWT from the approximation and detail coefficients structure of the SWT. 3-64 Discrete Stationary Wavelet Transform (SWT) We decompose a sequence of height numbers with the SWT, at level J = 3, using an orthogonal wavelet. The function swt calculates successively the following arrays, where A(j,ε1,...,εj) or D(j,ε1,...,εj) denotes an approximation or a detail coefficient at level j obtained for the ε-decimated DWT characterized by ε = [ε1,...,εj]. Step 0 (Original Data). A(0) A(0) A(0) A(0) A(0) A(0) A(0) A(0) Step 1. D(1,0) D(1,1) D(1,0) D(1,1) D(1,0) D(1,1) D(1,0) D(1,1) A(1,0) A(1,1) A(1,0) A(1,1) A(1,0) A(1,1) A(1,0) A(1,1) Step 2. D(1,0) D(1,1) D(1,0) D(1,1) D(1,0) D(1,1) D(1,0) D(1,1) D(2,0,0) D(2,1,0) D(2,0,1) D(2,1,1) D(2,0,0) D(2,1,0) D(2,0,1) D(2,1,1) A(2,0,0) A(2,1,0) A(2,0,1) A(2,1,1) A(2,0,0) A(2,1,0) A(2,0,1) A(2,1,1) Step 3. D(1,0) D(1,1) D(1,0) D(1,1) D(1,0) D(1,1) D(1,0) D(1,1) D(2,0,0) D(2,1,0) D(2,0,1) D(2,1,1) D(2,0,0) D(2,1,0) D(2,0,1) D(2,1,1) D(3,0,0,0) D(3,1,0,0) D(3,0,1,0) D(3,1,1,0) D(3,0,0,1) D(3,1,0,1) D(3,0,1,1) D(3,1,1,1) A(3,0,0,0) A(3,1,0,0) A(3,0,1,0) A(3,1,1,0) A(3,0,0,1) A(3,1,0,1) A(3,1,1,1) A(3,0,1,1) Let j denote the current level, where j is also the current step of the algorithm. Then we have the following abstract relations with εi = 0 or 1: [tmpAPP,tmpDET] = dwt(A(j, 1, , j),wname,'mode','per','shift', j+1); A(j+1, 1, , j, j+1) = wshift('1D',tmpAPP, j+1); D(j+1, 1, , j, j+1) = wshift('1D',tmpDET, j+1); 3-65 3 Discrete Wavelet Analysis where wshift performs a ε-circular shift of the input vector. Therefore, if εj+1 = 0, the wshift instruction is ineffective and can be suppressed. Let ε = [ε1,...,εJ] with εi = 0 or 1. We have 2J = 23 = eight different ε-decimated DWTs at level 3. Choosing ε, we can retrieve the corresponding ε-decimated DWT from the SWT array. Now, consider the last step, J = 3, and let [Cε,Lε] denote the wavelet decomposition structure of an ε-decimated DWT for a given ε. Then, it can be retrieved from the SWT decomposition structure by selecting the appropriate coefficients as follows: Cε = A(3, ε1, ε2, ε3) D(3, ε1, ε2, ε3) D(2, ε1, ε2) D(2, ε1, ε2) D(1, ε1) D(1, ε1) D(1, ε1) D(1, ε1) Lε = [1,1,2,4,8] For example, the ε-decimated DWT corresponding to ε = [ε1, ε2, ε3] = [1,0,1] is shown in bold in the sequence of arrays of the previous example. This can be extended to the 2-D case. The algorithm for the stationary wavelet transform for images is visualized in the following figure. 3-66 Discrete Stationary Wavelet Transform (SWT) Inverse Discrete Stationary Wavelet Transform (ISWT) Each ε-decimated DWT corresponding to a given ε can be inverted. 3-67 3 Discrete Wavelet Analysis To reconstruct the original signal using a given ε-decimated DWT characterized by [ε1,...,εJ], we can use the abstract algorithm FOR j = J:-1:1 A(j-1, 1, , j-1) = ... idwt(A(j, 1, , j),D(S, 1, , END j)],wname,'mode','per','shift', j); For each choice of ε = (ε1,...,εJ), we obtain the original signal A(0), starting from slightly different decompositions, and capturing in different ways the main features of the analyzed signal. The idea of the inverse discrete stationary wavelet transform is to average the inverses obtained for every ε-decimated DWT. This can be done recursively, starting from level J down to level 1. The ISWT is obtained with the following abstract algorithm: FOR j = J:-1:1 X0 = idwt(A(j, 1, , j),D(j, 1, , j)],wname, ... 'mode','per','shift',0); X1 = idwt(A(j, 1, , j),D(j, 1, , j)],wname, ... 'mode','per','shift',1); X1 = wshift('1D',X1,1); A(j-1, 1, , j-1) = (X0+X1)/2; END Along the same lines, this can be extended to the 2-D case. More About SWT Some useful references for the Stationary Wavelet Transform (SWT) are [CoiD95], [NasS95], and [PesKC96] in “References”. 3-68 One-Dimensional Discrete Stationary Wavelet Analysis One-Dimensional Discrete Stationary Wavelet Analysis This section takes you through the features of one-dimensional discrete stationary wavelet analysis using the Wavelet Toolbox software. For more information see “Discrete Stationary Wavelet Transform (SWT)” on page 3-62 in the Wavelet Toolbox User’s Guide. The toolbox provides these functions for one-dimensional discrete stationary wavelet analysis. For more information on the functions, see the reference pages. Analysis-Decomposition Functions Function Name Purpose swt Decomposition Synthesis-Reconstruction Functions Function Name Purpose iswt Reconstruction The stationary wavelet decomposition structure is more tractable than the wavelet one. So the utilities, useful for the wavelet case, are not necessary for the stationary wavelet transform (SWT). In this section, you’ll learn to • Load a signal • Perform a stationary wavelet decomposition of a signal • Construct approximations and details from the coefficients • Display the approximation and detail at level 1 • Regenerate a signal by using inverse stationary wavelet transform • Perform a multilevel stationary wavelet decomposition of a signal • Reconstruct the level 3 approximation 3-69 3 Discrete Wavelet Analysis • Reconstruct the level 1, 2, and 3 details • Reconstruct the level 1 and 2 approximations • Display the results of a decomposition • Reconstruct the original signal from the level 3 decomposition • Remove noise from a signal Since you can perform analyses either from the command line or using the graphical interface tools, this section has subsections covering each method. The final subsection discusses how to exchange signal and coefficient information between the disk and the graphical tools. One-Dimensional Analysis Using the Command Line This example involves a noisy Doppler test signal. 1 Load a signal. From the MATLAB prompt, type load noisdopp 2 Set the variables. Type s = noisdopp; For the SWT, if a decomposition at level k is needed, 2^k must divide evenly into the length of the signal. If your original signal does not have the correct length, you can use the Signal Extension GUI tool or the wextend function to extend it. 3 Perform a single-level Stationary Wavelet Decomposition. Perform a single-level decomposition of the signal using the db1 wavelet. Type [swa,swd] = swt(s,1,'db1'); 3-70 One-Dimensional Discrete Stationary Wavelet Analysis This generates the coefficients of the level 1 approximation (swa) and detail (swd). Both are of the same length as the signal. Type whos Name Size Bytes Class noisdopp 1x1024 8192 double array s 1x1024 8192 double array swa 1x1024 8192 double array swd 1x1024 8192 double array 4 Display the coefficients of approximation and detail. To display the coefficients of approximation and detail at level 1, type subplot(1,2,1), plot(swa); title('Approximation cfs') subplot(1,2,2), plot(swd); title('Detail cfs') 5 Regenerate the signal by Inverse Stationary Wavelet Transform. To find the inverse transform, type 3-71 3 Discrete Wavelet Analysis A0 = iswt(swa,swd,'db1'); To check the perfect reconstruction, type err = norm(s-A0) err = 2.1450e-14 6 Construct and display approximation and detail from the coefficients. To construct the level 1 approximation and detail (A1 and D1) from the coefficients swa and swd, type nulcfs = zeros(size(swa)); A1 = iswt(swa,nulcfs,'db1'); D1 = iswt(nulcfs,swd,'db1'); To display the approximation and detail at level 1, type subplot(1,2,1), plot(A1); title('Approximation A1'); subplot(1,2,2), plot(D1); title('Detail D1'); 3-72 One-Dimensional Discrete Stationary Wavelet Analysis 7 Perform a multilevel Stationary Wavelet Decomposition. To perform a decomposition at level 3 of the signal (again using the db1 wavelet), type [swa,swd] = swt(s,3,'db1'); This generates the coefficients of the approximations at levels 1, 2, and 3 (swa) and the coefficients of the details (swd). Observe that the rows of swa and swd are the same length as the signal length. Type clear A0 A1 D1 err nulcfs whos 3-73 3 Discrete Wavelet Analysis Name Size Bytes Class noisdopp 1x1024 8192 double array s 1x1024 8192 double array swa 3x1024 24576 double array swd 3x1024 24576 double array 8 Display the coefficients of approximations and details. To display the coefficients of approximations and details, type kp = 0; for i = 1:3 subplot(3,2,kp+1), plot(swa(i,:)); title(['Approx. cfs level ',num2str(i)]) subplot(3,2,kp+2), plot(swd(i,:)); title(['Detail cfs level ',num2str(i)]) kp = kp + 2; end 3-74 One-Dimensional Discrete Stationary Wavelet Analysis 9 Reconstruct approximation at Level 3 From coefficients. To reconstruct the approximation at level 3, type mzero = zeros(size(swd)); A = mzero; A(3,:) = iswt(swa,mzero,'db1'); 10 Reconstruct details from coefficients. To reconstruct the details at levels 1, 2 and 3, type D = mzero; for i = 1:3 3-75 3 Discrete Wavelet Analysis swcfs = mzero; swcfs(i,:) = swd(i,:); D(i,:) = iswt(mzero,swcfs,'db1'); end 11 Reconstruct and display approximations at Levels 1 and 2 from approximation at Level 3 and details at Levels 2 and 3. To reconstruct the approximations at levels 2 and 3, type A(2,:) = A(3,:) + D(3,:); A(1,:) = A(2,:) + D(2,:); To display the approximations and details at levels 1, 2 and 3, type kp = 0; for i = 1:3 subplot(3,2,kp+1), plot(A(i,:)); title(['Approx. level ',num2str(i)]) subplot(3,2,kp+2), plot(D(i,:)); title(['Detail level ',num2str(i)]) kp = kp + 2; end 3-76 One-Dimensional Discrete Stationary Wavelet Analysis 12 Remove noise by thresholding. To denoise the signal, use the ddencmp command to calculate a default global threshold. Use the wthresh command to perform the actual thresholding of the detail coefficients, and then use the iswt command to obtain the denoised signal. Note All methods for choosing thresholds in the 1-D Discrete Wavelet Transform case are also valid for the 1-D Stationary Wavelet Transform, which are also those used by the GUI tools. This is also true for the 2-D transforms. 3-77 3 Discrete Wavelet Analysis [thr,sorh] = ddencmp('den','wv',s); dswd = wthresh(swd,sorh,thr); clean = iswt(swa,dswd,'db1'); To display both the original and denoised signals, type subplot(2,1,1), title('Original subplot(2,1,2), title('denoised plot(s); signal') plot(clean); signal') The obtained signal remains a little bit noisy. The result can be improved by considering the decomposition of s at level 5 instead of level 3, and repeating steps 14 and 15. To improve the previous de-noising, type 3-78 One-Dimensional Discrete Stationary Wavelet Analysis [swa,swd] = swt(s,5,'db1'); [thr,sorh] = ddencmp('den','wv',s); dswd = wthresh(swd,sorh,thr); clean = iswt(swa,dswd,'db1'); subplot(2,1,1), plot(s); title('Original signal') subplot(2,1,2), plot(clean); title('denoised signal') A second syntax can be used for the swt and iswt functions, giving the same results: lev = 5; swc = swt(s,lev,'db1'); swcden = swc; swcden(1:end-1,:) = wthresh(swcden(1:end-1,:),sorh,thr); clean = iswt(swcden,'db1'); 3-79 3 Discrete Wavelet Analysis You can obtain the same plot by using the same plot commands as in step 16 above. Interactive 1-D Stationary Wavelet Transform Denoising Now we explore a strategy to denoise signals, based on the one-dimensional stationary wavelet analysis using the graphical interface tools. The basic idea is to average many slightly different discrete wavelet analyses. 1 Start the Stationary Wavelet Transform De-Noising 1-D Tool. From the MATLAB prompt, type wavemenu The Wavelet Toolbox Main Menu appears. 3-80 One-Dimensional Discrete Stationary Wavelet Analysis Click the SWT De-noising 1-D menu item. The discrete stationary wavelet transform de-noising tool for one-dimensional signals appears. 2 Load data. From the File menu, choose the Load Signal option. When the Load Signal dialog box appears, select the MAT-file noisbloc.mat, which should reside in the MATLAB folder toolbox/wavelet/wavedemo. Click the OK button. The noisy blocks signal is loaded into the SWT De-noising 1-D tool. 3-81 3 Discrete Wavelet Analysis 3 Perform a Stationary Wavelet Decomposition. Select the db1 wavelet from the Wavelet menu and select 5 from the Level menu, and then click the Decompose Signal button. After a pause for computation, the tool displays the stationary wavelet approximation and detail coefficients of the decomposition. These are also called nondecimated coefficients since they are obtained using the same scheme as for the DWT, but omitting the decimation step (see “Fast Wavelet Transform (FWT) Algorithm” on page 3-37 in the Wavelet Toolbox User’s Guide). 4 denoise the signal using the Stationary Wavelet Transform. While a number of options are available for fine-tuning the de-noising algorithm, we’ll accept the defaults of fixed form soft thresholding and 3-82 One-Dimensional Discrete Stationary Wavelet Analysis unscaled white noise. The sliders located on the right part of the window control the level-dependent thresholds, indicated by yellow dotted lines running horizontally through the graphs of the detail coefficients to the left of the window. The yellow dotted lines can also be dragged directly using the left mouse button over the graphs. Note that the approximation coefficients are not thresholded. Click the denoise button. The result is quite satisfactory, but seems to be oversmoothed around the discontinuities of the signal. This can be seen by looking at the residuals, and zooming on a breakdown point, for example around position 800. 3-83 3 Discrete Wavelet Analysis Selecting a Thresholding Method. Select hard for the thresholding mode instead of soft, and then click the denoise button. The result is of good quality and the residuals look like a white noise sample. To investigate this last point, you can get more information on residuals by clicking the Residuals button. Importing and Exporting from the GUI The tool lets you save the denoised signal to disk. The toolbox creates a MAT-file in the current folder with a name of your choice. To save the above denoised signal, use the menu option File > Save denoised Signal. A dialog box appears that lets you specify a folder and filename for storing the signal. Type the name dnoibloc. After saving the signal data to the file dnoibloc.mat, load the variables into your workspace: load dnoibloc whos 3-84 Name Size Bytes Class dnoibloc 1x1024 8192 double array thrParams 1x5 580 cell array wname 1x3 6 char array One-Dimensional Discrete Stationary Wavelet Analysis The denoised signal is given by dnoibloc. In addition, the parameters of the de-noising process are available. The wavelet name is contained in wname: wname wname = db1 and the level dependent thresholds are encoded in thrParams, which is a cell array of length 5 (the level of the decomposition). For i from 1 to 5, thrParams{i} contains the lower and upper bounds of the interval of thresholding and the threshold value (since interval dependent thresholds are allowed). For more information, see “One-Dimensional Adaptive Thresholding of Wavelet Coefficients” on page 5-17. For example, for level 1, thrParams{1} ans = 1.0e+03 * 0.0010 1.0240 0.0041 Here the lower bound is 1, the upper bound is 1024, and the threshold value is 4.1. The total time-interval is not segmented and the procedure does not use the interval dependent thresholds. 3-85 3 Discrete Wavelet Analysis One-Dimensional Multisignal Analysis This section takes you through the features of one-dimensional multisignal wavelet analysis, compression and denoising using the Wavelet Toolbox software. The rationale for each topic is the same as in the 1-D single signal case. The toolbox provides the following functions for multisignal analysis. Analysis-Decomposition and Synthesis-Reconstruction Functions Function Name Purpose mdwtdec Multisignal wavelet decomposition mdwtrec Multisignal wavelet reconstruction and extraction of approximation and detail coefficients Decomposition Structure Utilities Function Name Purpose chgwdeccfs Change multisignal 1-D decomposition coefficients wdecenergy Multisignal 1-D decomposition energy repartition Compression and Denoising Functions 3-86 Function Name Purpose mswcmp Multisignal 1-D compression using wavelets mswcmpscr Multisignal 1-D wavelet compression scores mswcmptp Multisignal 1-D compression thresholds and performance mswden Multisignal 1-D denoising using wavelets mswthresh Perform multisignal 1-D thresholding One-Dimensional Multisignal Analysis You can perform analyses from the MATLAB command line or by using the graphical interface tools. This section describes each method. The last section discusses how to exchange signal and coefficient information between the disk and the graphical tools. One-Dimensional Multisignal Analysis — Command Line 1 Load a file, from the MATLAB prompt, by typing load thinker The file thinker.mat contains a single variable X. Use whos to show information about X. whos Name Size Bytes Class X 192x96 147456 double array 2 Plot some signals. figure; plot(X(1:5,:)','r'); hold on plot(X(21:25,:)','b'); plot(X(31:35,:)','g') set(gca,'Xlim',[1,96]) grid 3-87 3 Discrete Wavelet Analysis 3 Perform a wavelet decomposition of signals at level 2 of row signals using the db2 wavelet. dec = mdwtdec('r',X,2,'db2') This generates the decomposition structure dec: dec = dirDec: level: wname: dwtFilters: dwtEXTM: dwtShift: dataSize: ca: cd: 'r' 2 'db2' [1x1 struct] 'sym' 0 [192 96] [192x26 double] {[192x49 double] [192x26 double]} 4 Change wavelet coefficients. For each signal change the wavelet coefficients by setting all the coefficients of the detail of level 1 to zero. decBIS = chgwdeccfs(dec,'cd',0,1); This generates a new decomposition structure decBIS. 3-88 One-Dimensional Multisignal Analysis 5 Perform a wavelet reconstruction of signals and plot some of the new signals. Xbis = mdwtrec(decBIS); figure; plot(Xbis(1:5,:)','r'); hold on plot(Xbis(21:25,:)','b'); plot(Xbis(31:35,:)','g') grid; set(gca,'Xlim',[1,96]) Compare old and new signals by plotting them together. figure; idxSIG = [1 31]; plot(X(idxSIG,:)','r','linewidth',2); hold on plot(Xbis(idxSIG,:)','b','linewidth',2); 3-89 3 Discrete Wavelet Analysis grid; set(gca,'Xlim',[1,96]) 6 Set the wavelet coefficients at level 1 and 2 for signals 31 to 35 to the value zero, perform a wavelet reconstruction of signal 31, and compare some of the old and new signals. decTER = chgwdeccfs(dec,'cd',0,1:2,31:35); Y = mdwtrec(decTER,'a',0,31); figure; plot(X([1 31],:)','r','linewidth',2); hold on plot([Xbis(1,:) ; Y]','b','linewidth',2); grid; set(gca,'Xlim',[1,96]) 3-90 One-Dimensional Multisignal Analysis 7 Compute the energy of signals and the percentage of energy for wavelet components. [E,PEC,PECFS] = wdecenergy(dec); Energy of signals 1 and 31: Ener_1_31 = E([1 31]) Ener_1_31 = 1.0e+006 * 3.7534 2.2411 8 Compute the percentage of energy for wavelet components of signals 1 and 31. PEC_1_31 = PEC([1 31],:) PEC_1_31 = 99.7760 99.3850 0.1718 0.2926 0.0522 0.3225 The first column shows the percentage of energy for approximations at level 2. Columns 2 and 3 show the percentage of energy for details at level 2 and 1, respectively. 9 Display the percentage of energy for wavelet coefficients of signals 1 and 31. As we can see in the dec structure, there are 26 coefficients for the approximation and the detail at level 2, and 49 coefficients for the detail at level 1. PECFS_1 = PECFS(1,:); PECFS_31 = PECFS(31,:); figure; plot(PECFS_1,'r','linewidth',2); hold on plot(PECFS_31,'b','linewidth',2); grid; set(gca,'Xlim',[1,size(PECFS,2)]) 3-91 3 Discrete Wavelet Analysis 10 Compress the signals to obtain a percentage of zeros near 95% for the wavelet coefficients. [XC,decCMP,THRESH] = mswcmp('cmp',dec,'N0_perf',95); [Ecmp,PECcmp,PECFScmp] = wdecenergy(decCMP); Plot the original signals 1 and 31, and the corresponding compressed signals. figure; plot(X([1 31],:)','r','linewidth',2); plot(XC([1 31],:)','b','linewidth',2); grid; set(gca,'Xlim',[1,96]) 3-92 hold on One-Dimensional Multisignal Analysis Compute thresholds, percentage of energy preserved and percentage of zeros associated with the L2_perf method preserving at least 95% of energy. [THR_VAL,L2_Perf,N0_Perf] = mswcmptp(dec,'L2_perf',95); idxSIG = [1,31]; Thr = THR_VAL(idxSIG) Thr = 256.1914 158.6085 L2per = L2_Perf(idxSIG) L2per = 96.5488 94.7197 N0per = N0_Perf(idxSIG) N0per = 79.2079 86.1386 Compress the signals to obtain a percentage of zeros near 60% for the wavelet coefficients. [XC,decCMP,THRESH] = mswcmp('cmp',dec,'N0_perf',60); XC signals are the compressed versions of the original signals in the row direction. Compress the XC signals in the column direction XX = mswcmp('cmpsig','c',XC,'db2',2,'N0_perf',60); Plot original signals X and the compressed signals XX as images. figure; subplot(1,2,1); image(X) subplot(1,2,2); image(XX) colormap(pink(222)) 3-93 3 Discrete Wavelet Analysis 11 Denoise the signals using the universal threshold: [XD,decDEN,THRESH] = mswden('den',dec,'sqtwolog','sln'); figure; plot(X([1 31],:)','r','linewidth',2); hold on plot(XD([1 31],:)','b','linewidth',2); grid; set(gca,'Xlim',[1,96]) XD signals are the denoised versions of the original signals in the row direction. Denoise the XD signals in column direction 3-94 One-Dimensional Multisignal Analysis XX = mswden('densig','c',XD,'db2',2,'sqtwolog','sln'); Plot original signals X and the denoised signals XX as images. figure; subplot(1,2,1); image(X) subplot(1,2,2); image(XX) colormap(pink(222)) Interactive One-Dimensional Multisignal Analysis In this section, we explore the same signal as in the previous section, but use the graphical interface tools to analyze it. 1 Start the Wavelet 1-D Multisignal Analysis Tool. From the MATLAB prompt, type: wavemenu The Wavelet Toolbox Main Menu appears. 3-95 3 Discrete Wavelet Analysis Click Multisignal Analysis 1-D to open the Wavelet 1-D Multisignal Analysis tool. 3-96 One-Dimensional Multisignal Analysis The tool is divided into five panes. Two of them are the same as in all Wavelet Toolbox GUIs: the Command Frame on the right side of the figure and the Dynamic Visualization tool at the bottom. The Command Frame contains a special component found in all multisignal tools: the Selection of Data Sets pane which is used to manage two lists. The three new panes are the Visualization of Selected Data pane, the Information on Selected Data pane, and the Selection of Data pane. 2 Load the signals. From the File menu, select Load > Signals. When the Load Signal dialog box appears, select the MAT-file thinker.mat from the MATLAB folder toolbox/wavelet/wmultisig1d and click OK. The data matrix loads in the Wavelet 1-D Multisignal Analysis tool, and the first signal appears. 3-97 3 Discrete Wavelet Analysis The Selection of Data pane contains a list of selectable signals. At the beginning, only the originally loaded signals are available. You can generate and add new signals to the list by decomposing, compressing, or denoising original signals. Each row of the list displays the index of selectable signal (Idx Sel), the index of original signal (Idx Sig) and three wavelet transform attributes describing the process used to obtain the selectable signal from the original one. 3 View the signals and signal information. With signal 1 highlighted, Shift-click the mouse on signal 3 to select signals 1, 2, and 3. Ctrl-click the mouse on signals 7, 9, and 11. (The Select ALL button at the bottom of the Selection of Data pane selects all signals and the Clear button deselects all signals.) The selected signals (1, 2, 3, 7, 9 and 11) appear in the Visualization of Selected Data pane. The Information on Selected Data pane contains the box plots of the minimums, the means, and the maximums of these signals. 3-98 One-Dimensional Multisignal Analysis 4 Highlight a signal. Using the Highlight Sel button in the lower-left corner of the Visualization of Selected Data pane, select signal 3. 5 Select Different Views. In the Visualization of Selected Data pane, change the view mode using the pop-up in the lower-right corner. Choose Separate Mode. The selected signals appear. 6 Decompose a multisignal. 3-99 3 Discrete Wavelet Analysis Perform an analysis at level 4 using the db2 wavelet and the same file used in the command line section: thinker.mat. In the upper right portion of the Wavelet 1-D Multisignal Analysis tool, select db2 and level 4 in the Wavelet fields. Click Decompose. After a pause for computation, all the original signals are decomposed and signal 1 is automatically selected In the Selection of Data pane, new information is added for each original signal — the percentage of energy of the wavelet components (D1,...,D4 and A4) and the total energy. The Information on Selected Data pane contains information on the single selected signal: Min, Mean, Max and the energy distribution of the signal. 3-100 One-Dimensional Multisignal Analysis Since the original signals are decomposed, new objects appear and the Selection of Data Sets pane in the Command Frame updates. The Selection of Data Sets pane defines the available signals that are now selectable from the Selection of Data pane. The list on the left allows you to select sets of signals and the right list allows you to select sets of corresponding coefficients: original signals (Orig. Signals), approximations (APP 1,...) and details from levels 1 to 4 (DET 1,...). In the list on the right, the coefficients vectors can be of different lengths, but only components of the same length can be selected together. After a decomposition the original signals (Orig. appears automatically selected. Signals) data set Select signals 1, 2, 3, 7, 9 and 11. 3-101 3 Discrete Wavelet Analysis The energy of selected signals is primarily concentrated in the approximation A4, so the box plot is crushed (see following figure on the left). Deselect App. On/Off to see a better representation of details energy (see following figure on the right). 7 Display multisignal decompositions. In the Visualization of Selected Data pane, change the view mode using the pop-up below the plots and select Full Dec Mode. The decompositions of the selected signals display. 3-102 One-Dimensional Multisignal Analysis Change the Level to 2. Select the signal 7 in Highlight Sel. 8 Change the visualization modes. Using the second pop-up from the left at the bottom of the pane, select Full Dec Mode (Cfs). The coefficients of the decompositions of the selected signals display. At level k, coefficients are duplicated 2k times. Change the view mode to Stem Mode (Abs), and then, change to Tree Mode. The wavelet tree corresponding to the decompositions of the selected signals displays. Select the level 4 and click the node a3. Then highlight signal 7. 9 Select Different Wavelet Components. Ctrl-click Orig. Signals, APP 1, APP 3 and DET 1 to select these four sets of signals from the list on the left in the Selection of Data Sets pane. The total number of selected data (Number of Sig.) appears in the Selection of Data Sets pane: four sets of 192 signals each is a total of 768 signals. 3-103 3 Discrete Wavelet Analysis Click the Asc. button in the Sort pane. The selected data are sorted in ascending order with respect to the Idx Sig parameter Note that DWT attributes of each selectable signal have been updated where a stands for approximation, d for detail and s for signal. Click the Idx Sel 1 signal and then shift-click the Idx Sel 579 signal. 3-104 One-Dimensional Multisignal Analysis Choose Separate Mode. Ctrl-click to select two sets of signals from the right-most list of the Selection of Data Sets pane: APP 1 and DET 1. 3-105 3 Discrete Wavelet Analysis Note that in this list of coefficients sets, the selected vectors must be of same length, which means that you must select components of the same level. Click the Asc. button in the Sort pane. The selected data are sorted in ascending order with respect to Idx Sig parameter. Select the ten first signals. 10 Compress a multisignal. The graphical interface tools feature a compression option with automatic or manual thresholding. 3-106 One-Dimensional Multisignal Analysis Cick Compress, located in the lower-right side of the window. This displays the Compression window. Note The tool always compresses all the original signals when you click the Compress button. Before compressing, choose the particular strategy for computing the thresholds. Select the adapted parameters in the Select Compression Method frame. Then, apply this strategy to compute the thresholds according to the current method, either to the current selected signals by clicking the Selected button, or to all signals by clicking the ALL button. For this example, accept the defaults and click the ALL button. 3-107 3 Discrete Wavelet Analysis The thresholds for each level (ThrD1 to ThrD4), the energy ratio (En. Rat.) and the sparsity ratio (NbZ Rat.) are displayed in the Selection of Data pane. Click the Compress button at the bottom of the Thresholding pane. Now you can select new data sets: compressed Signals, the corresponding approximations, details and coefficients. Press the Ctrl key and click the Compressed item in the left list of the Selection of Data Sets pane. The original signals and their compressed versions are selected (2 x 192 = 384 signals). Click the Asc. button at the bottom of the Selection of Data pane to sort the signals using Idx Sig number. With the mouse, select the first four signals. They correspond to the original signals 1, 2 and the corresponding compressed signals 193, 194. 3-108 One-Dimensional Multisignal Analysis Click the Close button to close the Compression window. 11 Denoise a multisignal. The graphical user interface offers a denoising option with either a predefined thresholding strategy or a manual thresholding method. Using this tool makes very easy to remove noise from many signals in one step. Display the Denoising window by clicking the Denoise button located in the bottom part of the Command Frame on the right of the window. 3-109 3 Discrete Wavelet Analysis A number of options are available for fine-tuning the denoising algorithm. For this example, accept the defaults: soft type of thresholding, Fixed form threshold method, and Scaled white noise as noise structure. Click the ALL button in the Thresholding pane. The threshold for each level (ThrD1, ..., ThrD4) computes and displays in the Selection of Data pane. Then click the Denoise button at the bottom of the Thresholding pane. Ctrl-click the Denoised item in the list on the left of the Selection of Data Sets pane. The original signals and the corresponding denoised ones are selected (2 x 192 = 384 signals). Click the Asc. button at the bottom of the Selection of Data pane to sort the signals according to the Idx Sig parameter. With the mouse, select the first four signals. They correspond to the original signals 1, 2 and the corresponding denoised signals 193, 194 3-110 One-Dimensional Multisignal Analysis Choose Separate Mode. 12 To view residuals, Ctrl-click the Orig. Signal, the Denoised and the Residuals items in the list on the left of the Selection of Data Sets pane. Original, denoised and residual signals are selected (3 x 192 = 576 signals). Click the Asc. button at the bottom of the Selection of Data pane to sort the signals using the Idx Sig parameter. With the mouse, select the first six signals. They correspond to the original signals 1, 2, the corresponding denoised signals 193, 194 and the residuals 385, 386. Then, choose Separate Mode. 3-111 3 Discrete Wavelet Analysis 13 Click Close to close the denoising tool. Then, click the Yes button to update the synthesized signals. Manual Threshold Tuning 1 Choose a method, select one or several signals in the Selection of Data pane using the mouse and keys. Then click the Selected button. You can select another group of signals using the same method. Press the Denoise button to denoise the selected signal(s). 3-112 One-Dimensional Multisignal Analysis You can also use manual threshold tuning. Click the Enable Manual Thresholding Tuning button. The horizontal lines in the wavelet coefficient axes (cd1, ..., cd4) can be dragged using the mouse. This may be done individually, by group or all together depending on the values in the Select Signal and Selected Level fields in the Manual Threshold Tuning pane. 3-113 3 Discrete Wavelet Analysis 2 In the Wavelet 1-D Multisignal Analysis Compression tool, you can use two methods for threshold tuning: the By level thresholding method which is used in the Wavelet 1-D Multisignal Analysis Denoising tool, and the Global thresholding method. You can drag the vertical lines in the Energy and Nb. Zeros Performances axes using the mouse. This can be done individually or all together depending on the values of Select Signal in the Manual Threshold Tuning pane. 3-114 One-Dimensional Multisignal Analysis The threshold value, L2 performance, and number of zeros performance are updated in the corresponding edit buttons in the Manual Threshold Tuning pane. Statistics on Signals 1 You can display various statistical parameters related to the signals and their components. From the Wavelet 1-D Multisignal Analysis tool, click the Statistics button. Then select the signal 1 in the Selection of Data Sets pane. 3-115 3 Discrete Wavelet Analysis Select the signals 1, 2, 3, 7, 9 and 11 in the Selection of Data pane, and display the corresponding boxplots and correlation plots. 3-116 One-Dimensional Multisignal Analysis 2 To display statistics on many wavelet components, in the Selection Data Sets pane, in the left column, select Orig. Signals, APP 1, DET 1, Denoised and Residuals signals. Then choose Separate Mode, and click the Asc. button in the Sort pane. The selected data are sorted in ascending order with respect to Idx Sig parameter. In the Selection of Data pane, select data related to signal 1. Clustering Signals Note To use clustering, you must have Statistics Toolbox™ software installed. 1 Click the Clustering button located in the Command Frame, which is in the lower right of the Wavelet 1-D Multisignal Analysis window to open the Clustering tool. 3-117 3 Discrete Wavelet Analysis You can cluster various type of signals and wavelet components: original, denoised or compressed, residuals, and approximations or details (reconstructed or coefficients). Similarly, there are several methods for constructing partitions of data. Use the default parameters (Original and Signal in Data to Cluster, and in Ascending Hierarchical, euclidean, ward, and 6 in Culstering) and click the Compute Clusters button. 3-118 One-Dimensional Multisignal Analysis A full dendrogram and a restricted dendrogram display in the Selection by Dendrogram pane. For each signal, the cluster number displays in the Selection of Data pane. 2 Select one cluster, several clusters, or a part of a cluster. Click the xticklabel 3 at the bottom of the restricted dendrogram. The links of the third cluster blink in the full dendrogram and the 24 signals of this class display in the Visualization of Selected Data pane. You can see their numbers in the Selection of Data pane. Clicking the line in the restricted or in the full dendrogram lets you select one cluster, several linked clusters, or a part of a cluster. For a more accurate selection, use the Dilate X and the Translate X sliders under the full dendrogram. You can also use the Yscale button located above the full dendrogram. The corresponding signals display in the Visualization of Selected Data pane and in the list of the Selection of Data pane. 3-119 3 Discrete Wavelet Analysis You can use the horizontal line in the full dendrogram to change the number of clusters. Use the left mouse button to drag the line up or down. 3 Use the Show Clusters button to examine the clusters of the current partition. You can display the mean (or the median) of each cluster, the global standard deviation and the pointwise standard deviation distance around the mean (or the median). The number of the cluster, the number of elements, the percentage of signals, and two indices of quality display for each cluster. 3-120 One-Dimensional Multisignal Analysis 4 Click the Store Current Partition button below the Clustering pane to store the current partition for further comparisons. A default name is suggested. Note that the 1-D Wavelet Multisignal Analysis tool stores the partitions and they are not saved on the disk. Partitions 1 Build and store several partitions (for example, partitions with signals, denoised signals approximations at level 1, 2 and 3, and denoised signals). Then, click the Open Partition Manager button below the Store Current Partition button. The Partitions Management pane appears. The names of all stored partitions are listed. 3-121 3 Discrete Wavelet Analysis Now, you can show, clear, or save the partitions (individually, selected ones, or all together). 2 To display partitions, select the Ori Signals and the Den Signals partitions, and click the Selected button next to the Show Partitions label. The clusters are almost the same, but it is difficult to see this on the Selected Partitions axis, due to the scaling difference. Press the Apply button to renumber the clusters (starting from the selected partition as basic numbering) to compare the two partitions. 3-122 One-Dimensional Multisignal Analysis Only three signals are not classified in the same cluster for the two considered partitions. 3-123 3 Discrete Wavelet Analysis 3 Select the partitions you want to save and click the Save Partitions button below the Store Current Partition button in the Partitions Management pane. Partitions are saved as an array of integers, where each column corresponds to one partition and contains the indices of clusters. When you choose the Full Partitions option, an array object (wpartobj) is saved. 4 To load or clear stored partitions use File > Partitions in the Wavelet 1-D Multisignal Analysis tool. (File > Partitions is also available in the Wavelet 1-D Multisignal Analysis Clustering tool and you can also save the current partition.) 3-124 One-Dimensional Multisignal Analysis To clear one or more stored partitions, select File > Partitions > Clear Partition. Select File > Partitions > Load Partition to load one or several partitions from the disk. The loaded partitions are stored in Wavelet 1-D Multisignal Analysis tool with any previously stored partitions. A partition can also be a manually created column vector. 3-125 3 Discrete Wavelet Analysis Note The number of signals in loaded partitions must be equal to the number of signals in the Wavelet 1-D Multisignal Analysis tool. A warning appears if this condition is not true. 5 In each subcomponent of the Wavelet 1-D Multisignal Analysis tool (main, statistics, denoising, compression, clustering), you can import a stored partition from the list in the Selection of Data pane. Click the Import Part button at the bottom of the Selection of Data pane, the Partition Set Manager window appears. Select one partition and click the Import button. For this example, go back to the main window, import the Ori Signals partition and sort the signals in descending order with respect to A4 energy percentage. 3-126 One-Dimensional Multisignal Analysis 6 Click the More on Partitions button at the bottom of the Partitions Management pane to display the Partition tool. 7 Select the Den Signals in Sel P2 in the upper-right corner of the window. Then, in the lower left axis, click the yellow text containing the value 2 (the 3-127 3 Discrete Wavelet Analysis coordinates of the corresponding point are (4,5)). The corresponding signals are displayed together with all related information. More on Clustering Instead of the Ascending Hierarchical Tree clustering method, you can use the K-means method. For this case, the partition cannot be represented by a dendrogram and other representations should be used. In the image representation (see figure below on the left), you can select a cluster by clicking on the corresponding color on the colorbar. You can also select a cluster or part of a cluster by clicking on the image. In the center representation (see figure below on the right) you can select a cluster by clicking on the corresponding colored center. 3-128 One-Dimensional Multisignal Analysis Importing and Exporting Information from the Graphical Interface The Wavelet 1-D Multisignal Analysis tool lets you move data to and from disk. Saving Information to Disk You can save decompositions and denoised or compressed signals (including the corresponding decompositions from Wavelet 1-D Multisignal Analysis tools) to disk. You then can manipulate the data and later import it again into the graphical tools. 3-129 3 Discrete Wavelet Analysis Saving Decompositions The Wavelet 1-D Multisignal Analysis main tool lets you save the entire set of data from a wavelet analysis to disk. The toolbox creates a MAT-file in the current folder with a name you choose. 1 Open the Wavelet 1-D Multisignal Analysis main tool and load the example analysis by selecting File > Example > Ex 21: Thinker (rows). 2 Save the data from this analysis, using the menu option File > Save Decompositions. A dialog box appears that lets you specify a folder and filename for storing the decomposition data. For this example, use the name decORI.mat. 3 Type the name decORI. 4 After saving the decomposition data to the file decORI.mat, load the variables into your workspace: load decORI whos Name Size Bytes Class dec 1x1 163306 struct dec dec = dirDec: 'r' 3-130 One-Dimensional Multisignal Analysis level: wname: dwtFilters: dwtEXTM: dwtShift: dataSize: ca: cd: 4 'db2' [1x1 struct] 'sym' 0 [192 96] [192x8 double] {1x4 cell} The field ca of the structure dec gives the coefficients of approximation at level 4, the field cd is a cell array which contains the coefficients of details. size(dec.cd{1}) ans = 192 49 size(dec.cd{2}) ans = 192 26 size(dec.cd{3}) ans = 192 14 size(dec.cd{4}) ans = 192 8 You can change the coefficients using the chgwdeccfs function. Note For a complete description of the dec structure, see “Loading Decompositions” on page 3-133. Loading Information into the Wavelet 1-D Multisignal Analysis Tool You can load signals or decompositions into the graphical interface. The information you load may be previously exported from the graphical interface, and then manipulated in the workspace; or it may be information you initially generated from the command line. In either case, you must observe the strict 3-131 3 Discrete Wavelet Analysis file formats and data structures used by the Wavelet 1-D Multisignal Analysis tools or errors will occur when you try to load information. Loading Signals. To load a signal you constructed in your MATLAB workspace into the Wavelet 1-D Multisignal Analysis tool, save the signal in a MAT-file (with extension .mat). For example, if you design a signal called magic128 and want to analyze it in the Wavelet 1-D Multisignal Analysis tool, type save magic128 magic128 Note The workspace variable magic128 must be a matrix and the number of rows and columns must be greater than 1. sizmag = size(magic128) sizmag = 128 128 To load this signal into the Wavelet 1-D Multisignal Analysis tool, use the File > Load Signal menu item. A dialog box appears in which you select the appropriate MAT-file to be loaded. 3-132 One-Dimensional Multisignal Analysis Note When you load a matrix of signals from the disk, the name of 2-D variables are inspected in the following order: x, X, sigDATA, and signals. Then, the 2-D variables encountered in the file are inspected in alphabetical order. Loading Decompositions. To load decompositions that you constructed in the MATLAB workspace into the Wavelet 1-D Multisignal Analysis tool, save the signal in a MAT-file (with extension mat). For instance, if you design a signal called magic128 and want to analyze it in the Wavelet 1-D Multisignal Analysis too, the structure dec must have the following fields: 'dirDec' Direction indicator with 'r' for row or 'c' for column 'level' Level of DWT decomposition 'wname' Wavelet name 'dwtFilters' Structure with four fields: LoD, HiD, LoR, HiR 'dwtEXTM' DWT extension mode (see dwtmode) 'dwtShift' DWT shift parameter (0 or 1) 'dataSize' Size of original matrix X 'ca' Approximation coefficients at level dec.level 'cd' Cell array of detail coefficients, from 1 to dec.level The coefficients cA and cD{k}, for (k = 1 to dec.level), are matrices and are stored rowwise if dec.dirDec is equal to 'r' or columnwise if dec.dirDec is equal to 'c'. 3-133 3 Discrete Wavelet Analysis Note The fields 'wname' and 'dwtFilters' have to be compatible (see the wfilters function). The sizes of cA and cD{k}, (for k = 1 to dec.level) must be compatible with the direction, the level of the decomposition, and the extension mode. Loading and Saving Partitions. Loading. The Wavelet 1-D Multisignal Analysis main tool and clustering tool let you load a set of partitions from disk. Saving Partitions. The Wavelet 1-D Multisignal Analysis clustering tool lets you save a set of partitions to disk. 3-134 One-Dimensional Multisignal Analysis For example: 1 Open the Wavelet 1-D Multisignal Analysis main tool and load the example analysis using File > Example > Ex 21: Thinker (rows). 2 Click the Clustering button. The Wavelet 1-D Multisignal Analysis Clustering window appears. 3 Click the Compute Clusters button, and then save the current partition using menu option File > Partitions > Save Current Partition. A dialog box appears that lets you specify the type of data to save. 4 Click the Save Curr. button. 5 Another dialog box appears that lets you specify a folder and filename for storing the partition data. Type the name curPART. 6 After saving the partition data to the file curPART.mat, load the variables into your workspace: load curPART whos Name Size Bytes Class tab_IdxCLU 192x1 1536 double 3-135 3 Discrete Wavelet Analysis 7 You can modify the array tab_IdxCLU in the workspace, and save the partition data in a file. For example to create two new partitions with four and two clusters, type the following lines: tab_IdxCLU(:,2) = rem((1:192)',4) + 1; tab_IdxCLU(:,3) = double((1:192)'>96) + 1; save newpart tab_IdxCLU Now you can use three partitions for the example Ex 21: Thinker (rows). Then, you can load the partitions stored in the file newPART.mat in the Wavelet 1-D Multisignal Analysis main tool and clustering tool. Note A partition is a column vector of integers. The values must vary from 1 to NbClusters (NbClusters > 1), and each cluster must contain at least one element. The number of rows must be equal to the number of signals. 3-136 Two-Dimensional Discrete Wavelet Analysis Two-Dimensional Discrete Wavelet Analysis This section takes you through the features of two-dimensional discrete wavelet analysis using the Wavelet Toolbox software. The toolbox provides these functions for image analysis. For more information, see the function reference pages. Note In this section the presentation and examples use two-dimensional arrays corresponding to indexed image representations. However, the functions described are also available when using truecolor images, which are represented by m-by-n-by-3 arrays of uint8. For more information on image formats, see “Wavelets: Working with Images”. Analysis-Decomposition Functions Function Name Purpose dwt2 Single-level decomposition wavedec2 Decomposition wmaxlev Maximum wavelet decomposition level Synthesis-Reconstruction Functions Function Name Purpose idwt2 Single-level reconstruction waverec2 Full reconstruction wrcoef2 Selective reconstruction upcoef2 Single reconstruction 3-137 3 Discrete Wavelet Analysis Decomposition Structure Utilities Function Name Purpose detcoef2 Extraction of detail coefficients appcoef2 Extraction of approximation coefficients upwlev2 Recomposition of decomposition structure De-Noising and Compression Function Name Purpose ddencmp Provide default values for de-noising and compression wbmpen Penalized threshold for wavelet 1-D or 2-D de-noising wdcbm2 Thresholds for wavelet 2-D using Birgé-Massart strategy wdencmp Wavelet de-noising and compression wthrmngr Threshold settings manager In this section, you’ll learn • How to load an image • How to analyze an image • How to perform single-level and multilevel image decompositions and reconstructions (command line only) • How to use Square and Tree mode features (GUI only) • How to zoom in on detail (GUI only) • How to compress an image Two-Dimensional Analysis — Command Line In this example we’ll show how you can use two-dimensional wavelet analysis to compress an image efficiently without sacrificing its clarity. 3-138 Two-Dimensional Discrete Wavelet Analysis Note Instead of directly using image(I) to visualize the image I, we use image(wcodemat(I)), which displays a rescaled version of I leading to a clearer presentation of the details and approximations (see wcodemat reference page). 1 Load an image. From the MATLAB prompt, type load wbarb; whos Name Size Bytes Class X 256x256 524288 double array map 192x3 4608 double array 2 Display the image. Type image(X); colormap(map); colorbar; 3-139 3 Discrete Wavelet Analysis 3 Convert an indexed image to a grayscale image. If the colormap is smooth, the wavelet transform can be directly applied to the indexed image; otherwise the indexed image should be converted to grayscale format. For more information, see “Wavelets: Working with Images”. Since the colormap is smooth in this image, you can now perform the decomposition. 4 Perform a single-level wavelet decomposition. To perform a single-level decomposition of the image using the bior3.7 wavelet, type 3-140 Two-Dimensional Discrete Wavelet Analysis [cA1,cH1,cV1,cD1] = dwt2(X,'bior3.7'); This generates the coefficient matrices of the level-one approximation (cA1) and horizontal, vertical and diagonal details (cH1,cV1,cD1, respectively). 5 Construct and display approximations and details from the coefficients. To construct the level-one approximation and details (A1, H1, V1, and D1) from the coefficients cA1, cH1, cV1, and cD1, type A1 H1 V1 D1 = = = = upcoef2('a',cA1,'bior3.7',1); upcoef2('h',cH1,'bior3.7',1); upcoef2('v',cV1,'bior3.7',1); upcoef2('d',cD1,'bior3.7',1); = = = = = size(X); idwt2(cA1,[],[],[],'bior3.7',sx); idwt2([],cH1,[],[],'bior3.7',sx); idwt2([],[],cV1,[],'bior3.7',sx); idwt2([],[],[],cD1,'bior3.7',sx); or sx A1 H1 V1 D1 To display the results of the level 1 decomposition, type colormap(map); subplot(2,2,1); image(wcodemat(A1,192)); title('Approximation A1') subplot(2,2,2); image(wcodemat(H1,192)); title('Horizontal Detail H1') subplot(2,2,3); image(wcodemat(V1,192)); title('Vertical Detail V1') subplot(2,2,4); image(wcodemat(D1,192)); title('Diagonal Detail D1') 3-141 3 Discrete Wavelet Analysis 6 Regenerate an image by single-level Inverse Wavelet Transform. To find the inverse transform, type Xsyn = idwt2(cA1,cH1,cV1,cD1,'bior3.7'); This reconstructs or synthesizes the original image from the coefficients of the level 1 approximation and details. 7 Perform a multilevel wavelet decomposition. To perform a level 2 decomposition of the image (again using the bior3.7 wavelet), type [C,S] = wavedec2(X,2,'bior3.7'); where X is the original image matrix, and 2 is the level of decomposition. 3-142 Two-Dimensional Discrete Wavelet Analysis The coefficients of all the components of a second-level decomposition (that is, the second-level approximation and the first two levels of detail) are returned concatenated into one vector, C. Argument S is a bookkeeping matrix that keeps track of the sizes of each component. 8 Extract approximation and detail coefficients. To extract the level 2 approximation coefficients from C, type cA2 = appcoef2(C,S,'bior3.7',2); To extract the first- and second-level detail coefficients from C, type cH2 cV2 cD2 cH1 cV1 cD1 = = = = = = detcoef2('h',C,S,2); detcoef2('v',C,S,2); detcoef2('d',C,S,2); detcoef2('h',C,S,1); detcoef2('v',C,S,1); detcoef2('d',C,S,1); or [cH2,cV2,cD2] = detcoef2('all',C,S,2); [cH1,cV1,cD1] = detcoef2('all',C,S,1); where the first argument ('h', 'v', or 'd') determines the type of detail (horizontal, vertical, diagonal) extracted, and the last argument determines the level. 9 Reconstruct the Level 2 approximation and the Level 1 and 2 details. To reconstruct the level 2 approximation from C, type A2 = wrcoef2('a',C,S,'bior3.7',2); To reconstruct the level 1 and 2 details from C, type H1 V1 D1 H2 V2 = = = = = wrcoef2('h',C,S,'bior3.7',1); wrcoef2('v',C,S,'bior3.7',1); wrcoef2('d',C,S,'bior3.7',1); wrcoef2('h',C,S,'bior3.7',2); wrcoef2('v',C,S,'bior3.7',2); 3-143 3 Discrete Wavelet Analysis D2 = wrcoef2('d',C,S,'bior3.7',2); 10 Display the results of a multilevel decomposition. Note With all the details involved in a multilevel image decomposition, it makes sense to import the decomposition into the Wavelet 2-D graphical tool in order to more easily display it. For information on how to do this, see “Loading Decompositions” on page 3-164. To display the results of the level 2 decomposition, type colormap(map); subplot(2,4,1);image(wcodemat(A1,192)); title('Approximation A1') subplot(2,4,2);image(wcodemat(H1,192)); title('Horizontal Detail H1') subplot(2,4,3);image(wcodemat(V1,192)); title('Vertical Detail V1') subplot(2,4,4);image(wcodemat(D1,192)); title('Diagonal Detail D1') subplot(2,4,5);image(wcodemat(A2,192)); title('Approximation A2') subplot(2,4,6);image(wcodemat(H2,192)); title('Horizontal Detail H2') subplot(2,4,7);image(wcodemat(V2,192)); title('Vertical Detail V2') subplot(2,4,8);image(wcodemat(D2,192)); title('Diagonal Detail D2') 11 Reconstruct the original image from the multilevel decomposition. To reconstruct the original image from the wavelet decomposition structure, type X0 = waverec2(C,S,'bior3.7'); This reconstructs or synthesizes the original image from the coefficients C of the multilevel decomposition. 3-144 Two-Dimensional Discrete Wavelet Analysis 12 Compress the image and display it. To compress the original image X, use the ddencmp command to calculate the default parameters and the wdencmp command to perform the actual compression. Type [thr,sorh,keepapp]= ddencmp('cmp','wv',X); [Xcomp,CXC,LXC,PERF0,PERFL2] = ... wdencmp('gbl',C,S,'bior3.7',2,thr,sorh,keepapp); Note that we pass in to wdencmp the results of the decomposition (C and S) we calculated in step 7. We also specify the bior3.7 wavelets, because we used this wavelet to perform the original analysis. Finally, we specify the global thresholding option 'gbl'. See ddencmp and wdencmp reference pages for more information about the use of these commands. To view the compressed image side by side with the original, type colormap(map); subplot(121); image(X); title('Original Image'); axis square subplot(122); image(Xcomp); title('Compressed Image'); axis square PERF0 PERF0 = 49.8076 3-145 3 Discrete Wavelet Analysis PERFL2 PERFL2 = 99.9817 These returned values tell, respectively, what percentage of the wavelet coefficients was set to zero and what percentage of the image’s energy was preserved in the compression process. Note that, even though the compressed image is constructed from only about half as many nonzero wavelet coefficients as the original, there is almost no detectable deterioration in the image quality. 3-146 Two-Dimensional Discrete Wavelet Analysis Interactive Two-Dimensional Wavelet Analysis In this section we explore the same image as in the previous section, but we use the graphical interface tools to analyze the image. 1 Start the 2-D Wavelet Analysis Tool. From the MATLAB prompt, type wavemenu The Wavelet Tool Main Menu appears. Click the Wavelet 2-D menu item. The discrete wavelet analysis tool for two-dimensional image data appears. 3-147 3 Discrete Wavelet Analysis 2 Load an image. From the File menu, choose the Load > Image option. 3-148 Two-Dimensional Discrete Wavelet Analysis When the Load Image dialog box appears, select the MAT-file wbarb.mat, which is in the MATLAB folder toolbox/wavelet/wavedemo. Click the OK button. The image is loaded into the Wavelet 2-D tool. 3 Analyze the image. Using the Wavelet and Level menus located to the upper right, determine the wavelet family, the wavelet type, and the number of levels to be used for the analysis. For this analysis, select the bior3.7 wavelet at level 2. Click the Analyze button. After a pause for computation, the Wavelet 2-D tool displays its analysis. 3-149 3 Discrete Wavelet Analysis Using Square Mode Features By default, the analysis appears in “Square Mode.” This mode includes four different displays. In the upper left is the original image. Below that is the image reconstructed from the various approximations and details. To the lower right is a decomposition showing the coarsest approximation coefficients and all the horizontal, diagonal, and vertical detail coefficients. 3-150 Two-Dimensional Discrete Wavelet Analysis Finally, the visualization space at the top right displays any component of the analysis that you want to look at more closely. Click on any decomposition component in the lower right window. A green border highlights the selected component. At the lower right of the Wavelet 2-D window, there is a set of three buttons labeled “Operations on selected image.” Note that if you click again on the same component, you’ll deselect it and the green border disappears. Click the Visualize button. The selected image is displayed in the visualization area. You are seeing the raw, unreconstructed two-dimensional wavelet coefficients. Using the other buttons, you can display the reconstructed version of the selected image component, or you can view the selected component at full screen resolution. Using Tree Mode Features Choose Tree from the View Mode menu. Your display changes to reveal the following. 3-151 3 Discrete Wavelet Analysis This is the same information shown in square mode, with in addition all the approximation coefficients, but arranged to emphasize the tree structure of the decomposition. The various buttons and menus work just the same as they do in square mode. Zooming in on Detail 3-152 Two-Dimensional Discrete Wavelet Analysis Drag a rubber band box (by holding down the left mouse button) over the portion of the image you want to magnify. Click the XY+ button (located at the bottom of the screen) to zoom horizontally and vertically. The Wavelet 2-D tool enlarges the displayed images. To zoom back to original magnification, click the History <<- button. 4 Compress the image Click the Compress button, located to the upper right of the Wavelet 2-D window. The Wavelet 2-D Compression window appears. 3-153 3 Discrete Wavelet Analysis The tool automatically selects thresholding levels to provide a good initial balance between retaining the image’s energy while minimizing the number of coefficients needed to represent the image. However, you can also adjust thresholds manually using the By Level thresholding option, and then the sliders or edits corresponding to each level. 3-154 Two-Dimensional Discrete Wavelet Analysis For this example, select the By Level thresholding option and select the Remove near 0 method from the Select thresholding method menu. The following window is displayed. Select from the direction menu whether you want to adjust thresholds for horizontal, diagonal or vertical details. To make the actual adjustments for each level, use the sliders or use the left mouse button to directly drag the yellow vertical lines. To compress the original image, click the Compress button. After a pause for computation, the compressed image is displayed beside the original. Notice that compression eliminates almost half the coefficients, yet no detectable deterioration of the image appears. 3-155 3 Discrete Wavelet Analysis 5 Show the residuals. From the Wavelet 2-D Compression tool, click the Residuals button. The More on Residuals for Wavelet 2-D Compression window appears. Displayed statistics include measures of tendency (mean, mode, median) and dispersion (range, standard deviation). In addition, the tool provides frequency-distribution diagrams (histograms and cumulative histograms). The same tool exists for the Wavelet 2-D De-noising tool. Note The statistics displayed in the above figure are related to the displayed image but not to the original one. Usually this information is the same, but in some cases, edge effects may cause the original image to be cropped slightly. To see the exact statistics, use the command line functions to get the desired image and then apply the desired MATLAB statistical function(s). Importing and Exporting Information from the Graphical Interface The Wavelet 2-D graphical tool lets you import information from and export information to disk, if you adhere to the proper file formats. 3-156 Two-Dimensional Discrete Wavelet Analysis Saving Information to Disk You can save synthesized images, coefficients, and decompositions from the Wavelet 2-D tool to disk, where the information can be manipulated and later reimported into the graphical tool. Saving Synthesized Images. You can process an image in the Wavelet 2-D tool, and then save the processed image to a MAT-file (with extension mat or other). For example, load the example analysis: File > Example Analysis > at level 3, with sym4 → detail Durer and perform a compression on the original image. When you close the Wavelet 2-D Compression window, update the synthesized image by clicking Yes in the dialog box that appears. Then, from the Wavelet 2-D tool, select the File > Save > Synthesized Image menu option. A dialog box appears allowing you to select a folder and filename for the MAT-file (with extension mat or other). For this example, choose the name symage. To load the image into your workspace, type load symage whos 3-157 3 Discrete Wavelet Analysis Name Size Bytes Class X 359x371 1065512 double array map 64x3 1536 double array valTHR 1x1 8 double array wname 1x4 8 char array The synthesized image is given by X and map contains the colormap. In addition, the parameters of the de-noising or compression process are given by the wavelet name (wname) and the global threshold (valTHR). Saving Discrete Wavelet Transform Coefficients. The Wavelet 2-D tool lets you save the coefficients of a discrete wavelet transform (DWT) to disk. The toolbox creates a MAT-file in the current folder with a name you choose. To save the DWT coefficients from the present analysis, use the menu option File > Save > Coefficients. A dialog box appears that lets you specify a folder and filename for storing the coefficients. Consider the example analysis: File > Example Analysis > at level 3, with sym4 → Detail Durer After saving the discrete wavelet coefficients to the file cfsdurer.mat, load the variables into your workspace: load cfsdurer whos 3-158 Name Size Bytes Class coefs 1x142299 1138392 double array map 64x3 1536 double array sizes 5x2 80 double array Two-Dimensional Discrete Wavelet Analysis Name Size Bytes Class valTHR 0x0 0 double array wname 1x4 8 char array Variable map contains the colormap. Variable wname contains the wavelet name and valTHR is empty since the synthesized image is the same as the original one. Variables coefs and sizes contain the discrete wavelet coefficients and the associated matrix sizes. More precisely, in the above example, coefs is a 1-by-142299 vector of concatenated coefficients, and sizes gives the length of each component. Saving Decompositions. The Wavelet 2-D tool lets you save the entire set of data from a discrete wavelet analysis to disk. The toolbox creates a MAT-file in the current folder with a name you choose, followed by the extension wa2 (wavelet analysis 2-D). Open the Wavelet 2-D tool and load the example analysis: File > Example Analysis > at level 3, with sym4 → Detail Durer. To save the data from this analysis, use the menu option File > Save > Decomposition. A dialog box appears that lets you specify a folder and filename for storing the decomposition data. Type the name decdurer. After saving the decomposition data to the file decdurer.wa2, load the variables into your workspace: load decdurer.wa2 -mat whos Name Size Bytes Class coefs 1x142299 1138392 double array data_name 1x6 12 char array 3-159 3 Discrete Wavelet Analysis Name Size Bytes Class map 64x3 1536 double array sizes 5x2 80 double array valTHR 0x0 0 double array wave_name 1x4 8 char array Variables coefs and sizes contain the wavelet decomposition structure. Other variables contain the wavelet name, the colormap, and the filename containing the data. Variable valTHR is empty since the synthesized image is the same as the original one. Note Save options are also available when performing de-noising or compression inside the Wavelet 2-D tool. In the Wavelet 2-D De-noising window, you can save denoised image and decomposition. The same holds true for the Wavelet 2-D Compression window. This way, you can save many different trials from inside the De-noising and Compression windows without going back to the main Wavelet 2-D window during a fine-tuning process. When saving a synthesized signal, a decomposition or coefficients to a MAT-file, the mat file extension is not necessary. You can save approximations individually for each level or save them all at once. Loading Information into the Wavelet 2-D Tool You can load images, coefficients, or decompositions into the graphical interface. The information you load may have been previously exported from the graphical interface, and then manipulated in the workspace; or it may have been information you generated initially from the command line. In either case, you must observe the strict file formats and data structures used by the Wavelet 2-D tool, or else errors will result when you try to load information. 3-160 Two-Dimensional Discrete Wavelet Analysis Loading Images. This toolbox supports only indexed images. An indexed image is a matrix containing only integers from 1 to n, where n is the number of colors in the image. This image may optionally be accompanied by an n-by-3 matrix called map. This is the colormap associated with the image. When MATLAB displays such an image, it uses the values of the matrix to look up the desired color in this colormap. If the colormap is not given, the Wavelet 2-D tool uses a monotonic colormap with max(max(X))min(min(X))+1 colors. To load an image you’ve constructed in your MATLAB workspace into the Wavelet 2-D tool, save the image (and optionally, the variable map) in a MAT-file (with extension mat or other). For instance, suppose you’ve created an image called brain and want to analyze it in the Wavelet 2-D tool. Type X = brain; map = pink(256); save myfile X map To load this image into the Wavelet 2-D tool, use the menu option File > Load > Image. 3-161 3 Discrete Wavelet Analysis A dialog box appears that lets you select the appropriate MAT-file to be loaded. Note The graphical tools allow you to load an image that does not contain integers from 1 to n. The computations are correct because they act directly on the matrix, but the display of the image is strange. The values less than 1 are evaluated as 1, the values greater than n are evaluated as n, and a real value within the interval [1,n] is evaluated as the closest integer. The coefficients, approximations, and details produced by wavelet decomposition are not indexed image matrices. To display these images in a suitable way, the Wavelet 2-D tool follows these rules: • Reconstructed approximations are displayed using the colormap map. • The coefficients and the reconstructed details are displayed using the colormap map applied to a rescaled version of the matrices. Note The first two-dimensional variable encountered in the file (except the variable map, which is reserved for the colormap) is considered the image. Variables are inspected in alphabetical order. Loading Discrete Wavelet Transform Coefficients. To load discrete wavelet transform (DWT) coefficients into the Wavelet 2-D tool, first save the appropriate data in a MAT-file, which must contain at least the two variables: • coefs, the coefficients vector • sizes, the bookkeeping matrix For an indexed image the matrix sizes is a (n+2-by-2) array: 3-162 Two-Dimensional Discrete Wavelet Analysis For a truecolor image, the matrix sizes is a (n+2-by-3): Variable coefs must be a vector of concatenated DWT coefficients. The coefs vector for an n-level decomposition contains 3n+1 sections, consisting of the level-n approximation coefficients, followed by the horizontal, vertical, and diagonal detail coefficients, in that order, for each level. Variable sizes is a matrix, the rows of which specify the size of cAn, the size of cHn (or cVn, or cDn),..., the size of cH1 (or cV1, or cD1), and the size of the original image X. The sizes of vertical and diagonal details are the same as the horizontal detail. After constructing or editing the appropriate data in your workspace, type 3-163 3 Discrete Wavelet Analysis save myfile coefs sizes Use the File > Load > Coefficients menu option from the Wavelet 2-D tool to load the data into the graphical tool. A dialog box appears, allowing you to choose the folder and file in which your data reside. Loading Decompositions. To load discrete wavelet transform decomposition data into the Wavelet 2-D tool, you must first save the appropriate data in a MAT-file (with extension wa2 or other). The MAT-file contains these variables. Variable Status Description coefs Required Vector of concatenated DWT coefficients sizes Required Matrix specifying sizes of components of coefs and of the original image wave_name Required String specifying name of wavelet used for decomposition (e.g., db3) map Optional n-by-3 colormap matrix. data_name Optional String specifying name of decomposition After constructing or editing the appropriate data in your workspace, type save myfile.wa2 coefs sizes wave_name Use the File > Load > Decomposition menu option from the Wavelet 2-D tool to load the image decomposition data. A dialog box appears, allowing you to choose the folder and file in which your data reside. 3-164 Two-Dimensional Discrete Wavelet Analysis Note When loading an image, a decomposition, or coefficients from a MAT-file, the extension of this file is free. The mat extension is not necessary. 3-165 3 Discrete Wavelet Analysis Two-Dimensional Discrete Stationary Wavelet Analysis This section takes you through the features of two-dimensional discrete stationary wavelet analysis using the Wavelet Toolbox software. Analysis-Decomposition Function Function Name Purpose swt2 Decomposition Synthesis-Reconstruction Function Function Name Purpose iswt2 Reconstruction The stationary wavelet decomposition structure is more tractable than the wavelet one. So, the utilities useful for the wavelet case are not necessary for the Stationary Wavelet Transform (SWT). In this section, you’ll learn to • Load an image • Analyze an image • Perform single-level and multilevel image decompositions and reconstructions (command line only) • denoise an image Two-Dimensional Analysis Using the Command Line In this example, we’ll show how you can use two-dimensional stationary wavelet analysis to denoise an image. 3-166 Two-Dimensional Discrete Stationary Wavelet Analysis Note Instead of using image(I) to visualize the image I, we use image(wcodemat(I)), which displays a rescaled version of I leading to a clearer presentation of the details and approximations (see the wcodemat reference page). This example involves a image containing noise. 1 Load an image. From the MATLAB prompt, type load noiswom whos Name Size Bytes Class X 96x96 73728 double array map 255x3 6120 double array For the SWT, if a decomposition at level k is needed, 2^k must divide evenly into size(X,1) and size(X,2). If your original image is not of correct size, you can use the Image Extension GUI tool or the function wextend to extend it. 2 Perform a single-level Stationary Wavelet Decomposition. Perform a single-level decomposition of the image using the db1 wavelet. Type [swa,swh,swv,swd] = swt2(X,1,'db1'); This generates the coefficients matrices of the level-one approximation (swa) and horizontal, vertical and diagonal details (swh, swv, and swd, respectively). Both are of size-the-image size. Type whos 3-167 3 Discrete Wavelet Analysis Name Size Bytes Class X 96x96 73728 double array map 255x3 6120 double array swa 96x96 73728 double array swh 96x96 73728 double array swv 96x96 73728 double array swd 96x96 73728 double array 3 Display the coefficients of approximation and details. To display the coefficients of approximation and details at level 1, type map = pink(size(map,1)); colormap(map) subplot(2,2,1), image(wcodemat(swa,192)); title('Approximation swa') subplot(2,2,2), image(wcodemat(swh,192)); title('Horiz. Detail swh') subplot(2,2,3), image(wcodemat(swv,192)); title('Vertical Detail swv') subplot(2,2,4), image(wcodemat(swd,192)); title('Diag. Detail swd'); 3-168 Two-Dimensional Discrete Stationary Wavelet Analysis 4 Regenerate the image by Inverse Stationary Wavelet Transform. To find the inverse transform, type A0 = iswt2(swa,swh,swv,swd,'db1'); To check the perfect reconstruction, type err = max(max(abs(X-A0))) err = 1.1369e-13 5 Construct and display approximation and details from the coefficients. 3-169 3 Discrete Wavelet Analysis To construct the level 1 approximation and details (A1, H1, V1 and D1) from the coefficients swa, swh, swv and swd, type nulcfs = zeros(size(swa)); A1 = iswt2(swa,nulcfs,nulcfs,nulcfs,'db1'); H1 = iswt2(nulcfs,swh,nulcfs,nulcfs,'db1'); V1 = iswt2(nulcfs,nulcfs,swv,nulcfs,'db1'); D1 = iswt2(nulcfs,nulcfs,nulcfs,swd,'db1'); To display the approximation and details at level 1, type colormap(map) subplot(2,2,1), image(wcodemat(A1,192)); title('Approximation A1') subplot(2,2,2), image(wcodemat(H1,192)); title('Horiz. Detail H1') subplot(2,2,3), image(wcodemat(V1,192)); title('Vertical Detail V1') subplot(2,2,4), image(wcodemat(D1,192)); title('Diag. Detail D1') 3-170 Two-Dimensional Discrete Stationary Wavelet Analysis 6 Perform a multilevel Stationary Wavelet Decomposition. To perform a decomposition at level 3 of the image (again using the db1 wavelet), type [swa,swh,swv,swd] = swt2(X,3,'db1'); This generates the coefficients of the approximations at levels 1, 2, and 3 (swa) and the coefficients of the details (swh, swv and swd). Observe that the matrices swa(:,:,i), swh(:,:,i), swv(:,:,i), and swd(:,:,i) for a given level i are of size-the-image size. Type clear A0 A1 D1 H1 V1 err nulcfs whos 3-171 3 Discrete Wavelet Analysis Name Size Bytes Class X 96x96 73728 double array map 255x3 6120 double array swa 96x96x3 221184 double array swh 96x96x3 221184 double array swv 96x96x3 221184 double array swd 96x96x3 221184 double array 7 Display the coefficients of approximations and details. To display the coefficients of approximations and details, type colormap(map) kp = 0; for i = 1:3 subplot(3,4,kp+1), image(wcodemat(swa(:,:,i),192)); title(['Approx. cfs level ',num2str(i)]) subplot(3,4,kp+2), image(wcodemat(swh(:,:,i),192)); title(['Horiz. Det. cfs level ',num2str(i)]) subplot(3,4,kp+3), image(wcodemat(swv(:,:,i),192)); title(['Vert. Det. cfs level ',num2str(i)]) subplot(3,4,kp+4), image(wcodemat(swd(:,:,i),192)); title(['Diag. Det. cfs level ',num2str(i)]) kp = kp + 4; end 8 Reconstruct approximation at Level 3 and details from coefficients. To reconstruct the approximation at level 3, type mzero = zeros(size(swd)); A = mzero; A(:,:,3) = iswt2(swa,mzero,mzero,mzero,'db1'); To reconstruct the details at levels 1, 2 and 3, type H = mzero; V = mzero; 3-172 Two-Dimensional Discrete Stationary Wavelet Analysis D = mzero; for i = 1:3 swcfs = mzero; swcfs(:,:,i) = swh(:,:,i); H(:,:,i) = iswt2(mzero,swcfs,mzero,mzero,'db1'); swcfs = mzero; swcfs(:,:,i) = swv(:,:,i); V(:,:,i) = iswt2(mzero,mzero,swcfs,mzero,'db1'); swcfs = mzero; swcfs(:,:,i) = swd(:,:,i); D(:,:,i) = iswt2(mzero,mzero,mzero,swcfs,'db1'); end 9 Reconstruct and display approximations at Levels 1, 2 from approximation at Level 3 and details at Levels 1, 2, and 3. To reconstruct the approximations at levels 2 and 3, type A(:,:,2) = A(:,:,3) + H(:,:,3) + V(:,:,3) + D(:,:,3); A(:,:,1) = A(:,:,2) + H(:,:,2) + V(:,:,2) + D(:,:,2); To display the approximations and details at levels 1, 2, and 3, type colormap(map) kp = 0; for i = 1:3 subplot(3,4,kp+1), image(wcodemat(A(:,:,i),192)); title(['Approx. level ',num2str(i)]) subplot(3,4,kp+2), image(wcodemat(H(:,:,i),192)); title(['Horiz. Det. level ',num2str(i)]) subplot(3,4,kp+3), image(wcodemat(V(:,:,i),192)); title(['Vert. Det. level ',num2str(i)]) subplot(3,4,kp+4), image(wcodemat(D(:,:,i),192)); title(['Diag. Det. level ',num2str(i)]) kp = kp + 4; end 10 Remove noise by thresholding. To denoise an image, use the threshold value we find using the GUI tool (see the next section), use the wthresh command to perform the actual thresholding of the detail coefficients, and then use the iswt2 command to obtain the denoised image. 3-173 3 Discrete Wavelet Analysis thr = 44.5; sorh = 's'; dswh = wthresh(swh,sorh,thr); dswv = wthresh(swv,sorh,thr); dswd = wthresh(swd,sorh,thr); clean = iswt2(swa,dswh,dswv,dswd,'db1'); To display both the original and denoised images, type colormap(map) subplot(1,2,1), title('Original subplot(1,2,2), title('denoised 3-174 image(wcodemat(X,192)); image') image(wcodemat(clean,192)); image') Two-Dimensional Discrete Stationary Wavelet Analysis A second syntax can be used for the swt2 and iswt2 functions, giving the same results: lev= 4; swc = swt2(X,lev,'db1'); swcden = swc; swcden(:,:,1:end-1) = wthresh(swcden(:,:,1:end-1),sorh,thr); clean = iswt2(swcden,'db1'); You obtain the same plot by using the plot commands in step 9 above. Interactive 2-D Stationary Wavelet Transform Denoising In this section, we explore a strategy for de-noising images based on the two-dimensional stationary wavelet analysis using the graphical interface tools. The basic idea is to average many slightly different discrete wavelet analyses. 1 Start the Stationary Wavelet Transform De-Noising 2-D Tool. From the MATLAB prompt, type wavemenu The Wavelet Toolbox Main Menu appears: 3-175 3 Discrete Wavelet Analysis Click the SWT De-noising 2-D menu item. 3-176 Two-Dimensional Discrete Stationary Wavelet Analysis 2 Load data. From the File menu, choose the Load Image option. When the Load Image dialog box appears, select the MAT-file noiswom.mat, which should reside in the MATLAB folder toolbox/wavelet/wavedemo. Click the OK button. The noisy woman image is loaded into the SWT De-noising 2-D tool. 3 Perform a Stationary Wavelet Decomposition. Select the haar wavelet from the Wavelet menu, select 4 from the Level menu, and then click the Decompose Image button. The tool displays the histograms of the stationary wavelet detail coefficients of the image on the left of the window. These histograms are organized as follows: • From the bottom for level 1 to the top for level 4 3-177 3 Discrete Wavelet Analysis • On the left horizontal coefficients, in the middle diagonal coefficients, and on the right vertical coefficients 4 denoise the image using the Stationary Wavelet Transform. While a number of options are available for fine-tuning the de-noising algorithm, we’ll accept the defaults of fixed form soft thresholding and unscaled white noise. The sliders located to the right of the window control the level dependent thresholds indicated by yellow dotted lines running vertically through the histograms of the coefficients on the left of the window. Click the denoise button. The result seems to be oversmoothed and the selected thresholds too aggressive. Nevertheless, the histogram of the residuals is quite good since it is close to a Gaussian distribution, which is the noise introduced to produce the analyzed image noiswom.mat from a piece of the original image woman.mat. 3-178 Two-Dimensional Discrete Stationary Wavelet Analysis 5 Selecting a thresholding method. From the Select thresholding method menu, choose the Penalize low item. The associated default for the thresholding mode is automatically set to hard; accept it. Use the Sparsity slider to adjust the threshold value close to 45.5, and then click the denoise button. The result is quite satisfactory, although it is possible to improve it slightly. Select the sym6 wavelet and click the Decompose Image button. Use the Sparsity slider to adjust the threshold value close to 40.44, and then click the denoise button. Importing and Exporting Information from the Graphical Interface The tool lets you save the denoised image to disk. The toolbox creates a MAT-file in the current folder with a name you choose. 3-179 3 Discrete Wavelet Analysis To save the denoised image from the present de-noising process, use the menu File > Save denoised Image. A dialog box appears that lets you specify a folder and filename for storing the image. Type the name dnoiswom. After saving the image data to the file dnoiswom.mat, load the variables into your workspace: load dnoiswom whos Name Size Bytes Class X 96x96 73728 double array map 255x3 6120 double array valTHR 3x4 96 double array wname 1x4 8 char array The denoised image is X and map is the colormap. In addition, the parameters of the de-noising process are available. The wavelet name is contained in wname, and the level dependent thresholds are encoded in valTHR. The variable valTHR has four columns (the level of the decomposition) and three rows (one for each detail orientation). 3-180 Three-Dimensional Discrete Wavelet Analysis Three-Dimensional Discrete Wavelet Analysis This section demonstrates the features of three-dimensional discrete wavelet analysis using the Wavelet Toolbox software. The toolbox provides these functions for 3-D data analysis. (You use the Wavelet 3-D GUI to perform all tasks except the first task. • Getting information on the command line functions • Loading 3-D data • Analyzing a 3-D data • Selecting and displaying slices • Creating a slice movie • Creating true 3-D display • Importing and exporting information Performing Three-Dimensional Analysis Using the Command Line The example wavelet3ddemo and the documentation of the Analysis-Decomposition and Synthesis-Reconstruction functions show how you can analyze 3-D arrays efficiently using command line functions dedicated to the three-dimensional wavelet analysis. For more information, see the function reference pages. Analysis-Decomposition Functions Function Name Purpose dwt3 Single-level decomposition wavedec3 Decomposition 3-181 3 Discrete Wavelet Analysis Synthesis-Reconstruction Functions Function Name Purpose idwt3 Single-level reconstruction waverec3 Full reconstruction Performing Three-Dimensional Analysis Using the Graphical Interface In this section you explore the same 3-D-data as in the wavelet3ddemo example, but you use the graphical interface tools. 1 Start the 3-D Wavelet Analysis Tool. From the MATLAB prompt, type wavemenu The Wavelet Tool Main Menu appears. Click the Wavelet 3-D menu item. The discrete wavelet analysis tool for three-dimensional data opens. 3-182 Three-Dimensional Discrete Wavelet Analysis 2 Load a 3-D array. From the File menu, choose Load > Data. When the Load Data dialog box appears, select the MAT-file wmri.mat, which is in the MATLAB folder toolbox/wavelet/wavelet. Click OK to load the 3-D data into the Wavelet 3-D tool. 3 Analyze the 3-D array. Using the Wavelet and Level menus located in the upper part of the tool, specify: • The wavelet familes (one per direction X, Y and Z) • The decomposition level and the wavelet extension mode to be used for the analysis For this analysis, accept the defaults: db1 wavelet for each direction, decomposition at level 2 and symmetric extension mode (sym). 3-183 3 Discrete Wavelet Analysis Click Decompose. After a pause for computation, the Wavelet 3-D tool displays its analysis. 3-184 Three-Dimensional Discrete Wavelet Analysis Review the slices of data and wavelet components in the graphical display. These slices are orthogonal to the z-direction as indicated by Slice Orientation in the command part of the window. This option lets you choose the desired slice orientation. The first row of the graphical display area displays from left to right and for Z = 1: • The original data slice • The approximation at level 2 slice (low-pass component APP2) • The slice which is the sum of all the components from level 1 to level 2, different from the low-pass one. The x-labels of the three axes give you the name and the size of the displayed data. 3-185 3 Discrete Wavelet Analysis The next two lines of axes, display the wavelet coefficients at level 2, which is the desired level of the analysis. In the first line, the first graph contains the coefficients of approximation at level 2. The remaining seven axes display the seven types of wavelet coefficients at level 2. These coefficients contain the x-labels of the eight axes and display the name, type and size of the displayed data. For example, in the third graphic of the bottom line, you can see the Cfs-DAD coefficients at level 2, which correspond to an array of size 32 x 32 x 7. The name of the DAD coefficients group indicates that it is obtained using • The high-pass filter in x direction (D for detail) • The low-pass filter in y direction (A for approximation) • The high-pass filter in Z direction (D again for detail), leading to the DAD component 3-186 Three-Dimensional Discrete Wavelet Analysis You use the Displayed Level in the command part of the window to choose the level of the displayed component, from 1 to the decomposition level. You can modify characteristics of the display using the options in the command part of the window. Each pair of sliders controls part of graphical array, the original and the reconstructed slices with the first pair or the coefficients slices with the second pair. Above each slider you can see the number of slices in the current slice orientation. Using the slider (or by directly editing the values) of Rec. Z-Slice, choose slice number twelve. Similarly, choose slice number two using Cfs. Z-Slice. 3-187 3 Discrete Wavelet Analysis The Slice Movie button lets you see a movie of all the slices, first for the reconstructed slices and then for the coefficients slices. In this case, the movie contains 27 reconstructed images and 7 coefficients images. 3D Display lets you examine the original data and the wavelet components in true 3-D mode. Click 3D Display and select APP1. 3-188 Three-Dimensional Discrete Wavelet Analysis A rotated 3-D view of the approximation at level 1 opens in a new window. Use the sliders in the 3-D tool to examine the 3-D data. Importing and Exporting Information from the Graphical Interface You can import information from and export information either to disk or to the workspace using the Wavelet 3-D graphical tool. Loading Information into the Wavelet 3-D Tool To load 3-D data you have constructed in your MATLAB workspace into the Wavelet 3-D tool, save the 3-D data in a MAT-file, using M = magic(8); X = repmat(M,[1 1 8]); save magic3d X whos where M and X are Name Size Bytes Class M 8x8 512 double X 8x8x8 4096 double 3-189 3 Discrete Wavelet Analysis To load this 3-D data into the Wavelet 3-D tool, use the menu option File > Load Data. You then select the MAT-file to load. Similarly, you can load information from the workspace using File > Import Data. You then select the variable to load. Saving Information to a File You can save decompositions and approximations from the Wavelet 3-D tool to a file or to the workspace. Saving Decompositions. The Wavelet 3-D tool lets you save the entire set of data from a discrete wavelet analysis to a file. The toolbox creates a MAT-file in the current folder with a name you choose. 1 Open the Wavelet 3-D tool with File > Load Data, and select magic3d to load the 3-D data file. 3-190 Three-Dimensional Discrete Wavelet Analysis 2 After analyzing your data, save it by using File > Save > Decomposition. 3 In the dialog box that appears, specify a folder and file name for storing the decomposition data. Type the name dec_magic3d. 4 After saving the decomposition data to the file dec_magic3d.mat, load the variables into your workspace. load dec_magic3d whos where wdec is Name Size Bytes Class wdec 1x1 9182 struct The variable wdec contains the wavelet decomposition structure. wdec = sizeINI: [8 8 8] level: 2 filters: [1x1 struct] mode: 'sym' dec: {15x1 cell} sizes: [3x3 double] Saving Approximations. You can process a 3-D data in the Wavelet 3-D tool and then save any desired approximation, depending on the level chosen for the decomposition. 1 Open the Wavelet 3-D tool and load the file containing the 3-D data to analyze by using File > Load Data 2 Select magic3d. 3 Select the File > Save > Approximations > Approximation at level 2 menu option. 3-191 3 Discrete Wavelet Analysis 4 In the dialog box that appears, select a folder and file name for the MAT-file. For this example, choose the name App2_magic3D. 5 Load the image data into your workspace. load App2_magic3D whos where x is 3-192 Name Size Bytes Class x 8x8x8 4096 double 4 Wavelet Packets • “About Wavelet Packet Analysis” on page 4-2 • “One-Dimensional Wavelet Packet Analysis” on page 4-7 • “Two-Dimensional Wavelet Packet Analysis” on page 4-15 • “Importing and Exporting from Graphical Tools” on page 4-22 • “Wavelet Packets” on page 4-30 • “Introduction to Object-Oriented Features” on page 4-52 • “Objects in the Wavelet Toolbox Software” on page 4-53 • “Examples Using Objects” on page 4-55 • “Description of Objects in the Wavelet Toolbox Software” on page 4-67 • “Advanced Use of Objects” on page 4-74 4 Wavelet Packets About Wavelet Packet Analysis Wavelet Toolbox software contains graphical tools and command line functions that let you • Examine and explore characteristics of individual wavelet packets • Perform wavelet packet analysis of one- and two-dimensional data • Use wavelet packets to compress and remove noise from signals and images This chapter takes you step-by-step through examples that teach you how to use the Wavelet Packet 1-D and Wavelet Packet 2-D graphical tools. The last section discusses how to transfer information from the graphical tools into your disk, and back again. Note All the graphical user interface tools described in this chapter let you import information from and export information to either disk or workspace. Because of the inherent complexity of packing and unpacking complete wavelet packet decomposition tree structures, we recommend using the Wavelet Packet 1-D and Wavelet Packet 2-D graphical tools for performing exploratory analyses. The command line functions are also available and provide the same capabilities. However, it is most efficient to use the command line only for performing batch processing. Note For more background on the wavelet packets, you can see the section “Wavelet Packets” on page 4-30. Some object-oriented programming features are used for wavelet packet tree structures. For more detail, refer to “Introduction to Object-Oriented Features” on page 4-52. 4-2 About Wavelet Packet Analysis This chapter takes you through the features of one- and two-dimensional wavelet packet analysis using the Wavelet Toolbox software. You’ll learn how to • Load a signal or image • Perform a wavelet packet analysis of a signal or image • Compress a signal • Remove noise from a signal • Compress an image • Show statistics and histograms The toolbox provides these functions for wavelet packet analysis. For more information, see the reference pages. The reference entries for these functions include examples showing how to perform wavelet packet analysis via the command line. Some more advanced examples mixing command line and GUI functions can be found in the section “Examples Using Objects” on page 4-55. Analysis-Decomposition Functions Function Name Purpose wpcoef Wavelet packet coefficients wpdec and wpdec2 Full decomposition wpsplt Decompose packet Synthesis-Reconstruction Functions Function Name Purpose wprcoef Reconstruct coefficients wprec and wprec2 Full reconstruction wpjoin Recompose packet 4-3 4 Wavelet Packets Decomposition Structure Utilities Function Name Purpose besttree Find best tree bestlevt Find best level tree entrupd Update wavelet packets entropy get Get WPTREE object fields contents read Read values in WPTREE object fields wenergy Entropy wp2wtree Extract wavelet tree from wavelet packet tree wpcutree Cut wavelet packet tree De-Noising and Compression Function Name Purpose ddencmp Default values for de-noising and compression wpbmpen Penalized threshold for wavelet packet de-noising wpdencmp De-noising and compression using wavelet packets wpthcoef Wavelet packets coefficients thresholding wthrmngr Threshold settings manager In the wavelet packet framework, compression and de-noising ideas are exactly the same as those developed in the wavelet framework. The only difference is that wavelet packets offer a more complex and flexible analysis, because in wavelet packet analysis, the details as well as the approximations are split. 4-4 About Wavelet Packet Analysis A single wavelet packet decomposition gives a lot of bases from which you can look for the best representation with respect to a design objective. This can be done by finding the “best tree” based on an entropy criterion. De-noising and compression are interesting applications of wavelet packet analysis. The wavelet packet de-noising or compression procedure involves four steps: 1 Decomposition For a given wavelet, compute the wavelet packet decomposition of signal x at level N. 2 Computation of the best tree For a given entropy, compute the optimal wavelet packet tree. Of course, this step is optional. The graphical tools provide a Best Tree button for making this computation quick and easy. 3 Thresholding of wavelet packet coefficients For each packet (except for the approximation), select a threshold and apply thresholding to coefficients. The graphical tools automatically provide an initial threshold based on balancing the amount of compression and retained energy. This threshold is a reasonable first approximation for most cases. However, in general you will have to refine your threshold by trial and error so as to optimize the results to fit your particular analysis and design criteria. 4-5 4 Wavelet Packets The tools facilitate experimentation with different thresholds, and make it easy to alter the tradeoff between amount of compression and retained signal energy. 4 Reconstruction Compute wavelet packet reconstruction based on the original approximation coefficients at level N and the modified coefficients. In this example, we’ll show how you can use one-dimensional wavelet packet analysis to compress and to de-noise a signal. 4-6 One-Dimensional Wavelet Packet Analysis One-Dimensional Wavelet Packet Analysis We now turn to the Wavelet Packet 1-D tool to analyze a synthetic signal that is the sum of two linear chirps. Starting the Wavelet Packet 1-D Tool. 1 From the MATLAB prompt, type wavemenu. The Wavelet Toolbox Main Menu appears. Click the Wavelet Packet 1-D menu item. Loading a Signal. 1 From the File menu, choose the Load Signal option. 4-7 4 Wavelet Packets 2 When the Load Signal dialog box appears, select the MAT-file sumlichr.mat, which should reside in the MATLAB folder toolbox/wavelet/wavedemo. Click the OK button. The sumlichr signal is loaded into the Wavelet Packet 1-D tool. 4-8 One-Dimensional Wavelet Packet Analysis Analyzing a Signal. 1 Make the appropriate settings for the analysis. Select the db2 wavelet, level 4, entropy threshold, and for the threshold parameter type 1. Click the Analyze button. The available entropy types are listed below. Type Description Shannon Nonnormalized entropy involving the logarithm of the squared value of each signal sample — or, more formally, −∑ si2 log(si2 ). Threshold The number of samples for which the absolute value of the signal exceeds a threshold ε. Norm The concentration in l p norm with 1 ≤ p. Log Energy The logarithm of “energy,” defined as the sum over all samples: ∑ log(si2 ). 4-9 4 Wavelet Packets Type Description SURE (Stein’s Unbiased Risk Estimate) A threshold-based method in which the threshold equals 2 log e ( n log 2 (n) ) where n is the number of samples in the signal. User An entropy type criterion you define in a file. For more information about the available entropy types, user-defined entropy, and threshold parameters, see the wentropy reference page and “Choosing the Optimal Decomposition” on page 4-41. Note Many capabilities are available using the command area on the right of the Wavelet Packet 1-D window. Computing the Best Tree. Because there are so many ways to reconstruct the original signal from the wavelet packet decomposition tree, we select the best tree before attempting to compress the signal. 1 Click the Best Tree button. After a pause for computation, the Wavelet Packet 1-D tool displays the best tree. Use the top and bottom sliders to spread nodes apart and pan over to particular areas of the tree, respectively. Observe that, for this analysis, the best tree and the initial tree are almost the same. One branch at the far right of the tree was eliminated. 4-10 One-Dimensional Wavelet Packet Analysis Compressing a Signal Using Wavelet Packets Selecting a Threshold for Compression. 1 Click the Compress button. The Wavelet Packet 1-D Compression window appears with an approximate threshold value automatically selected. 4-11 4 Wavelet Packets The leftmost graph shows how the threshold (vertical yellow dotted line) has been chosen automatically (1.482) to balance the number of zeros in the compressed signal (blue curve that increases as the threshold increases) with the amount of energy retained in the compressed signal (purple curve that decreases as the threshold increases). This threshold means that any signal element whose value is less than 1.482 will be set to zero when we perform the compression. Threshold controls are located to the right (see the red box in the figure above). Note that the automatic threshold of 1.482 results in a retained energy of only 81.49%. This may cause unacceptable amounts of distortion, especially in the peak values of the oscillating signal. Depending on your design criteria, you may want to choose a threshold that retains more of the original signal’s energy. 2 Adjust the threshold by typing 0.8938 in the text field opposite the threshold slider, and then press the Enter key. The value 0.8938 is a number that we have discovered through trial and error yields more satisfactory results for this analysis. 4-12 One-Dimensional Wavelet Packet Analysis After a pause, the Wavelet Packet 1-D Compression window displays new information. Note that, as we have reduced the threshold from 1.482 to 0.8938, • The vertical yellow dotted line has shifted to the left. • The retained energy has increased from 81.49% to 90.96%. • The number of zeros (equivalent to the amount of compression) has decreased from 81.55% to 75.28%. Compressing a Signal. 1 Click the Compress button. The Wavelet Packet 1-D tool compresses the signal using the thresholding criterion we selected. The original (red) and compressed (yellow) signals are displayed superimposed. Visual inspection suggests the compression quality is quite good. Looking more closely at the compressed signal, we can see that the number of zeros in the wavelet packets representation of the compressed signal is about 75.3%, and the retained energy about 91%. If you try to compress the same signal using wavelets with exactly the same parameters, only 89% of the signal energy is retained, and only 59% of the 4-13 4 Wavelet Packets wavelet coefficients set to zero. This illustrates the superiority of wavelet packets for performing compression, at least on certain signals. You can demonstrate this to yourself by returning to the main Wavelet Packet 1-D window, computing the wavelet tree, and then repeating the compression. De-Noising a Signal Using Wavelet Packets We now use the Wavelet Packet 1-D tool to analyze a noisy chirp signal. This analysis illustrates the use of Stein’s Unbiased Estimate of Risk (SURE) as a principle for selecting a threshold to be used for de-noising. This technique calls for setting the threshold T to T = 2 log e ( n log 2 (n) ) where n is the length of the signal. A more thorough discussion of the SURE criterion appears in “Choosing the Optimal Decomposition” on page 4-41. For now, suffice it to say that this method works well if your signal is normalized in such a way that the data fit the model x(t) = f(t) + e(t), where e(t) is a Gaussian white noise with zero mean and unit variance. If you’ve already started the Wavelet Packet 1-D tool and it is active on your computer’s desktop, skip ahead to step 3. 4-14 Two-Dimensional Wavelet Packet Analysis Two-Dimensional Wavelet Packet Analysis In this section, we employ the Wavelet Packet 2-D tool to analyze and compress an image of a fingerprint. This is a real-world problem: the Federal Bureau of Investigation (FBI) maintains a large database of fingerprints — about 30 million sets of them. The cost of storing all this data runs to hundreds of millions of dollars. “The FBI uses eight bits per pixel to define the shade of gray and stores 500 pixels per inch, which works out to about 700,000 pixels and 0.7 megabytes per finger to store finger prints in electronic form.” (Wickerhauser, see the reference [Wic94] p. 387, listed in “References”). “The technique involves a two-dimensional DWT, uniform scalar quantization (a process that truncates, or quantizes, the precision of the floating-point DWT output) and Huffman entropy coding (i.e., encoding the quantized DWT output with a minimal number of bits).” (Brislawn, see the reference [Bris95] p. 1278, listed in “References”). By turning to wavelets, the FBI has achieved a 15:1 compression ratio. In this application, wavelet compression is better than the more traditional JPEG compression, as it avoids small square artifacts and is particularly well suited to detect discontinuities (lines) in the fingerprint. Note that the international standard JPEG 2000 will include the wavelets as a part of the compression and quantization process. This points out the present strength of the wavelets. Starting the Wavelet Packet 2-D Tool. 1 From the MATLAB prompt, type wavemenu The Wavelet Toolbox Main Menu appears. 4-15 4 Wavelet Packets Click the Wavelet Packet 2-D menu item. Loading an Image From the File menu, choose the Load Image option. 4-16 Two-Dimensional Wavelet Packet Analysis 2 When the Load Image dialog box appears, select the MAT-file detfingr.mat, which should reside in the MATLAB folder toolbox/wavelet/wavedemo. Click the OK button. The fingerprint image is loaded into the Wavelet Packet 2-D tool. Analyzing an Image 3 Make the appropriate settings for the analysis. Select the haar wavelet, level 3, and entropy type shannon. Click the Analyze button. 4-17 4 Wavelet Packets Note Many capabilities are available using the command area on the right of the Wavelet Packet 2-D window. 4 Click the Best Tree button to compute the best tree before compressing the image. Compressing an Image Using Wavelet Packets 1 Click the Compress button to bring up the Wavelet Packet 2-D Compression window. Select the Bal. sparsity-norm (sqrt) option from the Select thresholding method menu. 4-18 Two-Dimensional Wavelet Packet Analysis Notice that the default threshold (7.125) provides about 64% compression while retaining virtually all the energy of the original image. Depending on your criteria, it may be worthwhile experimenting with more aggressive thresholds to achieve a higher degree of compression. Recall that we are not doing any quantization of the image, merely setting specific coefficients to zero. This can be considered a precompression step in a broader compression system. 2 Alter the threshold: type the number 30 in the text field opposite the threshold slider located on the right side of the Wavelet Packet 2-D Compression window. Then press the Enter key. Setting all wavelet packet coefficients whose value falls below 30 to zero yields much better results. Note that the new threshold achieves around 92% of zeros, while still retaining nearly 98% of the image energy. 3 Click the Compress button to start the compression. You can see the result obtained by wavelet packet coefficients thresholding and image reconstruction. The visual recovery is correct, but not perfect. 4-19 4 Wavelet Packets The compressed image, shown side by side with the original, shows some artifacts. 4 Click the Close button located at the bottom of the Wavelet Packet 2-D Compression window. Update the synthesized image by clicking Yes when the dialog box appears. Take this opportunity to try out your own compression strategy. Adjust the threshold value, the entropy function, and the wavelet, and see if you can obtain better results. Hint The bior6.8 wavelet is better suited to this analysis than is haar, and can lead to a better compression ratio. When a biorthogonal wavelet is used, then instead of “Retained energy” the information displayed is “Energy ratio.” For more information, see “Compression Scores” on page 5-53. Before concluding this analysis, it is worth turning our attention to the “colored coefficients for terminal nodes plot” and considering the best tree decomposition for this image. 4-20 Two-Dimensional Wavelet Packet Analysis This plot is shown in the lower right side of the Wavelet Packet 2-D tool. The plot shows us which details have been decomposed and which have not. Larger squares represent details that have not been broken down to as many levels as smaller squares. Consider, for example, this level 2 decomposition pattern: Looking at the pattern of small and large squares in the fingerprint analysis shows that the best tree algorithm has apparently singled out the diagonal details, often sparing these from further decomposition. Why is this? If we consider the original image, we realize that much of its information is concentrated in the sharp edges that constitute the fingerprint’s pattern. Looking at these edges, we see that they are predominantly oriented horizontally and vertically. This explains why the best tree algorithm has “chosen” not to decompose the diagonal details — they do not provide very much information. 4-21 4 Wavelet Packets Importing and Exporting from Graphical Tools The Wavelet Packet 1-D and Wavelet Packet 2-D tools let you import information from and export information to your disk. If you adhere to the proper file formats, you can • Save decompositions as well as synthesized signals and images from the wavelet packet graphical tools to disk • Load signals, images, and one- and two-dimensional decompositions from disk into the Wavelet Packet 1-D and Wavelet Packet 2-D graphical tools Saving Information to Disk Using specific file formats, the graphical tools let you save synthesized signals or images, as well as one- or two-dimensional wavelet packet decomposition structures. This feature provides flexibility and allows you to combine command line and graphical interface operations. Saving Synthesized Signals You can process a signal in the Wavelet Packet 1-D tool, and then save the processed signal to a MAT-file. For example, load the example analysis: File > Example Analysis > db1 – depth: 2 – ent: shannon > sumsin and perform a compression or de-noising operation on the original signal. When you close the Wavelet Packet 1-D De-noising or Wavelet Packet 1-D Compression window, update the synthesized signal by clicking Yes in the dialog box. Then, from the Wavelet Packet 1-D tool, select the File > Save > Synthesized Signal menu option. A dialog box appears allowing you to select a folder and filename for the MAT-file. For this example, choose the name synthsig. 4-22 Importing and Exporting from Graphical Tools To load the signal into your workspace, simply type load synthsig whos Name Size Bytes Class synthsig 1x1000 8000 double array valTHR 1x1 8 double array wname 1x3 6 char array The synthesized signal is given by synthsig. In addition, the parameters of the de-noising or compression process are given by the wavelet name (wname) and the global threshold (valTHR). valTHR valTHR = 1.9961 Saving Synthesized Images You can process an image in the Wavelet Packet 2-D tool, and then save the processed image to a MAT-file (with extension mat or other). For example, load the example analysis: File > Example Analysis > db1 – depth: 1 – ent: shannon > woman and perform a compression on the original image. When you close the Wavelet Packet 2-D Compression window, update the synthesized image by clicking Yes in the dialog box that appears. Then, from the Wavelet 2-D tool, select the File > Save > Synthesized Image menu option. A dialog box appears allowing you to select a folder and filename for the MAT-file. For this example, choose the name wpsymage. 4-23 4 Wavelet Packets To load the image into your workspace, simply type load wpsymage whos Name Size Bytes Class X 256x256 524288 double array map 255x3 6120 double array valTHR 1x1 8 double array wname 1x3 6 char array The synthesized image is given by X. The variable map contains the associated colormap. In addition, the parameters of the de-noising or compression process are given by the wavelet name (wname) and the global threshold (valTHR). Saving One-Dimensional Decomposition Structures The Wavelet Packet 1-D tool lets you save an entire wavelet packet decomposition tree and related data to your disk. The toolbox creates a MAT-file in the current folder with a name you choose, followed by the extension wp1 (wavelet packet 1-D). Open the Wavelet Packet 1-D tool and load the example analysis: File > Example Analysis > db1 – depth: 2 – ent: shannon > sumsin To save the data from this analysis, use the menu option File > Save Decomposition. A dialog box appears that lets you specify a folder and file name for storing the decomposition data. Type the name wpdecex1d. After saving the decomposition data to the file wpdecex1d.wp1, load the variables into your workspace. load wpdecex1d.wp1 -mat whos 4-24 Importing and Exporting from Graphical Tools Name Size Bytes Class data_name 1x6 12 char array tree_struct 1x1 11176 wptree object valTHR 0x0 0 double array The variable tree_struct contains the wavelet packet tree structure. The variable data_name contains the data name and valTHR contains the global threshold, which is currently empty since the synthesized signal does not exist. Saving Two-Dimensional Decomposition Structures The file format, variables, and conventions are exactly the same as in the one-dimensional case except for the extension, which is wp2 (wavelet packet 2-D). The variables saved are the same as with the one-dimensional case, with the addition of the colormap matrix map: Name Size Bytes Class data_name 1x5 10 char array map 255x3 6120 double array tree_struct 1x1 527400 wptree object valTHR 1x1 8 double array Save options are also available when performing de-noising or compression inside the Wavelet Packet 1-D and Wavelet Packet 2-D tools. In the Wavelet Packet De-Noising windows, you can save the de-noised signal or image and the decomposition. The same holds true for the Wavelet Packet Compression windows. This way, you can save directly many different trials from inside the De-Noising and Compression windows without going back to the main Wavelet Packet windows during a fine-tuning process. 4-25 4 Wavelet Packets Note When saving a synthesized signal (1-D), a synthesized image (2-D) or a decomposition to a MAT-file, the extension of this file is free. The mat extension is not necessary. Loading Information into the Graphical Tools You can load signals, images, or one- and two-dimensional wavelet packet decompositions into the graphical interface tools. The information you load may have been previously exported from the graphical interface, and then manipulated in the workspace, or it may have been information you generated initially from the command line. In either case, you must observe the strict file formats and data structures used by the graphical tools, or else errors will result when you try to load information. Loading Signals To load a signal you’ve constructed in your MATLAB workspace into the Wavelet Packet 1-D tool, save the signal in a MAT-file (with extension mat or other). For instance, suppose you’ve designed a signal called warma and want to analyze it in the Wavelet Packet 1-D tool. save warma warma The workspace variable warma must be a vector. sizwarma = size(warma) sizwarma = 1 1000 To load this signal into the Wavelet Packet 1-D tool, use the menu option File > Load Signal. A dialog box appears that lets you select the appropriate MAT-file to be loaded. 4-26 Importing and Exporting from Graphical Tools Note The first one-dimensional variable encountered in the file is considered the signal. Variables are inspected in alphabetical order. Loading Images This toolbox supports only indexed images. An indexed image is a matrix containing only integers from 1 to n, where n is the number of colors in the image. This image may optionally be accompanied by a n-by-3 matrix called map. This is the colormap associated with the image. When MATLAB displays such an image, it uses the values of the matrix to look up the desired color in this colormap. If the colormap is not given, the Wavelet Packet 2-D graphical tool uses a monotonic colormap with max(max(X)) min(min(X))+1 colors. To load an image you’ve constructed in your MATLAB workspace into the Wavelet Packet 2-D tool, save the image (and optionally, the variable map) in a MAT-file (with extension mat or other). For instance, suppose you’ve created an image called brain and want to analyze it in the Wavelet Packet 2-D tool. Type X = brain; map = pink(256); save myfile X map To load this image into the Wavelet Packet 2-D tool, use the menu option File > Load Image. A dialog box appears that lets you select the appropriate MAT-file to be loaded. Note The first two-dimensional variable encountered in the file (except the variable map, which is reserved for the colormap) is considered the image. Variables are inspected in alphabetical order. 4-27 4 Wavelet Packets Caution The graphical tools allow you to load an image that does not contain integers from 1 to n. The computations will be correct since they act directly on the matrix, but the display of the image will be strange. The values less than 1 will be evaluated as 1, the values greater than n will be evaluated as n, and a real value within the interval [1,n] will be evaluated as the closest integer. Note that the coefficients, approximations, and details produced by wavelet packets decomposition are not indexed image matrices. To display these images in a suitable way, the Wavelet Packet 2-D tool follows these rules: • Reconstructed approximations are displayed using the colormap map. The same holds for the result of the reconstruction of selected nodes. • The coefficients and the reconstructed details are displayed using the colormap map applied to a rescaled version of the matrices. Loading Wavelet Packet Decomposition Structures You can load one- and two-dimensional wavelet packet decompositions into the graphical tools providing you have previously saved the decomposition data in a MAT-file of the appropriate format. While it is possible to edit data originally created using the graphical tools and then exported, you must be careful about doing so. Wavelet packet data structures are complex, and the graphical tools do not do any consistency checking. This can lead to errors if you try to load improperly formatted data. One-dimensional data file contains the following variables: 4-28 Variable Status tree_struct Required Object specifying the tree structure data_name Optional String specifying the name of the decomposition valTHR Optional Global threshold (can be empty if neither compression nor de-noising has been done) Description Importing and Exporting from Graphical Tools These variables must be saved in a MAT-file (with extension wp1 or other). Two-dimensional data file contains the following variables: Variable Status Description tree_struct Required Object specifying the tree structure data_name Optional String specifying the name of the decomposition map Optional Image map valTHR Optional Global threshold (can be empty if neither compression nor de-noising has been done) These variables must be saved in a MAT-file (with extension wp2 or other). To load the properly formatted data, use the menu option File > Load Decomposition Structure from the appropriate tool, and then select the desired MAT-file from the dialog box that appears. The Wavelet Packet 1-D or 2-D graphical tool then automatically updates its display to show the new analysis. Note When loading a signal (1-D), an image (2-D), or a decomposition (1-D or 2-D) from a MAT-file, the extension of this file is free. The mat extension is not necessary. 4-29 4 Wavelet Packets Wavelet Packets The wavelet packet method is a generalization of wavelet decomposition that offers a richer signal analysis. Wavelet packet atoms are waveforms indexed by three naturally interpreted parameters: position, scale (as in wavelet decomposition), and frequency. For a given orthogonal wavelet function, we generate a library of bases called wavelet packet bases. Each of these bases offers a particular way of coding signals, preserving global energy, and reconstructing exact features. The wavelet packets can be used for numerous expansions of a given signal. We then select the most suitable decomposition of a given signal with respect to an entropy-based criterion. There exist simple and efficient algorithms for both wavelet packet decomposition and optimal decomposition selection. We can then produce adaptive filtering algorithms with direct applications in optimal signal coding and data compression. From Wavelets to Wavelet Packets In the orthogonal wavelet decomposition procedure, the generic step splits the approximation coefficients into two parts. After splitting we obtain a vector of approximation coefficients and a vector of detail coefficients, both at a coarser scale. The information lost between two successive approximations is captured in the detail coefficients. Then the next step consists of splitting the new approximation coefficient vector; successive details are never reanalyzed. In the corresponding wavelet packet situation, each detail coefficient vector is also decomposed into two parts using the same approach as in approximation vector splitting. This offers the richest analysis: the complete binary tree is produced as shown in the following figure. 4-30 Wavelet Packets Wavelet Packet Decomposition Tree at Level 3 The idea of this decomposition is to start from a scale-oriented decomposition, and then to analyze the obtained signals on frequency subbands. Wavelet Packets in Action: An Introduction The following simple examples illustrate certain differences between wavelet analysis and wavelet packet analysis. Wavelet Packet Spectrum The spectral analysis of wide-sense stationary signals using the Fourier transform is well-established. For nonstationary signals, there exist local Fourier methods such as the short-time Fourier transform (STFT). See “Short-Time Fourier Transform” for a brief description. Because wavelets are localized in time and frequency, it is possible to use wavelet-based counterparts to the STFT for the time-frequency analysis of nonstationary signals. For example, it is possible to construct the scalogram (wscalogram) based on the continuous wavelet transform (CWT). However, a potential drawback of using the CWT is that it is computationally expensive. The discrete wavelet transform (DWT) permits a time-frequency decomposition of the input signal, but the degree of frequency resolution in the DWT is typically considered too coarse for practical time-frequency analysis. As a compromise between the DWT- and CWT-based techniques, wavelet packets provide a computationally-efficient alternative with sufficient frequency resolution. You can use wpspectrum to perform a time-frequency analysis of your signal using wavelet packets. 4-31 4 Wavelet Packets The following examples illustrate the use of wavelet packets to perform a local spectral analysis. The following examples also use spectrogram from the Signal Processing Toolbox™ software as a benchmark to compare against the wavelet packet spectrum. If you do not have the Signal Processing Toolbox software, you can simply run the wavelet packet spectrum examples. Wavelet packet spectrum of a sine wave. fs = 1000; % sampling rate t = 0:1/fs:2; % 2 secs at 1kHz sample rate y = sin(256*pi*t); % sine of period 128 level = 6; wpt = wpdec(y,level,'sym8'); [Spec,Time,Freq] = wpspectrum(wpt,fs,'plot'); If you have the Signal Processing Toolbox software, you can compute the short-time Fourier transform. figure; windowsize = 128; window = hanning(windowsize); nfft = windowsize; noverlap = windowsize-1; [S,F,T] = spectrogram(y,window,noverlap,nfft,fs); imagesc(T,F,log10(abs(S))) set(gca,'YDir','Normal') xlabel('Time (secs)') ylabel('Freq (Hz)') title('Short-time Fourier Transform spectrum') Sum of two sine waves with frequencies of 64 and 128 hertz. fs = 1000; t = 0:1/fs:2; y = sin(128*pi*t) + sin(256*pi*t); % sine of periods 64 and 128. level = 6; wpt = wpdec(y,level,'sym8'); [Spec,Time,Freq] = wpspectrum(wpt,fs,'plot'); 4-32 Wavelet Packets If you have the Signal Processing Toolbox software, you can compute the short-time Fourier transform. figure; windowsize = 128; window = hanning(windowsize); nfft = windowsize; noverlap = windowsize-1; [S,F,T] = spectrogram(y,window,noverlap,nfft,fs); imagesc(T,F,log10(abs(S))) set(gca,'YDir','Normal') xlabel('Time (secs)') ylabel('Freq (Hz)') title('Short-time Fourier Transform spectrum') Signal with an abrupt change in frequency from 16 to 64 hertz at two seconds. fs = 500; t = 0:1/fs:4; y = sin(32*pi*t).*(t<2) + sin(128*pi*t).*(t>=2); level = 6; wpt = wpdec(y,level,'sym8'); [Spec,Time,Freq] = wpspectrum(wpt,fs,'plot'); If you have the Signal Processing Toolbox software, you can compute the short-time Fourier transform. figure; windowsize = 128; window = hanning(windowsize); nfft = windowsize; noverlap = windowsize-1; [S,F,T] = spectrogram(y,window,noverlap,nfft,fs); imagesc(T,F,log10(abs(S))) set(gca,'YDir','Normal') xlabel('Time (secs)') ylabel('Freq (Hz)') title('Short-time Fourier Transform spectrum') 4-33 4 Wavelet Packets Wavelet packet spectrum of a linear chirp. fs = 1000; t = 0:1/fs:2; y = sin(256*pi*t.^2); level = 6; wpt = wpdec(y,level,'sym8'); [Spec,Time,Freq] = wpspectrum(wpt,fs,'plot'); If you have the Signal Processing Toolbox software, you can compute the short-time Fourier transform. figure; windowsize = 128; window = hanning(windowsize); nfft = windowsize; noverlap = windowsize-1; [S,F,T] = spectrogram(y,window,noverlap,nfft,fs); imagesc(T,F,log10(abs(S))) set(gca,'YDir','Normal') xlabel('Time (secs)') ylabel('Freq (Hz)') title('Short-time Fourier Transform spectrum') Wavelet packet spectrum of quadratic chirp. y = wnoise('quadchirp',10); len = length(y); t = linspace(0,5,len); fs = 1/t(2); level = 6; wpt = wpdec(y,level,'sym8'); [Spec,Time,Freq] = wpspectrum(wpt,fs,'plot'); If you have the Signal Processing Toolbox software, you can compute the short-time Fourier transform. 4-34 Wavelet Packets windowsize = 128; window = hanning(windowsize); nfft = windowsize; noverlap = windowsize-1; imagesc(T,F,log10(abs(S))) set(gca,'YDir','Normal') xlabel('Time (secs)') ylabel('Freq (Hz)') title('Short-time Fourier Transform spectrum') Building Wavelet Packets The computation scheme for wavelet packets generation is easy when using an orthogonal wavelet. We start with the two filters of length 2N, where h(n) and g(n), corresponding to the wavelet. Now by induction let us define the following sequence of functions: (Wn(x), n = 0, 1, 2, ...) by W2n ( x) = 2 2 N −1 ∑ h(k)Wn (2 x − k) k =0 W2n+1 ( x) = 2 2 N −1 ∑ g(k)Wn (2 x − k) k =0 where W0(x) = φ(x) is the scaling function and W1(x) = ψ(x) is the wavelet function. For example for the Haar wavelet we have N = 1, h(0) = h(1) = 1 2 and 4-35 4 Wavelet Packets g(0) = − g(1) = 1 2 The equations become W2n ( x) = Wn (2 x) + Wn (2 x − 1) and W2n+1 ( x) = Wn (2 x) − Wn (2 x − 1) W0(x) = φ(x) is the Haar scaling function and W1(x) = ψ(x) is the Haar wavelet, both supported in [0, 1]. Then we can obtain W2n by adding two 1/2-scaled versions of Wn with distinct supports [0,1/2] and [1/2,1] and obtain W2n+1 by subtracting the same versions of Wn. For n = 0 to 7, we have the W-functions shown in the figure Haar Wavelet Packets on page 4-37. 4-36 Wavelet Packets Haar Wavelet Packets This can be obtained using the following command: [wfun,xgrid] = wpfun('db1',7,5); which returns in wfun the approximate values of Wn for n = 0 to 7, computed on a 1/25 grid of the support xgrid. Starting from more regular original wavelets and using a similar construction, we obtain smoothed versions of this system of W-functions, all with support in the interval [0, 2N–1]. The figure db2 Wavelet Packets on page 4-38 presents the system of W-functions for the original db2 wavelet. 4-37 4 Wavelet Packets db2 Wavelet Packets Wavelet Packet Atoms Starting from the functions (Wn ( x), n ∈ N ) and following the same line leading to orthogonal wavelets, we consider the three-indexed family of analyzing functions (the waveforms): (W j,n,k ( x) = 2− j / 2 Wn (2− j x − k) where n N and (j,k) Z2. 4-38 Wavelet Packets As in the wavelet framework, k can be interpreted as a time-localization parameter and j as a scale parameter. So what is the interpretation of n? The basic idea of the wavelet packets is that for fixed values of j and k, Wj,n,k analyzes the fluctuations of the signal roughly around the position 2j· k, at the scale 2j and at various frequencies for the different admissible values of the last parameter n. In fact, examining carefully the wavelet packets displayed in Haar Wavelet Packets on page 4-37 and db2 Wavelet Packets on page 4-38, the naturally ordered Wn for n = 0, 1, ..., 7, does not match exactly the order defined by the number of oscillations. More precisely, counting the number of zero crossings (up-crossings and down-crossings) for the db1 wavelet packets, we have the following. Natural order n 0 1 2 3 4 5 6 7 Number of zero crossings for db1 Wn 2 3 5 4 9 8 6 7 So, to restore the property that the main frequency increases monotonically with the order, it is convenient to define the frequency order obtained from the natural one recursively. Natural order n 0 1 2 3 4 5 6 7 Frequency order r(n) 0 1 3 2 6 7 5 4 As can be seen in the previous figures, Wr(n)(x) “oscillates” approximately n times. To analyze a signal (the chirp of Example 2 for instance), it is better to plot the wavelet packet coefficients following the frequency order from the low frequencies at the bottom to the high frequencies at the top, rather than naturally ordered coefficients. When plotting the coefficients, the various options related to the “Frequency” or “Natural” order choice are available using the GUI tools. 4-39 4 Wavelet Packets These options are also available from command-line mode when using the wpviewcf function. Organizing the Wavelet Packets The set of functions Wj,n = (Wj,n,k(x), k Z) is the (j,n) wavelet packet. For positive values of integers j and n, wavelet packets are organized in trees. The tree in the figure Wavelet Packets Organized in a Tree; Scale j Defines Depth and Frequency n Defines Position in the Tree on page 4-40 is created to give a maximum level decomposition equal to 3. For each scale j, the possible values of parameter n are 0, 1, ..., 2 j–1. Wavelet Packets Organized in a Tree; Scale j Defines Depth and Frequency n Defines Position in the Tree The notation Wj,n, where j denotes scale parameter and n the frequency parameter, is consistent with the usual depth-position tree labeling. x We have W0,0 = (φ ( x − k), k ∈ Z) , and W1,1 = (ψ ( − k), k ∈ Z) . 2 It turns out that the library of wavelet packet bases contains the wavelet basis and also several other bases. Let us have a look at some of those bases. 4-40 Wavelet Packets More precisely, let V0 denote the space (spanned by the family W0,0 ) in which the signal to be analyzed lies; then (Wd,1; d ≥ 1) is an orthogonal basis of V0. For every strictly positive integer D, (WD,0, (Wd,1; 1 ≤ d ≤ D)) is an orthogonal basis of V0. We also know that the family of functions {(Wj+1,2n), (Wj+1,2n+1)} is an orthogonal basis of the space spanned by Wj,n, which is split into two subspaces: Wj+1,2n spans the first subspace, and Wj+1,2n+1 the second one. This last property gives a precise interpretation of splitting in the wavelet packet organization tree, because all the developed nodes are of the form shown in the figure Wavelet Packet Tree: Split and Merge on page 4-41. Wavelet Packet Tree: Split and Merge It follows that the leaves of every connected binary subtree of the complete tree correspond to an orthogonal basis of the initial space. For a finite energy signal belonging to V0, any wavelet packet basis will provide exact reconstruction and offer a specific way of coding the signal, using information allocation in frequency scale subbands. Choosing the Optimal Decomposition Based on the organization of the wavelet packet library, it is natural to count the decompositions issued from a given orthogonal wavelet. A signal of length N = 2L can be expanded in α different ways, where α is the number of binary subtrees of a complete binary tree of depth L. As a result, α ≥ 2 N / 2 (see [Mal98] page 323). As this number may be very large, and since explicit enumeration is generally unmanageable, it is interesting to find an optimal decomposition with respect 4-41 4 Wavelet Packets to a convenient criterion, computable by an efficient algorithm. We are looking for a minimum of the criterion. Functions verifying an additivity-type property are well suited for efficient searching of binary-tree structures and the fundamental splitting. Classical entropy-based criteria match these conditions and describe information-related properties for an accurate representation of a given signal. Entropy is a common concept in many fields, mainly in signal processing. Let us list four different entropy criteria (see [CoiW92]); many others are available and can be easily integrated (type help wentropy). In the following expressions s is the signal and (si) are the coefficients of s in an orthonormal basis. The entropy E must be an additive cost function such that E(0) = 0 and E(s) = ∑ E(si ) i • The (nonnormalized) Shannon entropy E1(si ) = − si2 log(si2 ) so E1(s) = −∑ si2 log(si2 ) i with the convention 0log(0) = 0. • The concentration in l p norm with 1 ℜ ≤ p p E2(si ) = si so E2(s) = ∑ si p i p = sp • The logarithm of the “energy” entropy E3(si ) = log(si2 ) 4-42 Wavelet Packets so E3(s) = ∑ log(si2 ) i with the convention log(0) = 0. • The threshold entropy E4(si ) = 1 if si > ε and 0 elsewhere, so E4(s) = # {i such that si > ε } is the number of time instants when the signal is greater than a threshold ε. These entropy functions are available using the wentropy file. Example 1: Compute Various Entropies. 1 Generate a signal of energy equal to 1. s = ones(1,16)*0.25; 2 Compute the Shannon entropy of s. e1 = wentropy(s,'shannon') e1 = 2.7726 3 Compute the l1.5 entropy of s, equivalent to norm(s,1.5)1.5. e2 = wentropy(s,'norm',1.5) e2 = 2 4 Compute the “log energy” entropy of s. e3 = wentropy(s,'log energy') e3 = -44.3614 5 Compute the threshold entropy of s, using a threshold value of 0.24. e4 = wentropy(s,'threshold', 0.24) e4 = 16 4-43 4 Wavelet Packets Example 2: Minimum-Entropy Decomposition. This simple example illustrates the use of entropy to determine whether a new splitting is of interest to obtain a minimum-entropy decomposition. 1 We start with a constant original signal. Two pieces of information are sufficient to define and to recover the signal (i.e., length and constant value). w00 = ones(1,16)*0.25; 2 Compute entropy of original signal. e00 = wentropy(w00,'shannon') e00 = 2.7726 3 Then split w00 using the haar wavelet. [w10,w11] = dwt(w00,'db1'); 4 Compute entropy of approximation at level 1. e10 = wentropy(w10,'shannon') e10 = 2.0794 The detail of level 1, w11, is zero; the entropy e11 is zero. Due to the additivity property the entropy of decomposition is given by e10+e11=2.0794. This has to be compared to the initial entropy e00=2.7726. We have e10 + e11 < e00, so the splitting is interesting. 5 Now split w10 (not w11 because the splitting of a null vector is without interest since the entropy is zero). [w20,w21] = dwt(w10,'db1'); 6 We have w20=0.5*ones(1,4) and w21 is zero. The entropy of the approximation level 2 is e20 = wentropy(w20,'shannon') e20 = 1.3863 4-44 Wavelet Packets Again we have e20 + 0 < e10, so splitting makes the entropy decrease. 7 Then [w30,w31] = dwt(w20,'db1'); e30 = wentropy(w30,'shannon') e30 = 0.6931 [w40,w41] = dwt(w30,'db1') w40 = 1.0000 w41 = 0 e40 = wentropy(w40,'shannon') e40 = 0 In the last splitting operation we find that only one piece of information is needed to reconstruct the original signal. The wavelet basis at level 4 is a best basis according to Shannon entropy (with null optimal entropy since e40+e41+e31+e21+e11 = 0). 8 Perform wavelet packets decomposition of the signal s defined in example 1. t = wpdec(s,4,'haar','shannon'); The wavelet packet tree in Entropy Values on page 4-45 shows the nodes labeled with original entropy numbers. Entropy Values 9 Compute the best tree. 4-45 4 Wavelet Packets bt = besttree(t); The best tree is shown in the following figure. In this case, the best tree corresponds to the wavelet tree. The nodes are labeled with optimal entropy. Optimal Entropy Values Some Interesting Subtrees Using wavelet packets requires tree-related actions and labeling. The implementation of the user interface is built around this consideration. For more information on the technical details, see the reference pages. The complete binary tree of depth D corresponding to a wavelet packet decomposition tree developed at level D is denoted by WPT. We have the following interesting subtrees. 4-46 Decomposition Tree Subtree Such That the Set of Leaves Is a Basis Wavelet packets decomposition tree Complete binary tree: WPT of depth D Wavelet packets optimal decomposition tree Binary subtree of WPT Wavelet packets best-level tree Complete binary subtree of WPT Wavelet decomposition tree Left unilateral binary subtree of WPT of depth D Wavelet best-basis tree Left unilateral binary subtree of WPT Wavelet Packets We deduce the following definitions of optimal decompositions, with respect to an entropy criterion E. Decompositions Optimal Decomposition Best-Level Decomposition Wavelet packet decompositions Search among 2 trees Search among D trees Wavelet decompositions Search among D trees Search among D trees D For any nonterminal node, we use the following basic step to find the optimal subtree with respect to a given entropy criterion E (where Eopt denotes the optimal entropy value). Entropy Condition E(node) ≤ Action on Tree and on Entropy Labeling ∑ Eopt(c) ∑ Eopt(c) If (node≠root), merge and set Eopt(node) = E(node) c child of node E(node) > c child of node Split and set Eopt(node) = ∑ Eopt(c) c child of node with the natural initial condition on the reference tree, Eopt(t) = E(t) for each terminal node t. Reconstructing a Signal Approximation from a Node You can use the function wprcoef to reconstruct an approximation to your signal from any node in the wavelet packet tree. This is true irrespective of whether you are working with a full wavelet packet tree, or a subtree 4-47 4 Wavelet Packets determined by an optimality criterion. Use wpcoef if you want to extract the wavelet packet coefficients from a node without reconstructing an approximation to the signal. Load the noisy Doppler signal. load noisdopp Compute the wavelet packet decomposition down to level 5 using the sym4 wavelet. Use the periodization mode. dwtmode('per'); T = wpdec(noisdopp,5,'sym4'); plot(T) Plot the binary wavelet packet tree and click on the (4,1) doublet (node 16). Extract the wavelet packet coefficients from node 16. wpc = wpcoef(T,16); % wpc is length 64 Obtain an approximation to the signal from node 16. rwpc = wprcoef(T,16); % rwpc is length 1024 plot(noisdopp,'k'); hold on; plot(rwpc,'b','linewidth',2); 4-48 Wavelet Packets axis tight; Determine the optimum binary wavelet packet tree. Topt = besttree(T); % plot the best tree plot(Topt) Reconstruct an approximation to the signal from the (3,0) doublet (node 7). rsig = wprcoef(Topt,7); % rsig is length 1024 plot(noisdopp,'k'); hold on; plot(rsig,'b','linewidth',2); axis tight; 4-49 4 Wavelet Packets If you know which doublet in the binary wavelet packet tree you want to extract, you can determine the node corresponding to that doublet with depo2ind. For example, to determine the node corresponding to the doublet (3,0), enter: Node = depo2ind(2,[3 0]); Wavelet Packets 2-D Decomposition Structure Exactly as in the wavelet decomposition case, the preceding one-dimensional framework can be extended to image analysis. Minor direct modifications lead to quaternary tree-related definitions. An example is shown the following figure for depth 2. Quaternary Tree of Depth 2 4-50 Wavelet Packets Wavelet Packets for Compression and De-Noising In the wavelet packet framework, compression and de-noising ideas are identical to those developed in the wavelet framework. The only new feature is a more complete analysis that provides increased flexibility. A single decomposition using wavelet packets generates a large number of bases. You can then look for the best representation with respect to a design objective, using the besttree with an entropy function. 4-51 4 Wavelet Packets Introduction to Object-Oriented Features In the Wavelet Toolbox software, some object-oriented programming features are used for wavelet packet tree structures. You may want to skip this appendix, if you prefer to use the command line functions and graphical user interface (GUI) without knowing about the underlying objects and classes. But, it is useful for Save and Load actions where objects are involved. This appendix lets you understand the objects used in the toolbox, use some functions that are not fully documented in the reference pages, and extend the toolbox functionality using the predefined tree structures and some object programming features. It is helpful to be familiar with the basic MATLAB object-oriented language and terminology. 4-52 Objects in the Wavelet Toolbox™ Software Objects in the Wavelet Toolbox Software Four classes of objects are defined in the Wavelet Toolbox software. The hierarchical organization of these objects is described in the following scheme: Only the Wavelet Packet tools (1-D and 2-D) use the previous objects. More precisely, WPTREE objects are used to build wavelet packets. A short description of this hierarchy of objects follows. The WTBO class is an abstract class. Any object in the toolbox is parented by a WTBO object and would inherit the methods and fields of the WTBO class. The NTREE class is dedicated to tree manipulation (node labels, node splitting, node merging, ...), and it is also an abstract class. The main methods are • nodejoin, which recomposes nodes • nodesplt, which decomposes nodes • wtreemgr, which lets you access most of tree and node information (order, depth, terminal nodes, ascendants of a node, ...) In fact, the wtreemgr method is not used directly, but you can use the functions treeord, treedpth, leaves, nodeasc, ..., and the method get. The DTREE class is dedicated to trees with associated data: vectors or matrices. This class is also an abstract class and some methods have to be overloaded. The aim of the WPTREE class is to manage wavelet packets 1-D and 2-D. Some methods of the DTREE class have been overloaded, for example: split, merge, and recons. 4-53 4 Wavelet Packets Most of the methods are specific to the class WPTREE; for example: bestlevt, besttree, and wp2wtree. By typing help wavelet you can see the available methods in the Tree Management Utilities and Wavelets Packets Algorithms sections. 4-54 Examples Using Objects Examples Using Objects You can use command line functions, GUI functions, or you can mix both of them to work with wavelet packet trees (WPTREE objects). The most useful commands are • plot, drawtree, and readtree, which let you plot and get a wavelet packet tree • wpjoin and wpsplt, which let you change a wavelet packet tree structure • get, read, and write, which let you read and write coefficients or information in a wavelet packet tree We can see some of these features in the following examples. • “plot and wpviewcf” on page 4-55 • “drawtree and readtree” on page 4-59 • “Change Terminal Node Coefficients” on page 4-61 • “Thresholding Wavelet Packets” on page 4-63 plot and wpviewcf load noisbump x = noisbump; t = wpdec(x,3,'db2'); fig = plot(t); Click on node 7. 4-55 4 Wavelet Packets Change Node Action from Visualize to Split-Merge and merge the second node. 4-56 Examples Using Objects % From the command line, you can get the new tree. newt = plot(t,'read',fig); % % % % % % % % % The first argument of the plot function in the last command is dummy. Then the general syntax is: newt = plot(DUMMY,'read',fig); where DUMMY is any object parented by an NTREE object. DUMMY can be any object constructor name, which returns an object parented by an NTREE object. For example: newt = plot(ntree,'read',fig); newt = plot(dtree,'read',fig); newt = plot(wptree,'read',fig); 4-57 4 Wavelet Packets % From % then newt = fig2 = the command line you can modify the new tree, plot it. wpjoin(newt,3); plot(newt); % Change Node Label from Depth_position to Index and % click the node (3). You get the following figure. % Using plot(newt,fig), the plot is done in the figure fig, % which already contains a tree object. % You can see the colored wavelet packets coefficients using % from the command line, the wpviewcf function (type help % wpviewcf for more information). wpviewcf(newt,1) % You get the following plot, which contains the terminal nodes % colored coefficients. 4-58 Examples Using Objects drawtree and readtree load noisbump x = noisbump; t = wpdec(x,3,'db2'); fig = drawtree(t); % % % % % % % The last command creates a GUI. The same GUI can be obtained using the main menu and: - clicking the Wavelet Packet 1-D button, - loading the signal noisbump, - choosing the level and the wavelet - clicking the decomposition button. You get the following figure. 4-59 4 Wavelet Packets % % % % % From the GUI, you can modify the tree. For example, change Node label from Depth_Position to Index, change Node Action from Visualize to Split_Merge and merge the node 2. You get the following figure. % From the command line, you can get the new tree. 4-60 Examples Using Objects newt = readtree(fig); % From the command line you can modify the new tree; % then plot it in the same figure. newt = wpjoin(newt,3); drawtree(newt,fig); You can mix previous commands. The GUI associated with the plot command is simpler and quicker, but more actions and information are available using the full GUI tools related to wavelet packets. The methods associated with WPTREE objects let you do more complicated actions. Namely, using read and write methods, you can change terminal node coefficients. Let’s illustrate this point with the following “funny” example. Change Terminal Node Coefficients load gatlin2 t = wpdec2(X,1,'haar'); plot(t); 4-61 4 Wavelet Packets % Change Node Label from Depth_position to Index and % click the node (0). You get the following figure. % Now modify the coefficients of the four terminal nodes. newt = t; NBcols = 40; for node = 1:4 cfs = read(t,'data',node); tmp = cfs(1:end,1:NBcols); cfs(1:end,1:NBcols) = cfs(1:end,end-NBcols+1:end); cfs(1:end,end-NBcols+1:end) = tmp; newt = write(newt,'data',node,cfs); end plot(newt) % Change Node Label from Depth_position to Index and % click on the node (0). You get the following figure. 4-62 Examples Using Objects You can use this method for a more useful purpose. Let’s see a de-noising example. Thresholding Wavelet Packets load noisbloc x = noisbloc; t = wpdec(x,3,'sym4'); plot(t); % Change Node Label from Depth_position to Index and % click the node (0). You get the following plot. 4-63 4 Wavelet Packets % Global thresholding. t1 = t; sorh = 'h'; thr = wthrmngr('wp1ddenoGBL','penalhi',t); cfs = read(t,'data'); cfs = wthresh(cfs,sorh,thr); t1 = write(t1,'data',cfs); plot(t1) % Change Node Label from Depth_position to Index and % click the node (0). You get the following plot. 4-64 Examples Using Objects % Node by node thresholding. t2 = t; sorh = 's'; thr(1) = wthrmngr('wp1ddenoGBL','penalhi',t); thr(2) = wthrmngr('wp1ddenoGBL','sqtwologswn',t); tn = leaves(t); for k=1:length(tn) node = tn(k); cfs = read(t,'data',node); numthr = rem(node,2)+1; cfs = wthresh(cfs,sorh,thr(numthr)); t2 = write(t2,'data',node,cfs); end plot(t2) % Change Node Label from Depth_position to Index and % click the node (0). You get the following plot. 4-65 4 4-66 Wavelet Packets Description of Objects in the Wavelet Toolbox™ Software Description of Objects in the Wavelet Toolbox Software The following sections describe the objects in the Wavelet Toolbox software: • “WTBO Object” on page 4-67 • “NTREE Object” on page 4-68 • “DTREE Object” on page 4-69 • “WPTREE Object” on page 4-71 WTBO Object Class WTBO (Wavelet Toolbox Object) -- Parent class: none Fields wtboInfo Object information (Not used) ud Userdata field Methods wtbo Constructor for the class WTBO. get Get WTBO object field contents. set Set WTBO object field contents. Comments Since any object in the toolbox is parented by a WTBO object, you can associate your own data to an object using the 'ud' field, and then access it. If Obj is an object (parented by a WTBO object), use Obj = set(Obj,'ud',MyData) to define the data. To retrieve the data, use 4-67 4 Wavelet Packets MyData = get(O,'ud') NTREE Object Class NTREE (New Tree) -- Parent class: WTBO Fields wtbo Parent object order Tree order depth Tree depth spsch Split scheme for nodes tn Column vector with terminal nodes indices Methods ntree Constructor for the class NTREE. findactn Find active nodes. get Get NTREE object field contents. nodejoin Recompose node(s). nodesplt Split (decompose) node(s). plot Plot NTREE object. set Set NTREE object field contents. tlabels Labels for the nodes of a tree. wtreemgr Manager for NTREE object. Private 4-68 locnumcn Local number for a child node tabofasc Table of ascendants of nodes Description of Objects in the Wavelet Toolbox™ Software DTREE Object Class DTREE (Data Tree) -- Parent class: NTREE Fields ntree Parent object allNI All Nodes Information terNI Terminal Nodes Information Fields Description allNI is a NBnodes-by-3 array such that allNI(N,:) = [ind,size(1,1),size(1,2)] • ind = index of the node N • size = size of data associated with the node N terNI is a 1-by-2 cell array such that • terNI{1} is an NB_TerminalNodes-by-2 array such that - terNI{1}(N,:) is the size of coefficients associated with the N-th terminal node. The nodes are numbered from left to right and from top to bottom. The root index is 0. • terNI{2} is a row vector containing the previous coefficients stored row-wise in the above specified order. Methods dtree Constructor for the class DTREE. expand Expand data tree. fmdtree Field manager for DTREE object. nodejoin Recompose node. 4-69 4 Wavelet Packets nodesplt Split (decompose) node. rnodcoef Reconstruct node coefficients. defaninf Define node information (all nodes). get Get DTREE object field contents. plot Plot DTREE object. read Read values in DTREE object fields. set Set DTREE object field contents. write Write values in DTREE object fields. merge Merge (recompose) the data of a node. recons Reconstruct node coefficients. split Split (decompose) the data of a terminal node. Comments • After the constructor, the first set of methods (between line separators) might not be overloaded (or only with great care). The second set of methods can be overloaded. The third set of methods must be overloaded to recompose, reconstruct, or decompose nodes data. • The method nodejoin calls the method merge, the method nodesplt calls the method split, and the method rnodcoef calls the method recons. • To define nodes information, you must overload the method defaninf. For each node N, the basic information is given by allNI(N,1:3): [index,size(1,1),size(1,2)]; You can add other information by adding columns to allNI. See the WPTREE object method for an example. • If the method get is not overloaded, using the DTREE get method you can get some object field contents (but not all). 4-70 Description of Objects in the Wavelet Toolbox™ Software For example, if T is parented by a DTREE object of order 2 and if 'Tfield' is a field of T, whose content is Tval, [a,b] = get(t,'order','Tfield') returns a = 2 and b = 'errorWTBX'. Nevertheless, using a nondocumented method you can get the right values. Namely: [a,b] = getwtbo(t,'order','Tfield') returns a = 2 and b=Tval. WPTREE Object Class WPTREE (Wavelet Packet Tree) -- Parent class: DTREE Fields dtree Parent object wavInfo Structure (wavelet information) entInfo Structure (entropy information) Fields Description wavInfo wavName Wavelet Name Lo_D Low Decomposition filter Hi_D High Decomposition filter Lo_R Low Reconstruction filter Hi_R High Reconstruction filter entInfo entName Entropy Name entPar Entropy Parameter allNI Array(nbnode,5) (field of the dtree parent object) [ind,size,ent,ento] 4-71 4 Wavelet Packets ind Index size Size of data ent Entropy ento Optimal Entropy Methods Constructor. Method Description wptree Constructor for the class WPTREE Methods That Overload Those of DTREE Class. Method Description defaninf Define node information (all nodes). get Get WPTREE object field contents. merge Merge (recompose) the data of a node. read Read values in WPTREE object fields. recons Reconstruct wavelet packet coefficients. set Set WPTREE object field contents. split Split (decompose) the data of a terminal node. tlabels Labels for the nodes of a wavelet packet tree. write Write values in WPTREE object fields. Proper Methods of WPTREE Class. 4-72 Method Description bestlevt Best level of a wavelet packet tree. besttree Best wavelet packet tree. Description of Objects in the Wavelet Toolbox™ Software Method Description entrupd Entropy update (wavelet packet tree). wp2wtree Extract wavelet tree from wavelet packet tree. wpcoef Wavelet packet coefficients. wpcutree Cut wavelet packet tree. wpjoin Recompose wavelet packet. wpplotcf Plot wavelet packets colored coefficients. wprcoef Reconstruct wavelet packet coefficients. wprec Wavelet packet reconstruction 1-D. wprec2 Wavelet packet reconstruction 2-D. wpsplt Split (decompose) wavelet packet. wpthcoef Wavelet packet coefficients thresholding. wpviewcf Plot wavelet packets colored coefficients. 4-73 4 Wavelet Packets Advanced Use of Objects The following sections explain how to extend the toolbox with new objects through four examples. • “Building a Wavelet Tree Object (WTREE)” on page 4-74 • “Building a Right Wavelet Tree Object (RWVTREE)” on page 4-75 • “Building a Wavelet Tree Object (WVTREE)” on page 4-77 • “Building a Wavelet Tree Object (EDWTTREE)” on page 4-79 Building a Wavelet Tree Object (WTREE) This example creates a new class of objects: WTREE. Starting from the class DTREE and overloading the methods split and merge, we define a wavelet tree class. To plot a WTREE, the DTREE plot method is used. You can have a look at a one-dimensional example in the ex1_wt file and at a two-dimensional example in the ex2_wt file located in the toolbox/wavelet/wavedemo folder. These examples can be used directly, but they are also useful to learn how to build new object-oriented programming functions. The definition of the new class is described below. Class WTREE (parent class: DTREE) Fields 4-74 dtree Parent object dwtMode DWT extension mode wavInfo Structure (wavelet information) Advanced Use of Objects wavInfo Structure information wavName Wavelet Name Lo_D Low Decomposition filter Hi_D High Decomposition filter Lo_R Low Reconstruction filter Hi_R High Reconstruction filter Methods wtree Constructor for the class WTREE. merge Merge (recompose) the data of a node. split Split (decompose) the data of a terminal node. Building a Right Wavelet Tree Object (RWVTREE) This example creates a new class of objects: RWVTREE. We define a right wavelet tree class starting from the class WTREE and overloading the methods split, merge, and plot (inherited from DTREE). The plot method shows how to add Node Labels. You can have a look at a one-dimensional example in the ex1_rwvt file and at a two-dimensional example in the ex2_rwvt file located in the toolbox/wavelet/wavedemo folder. These programs can be used directly, but they are also useful to learn how to build new object-oriented programming functions. The definition of the new class is described below. Class RWVTREE (parent class: WTREE) 4-75 4 Wavelet Packets Fields dummy Not used wtree Parent object Methods rwvtree Constructor for the class RWVTREE. merge Merge (recompose) the data of a node. plot Plot RWVTREE object. split Split (decompose) the data of a terminal node. Running This Example The following figure is obtained using the example ex1_rwvt and clicking the node 14. The approximations are labeled in yellow and the details are labeled in red. The last nodes cannot be split. 4-76 Advanced Use of Objects Building a Wavelet Tree Object (WVTREE) This example creates a new class of objects: WVTREE. We define a wavelet tree class starting from the class WTREE and overloading the methods get, plot, and recons (all inherited from DTREE). The split and merge methods of the class WTREE are used. The plot method shows how to add Node Labels and Node Actions. You can have a look at a one-dimensional example in the ex1_wvt file and at a two-dimensional example in the ex2_wvt file located in the 4-77 4 Wavelet Packets toolbox/wavelet/wavedemo folder. These programs can be used directly, but they are also useful to learn how to build new object-oriented programming functions. The definition of the new class is described below. Class WVTREE (parent class: WTREE) Fields dummy Not used wtree Parent object Methods wvtree Constructor for the class WVTREE. get Get WVTREE object field contents. plot Plot WVTREE object. recons Reconstruct node coefficients. Running This Example The following figure is obtained using the example ex2_wvt and clicking the node 2. The approximations are labeled in yellow and the details are labeled in red. The last nodes cannot be split. The title of the figure contains the DWT extension mode used ('sym' in the present example). 4-78 Advanced Use of Objects Building a Wavelet Tree Object (EDWTTREE) This example creates a new class of objects: EDWTTREE. We define an ε-DWT tree class starting from the class DTREE and overloading the methods merge, plot, recons, and split. For more information on the ε-DWT, see the section “ ε -Decimated DWT” on page 3-62. The plot method shows how to add Node Labels, Node Actions, and Tree Actions. 4-79 4 Wavelet Packets You can have a look at the example in the ex1_edwt file located in the toolbox/wavelet/wavedemo folder. This program can be used directly, but it is also useful to learn how to build new object-oriented programming functions. The definition of the new class is described below. Class EDWTTREE (parent class: DTREE) Fields dtree Parent object dwtMode DWT extension mode wavInfo Structure (wavelet information) Fields Description wavInfo wavName Wavelet Name Lo_D Low Decomposition filter Hi_D High Decomposition filter Lo_R Low Reconstruction filter Hi_R High Reconstruction filter Methods 4-80 edwttree Constructor for the class EDWTTREE. merge Merge (recompose) the data of a node. plot Plot EDWTTREE object. recons Reconstruct node coefficients. split Split (decompose) the data of a terminal node. Advanced Use of Objects Running This Example The following figure is obtained using the example ex1_edwt, selecting the De-noise option in the Tree Action menu and clicking the node 0. The approximations are labeled in yellow and the details are labeled in red. The last nodes cannot be split. The title of the figure contains the DWT extension mode used ('sym' in the present example) and the name of the de-noising method. 4-81 4 4-82 Wavelet Packets 5 Denoising, Nonparametric Function Estimation, and Compression • “Denoising and Nonparametric Function Estimation” on page 5-2 • “One-Dimensional Adaptive Thresholding of Wavelet Coefficients” on page 5-17 • “Multivariate Wavelet Denoising” on page 5-27 • “Multiscale Principal Components Analysis” on page 5-41 • “Data Compression” on page 5-51 • “True Compression for Images” on page 5-55 • “Two-Dimensional True Compression” on page 5-61 • “One-Dimensional Wavelet Regression Estimation” on page 5-80 5 Denoising, Nonparametric Function Estimation, and Compression Denoising and Nonparametric Function Estimation The Wavelet Toolbox provides a number of functions for the estimation of an unknown function (signal or image) in noise. The most general 1-D model for this is s(n) = f(n) + σe(n) where n = 0,1,2,...N-1. The e(n) are Gaussian random variables distributed as N(0,1). The variance of the σe(n) is σ2. In practice, s(n) is often a discrete-time signal with equal time steps corrupted by additive noise and you are attempting to recover that signal. More generally, you can view s(n) as an N-dimensional random vector f (0) e(0) f (1) e(1) f (2) e(2) . . . f ( N 1) e( N 1) f (0) f (1) f (2) . . . f ( N 1) e(0) e(1) e(2) . . . e( N 1) In this general context, the relationship between denoising and regression is clear. You can replace the N-by-1 random vector by N-by-M random matrices to obtain the problem of recovering an image corrupted by additive noise. You can obtain a 1-D example of this model with the following code. load cuspamax; y = cuspamax+0.5*randn(size(cuspamax)); plot(y); hold on; plot(cuspamax,'r','linewidth',2); axis tight; 5-2 Denoising and Nonparametric Function Estimation legend('f(n)+\sigma e(n)','f(n)', 'Location', 'NorthWest'); For a broad class of functions (signals, images) that possess certain smoothness properties, wavelet techniques are optimal or near optimal for function recovery. Specifically, the method is efficient for families of functions f that have only a few nonzero wavelet coefficients. These functions have a sparse wavelet representation. For example, a smooth function almost everywhere, with only a few abrupt changes, has such a property. 5-3 5 Denoising, Nonparametric Function Estimation, and Compression The general wavelet–based method for denoising and nonparametric function estimation is to transform the data into the wavelet domain, threshold the wavelet coefficients, and invert the transform. You can summarize these steps as: 1 Decompose Choose a wavelet and a level N. Compute the wavelet decomposition of the signal s down to level N. 2 Threshold detail coefficients For each level from 1 to N, threshold the detail coefficients. 3 Reconstruct Compute wavelet reconstruction using the original approximation coefficients of level N and the modified detail coefficients of levels from 1 to N. Threshold Selection Rules The Wavelet Toolbox supports a number of threshold selection rules. Four threshold selection rules are implemented in the thselect. Each rule corresponds to a tptr option in the command thr = thselect(y,tptr) which returns the threshold value. Option Threshold Selection Rule 'rigrsure' Selection using principle of Stein’s Unbiased Risk Estimate (SURE) 'sqtwolog' Fixed form (universal) threshold equal to 2 ln( N ) with N the length of the signal. 5-4 Denoising and Nonparametric Function Estimation Option Threshold Selection Rule 'heursure' Selection using a mixture of the first two options 'minimaxi' Selection using minimax principle • Option 'rigrsure' uses for the soft threshold estimator a threshold selection rule based on Stein’s Unbiased Estimate of Risk (quadratic loss function). You get an estimate of the risk for a particular threshold value t. Minimizing the risks in t gives a selection of the threshold value. • Option 'sqtwolog' uses a fixed form threshold yielding minimax performance multiplied by a small factor proportional to log(length(s)). • Option 'heursure' is a mixture of the two previous options. As a result, if the signal-to-noise ratio is very small, the SURE estimate is very noisy. So if such a situation is detected, the fixed form threshold is used. • Option 'minimaxi' uses a fixed threshold chosen to yield minimax performance for mean square error against an ideal procedure. The minimax principle is used in statistics to design estimators. Since the denoised signal can be assimilated to the estimator of the unknown regression function, the minimax estimator is the option that realizes the minimum, over a given set of functions, of the maximum mean square error. The following example shows the threshold rules for a 1000-by-1 N(0,1) vector. The signal here is f (n) e(n) e(n) ~ N (0, 1) with f(n) = 0. rng default; sig = randn(1e3,1); thr_rigrsure = thselect(sig,'rigrsure') thr_univthresh = thselect(sig,'sqtwolog') thr_heursure = thselect(sig,'heursure') thr_minimaxi = thselect(sig,'minimaxi') hist(sig); h = findobj(gca,'Type','patch'); set(h,'FaceColor',[0.7 0.7 0.7],'EdgeColor','w'); hold on; plot([thr_rigrsure thr_rigrsure], [0 300],'linewidth',2); 5-5 5 Denoising, Nonparametric Function Estimation, and Compression plot([thr_univthresh thr_univthresh], [0 300],'r','linewidth',2); plot([thr_minimaxi thr_minimaxi], [0 300],'k','linewidth',2); plot([-thr_rigrsure -thr_rigrsure], [0 300],'linewidth',2); plot([-thr_univthresh -thr_univthresh], [0 300],'r','linewidth',2); plot([-thr_minimaxi -thr_minimaxi], [0 300],'k','linewidth',2); For Stein’s Unbiased Risk Estimate (SURE) and minimax thresholds, approximately 3% of coefficients are retained. In the case of the universal threshold, all values are rejected. We know that the detail coefficients vector is the superposition of the coefficients of f and the coefficients of e, and that the decomposition of e leads to detail coefficients, which are standard Gaussian white noises. 5-6 Denoising and Nonparametric Function Estimation After you use thselect to determine a threshold, you can threshold each level of a . This second step can be done using wthcoef, directly handling the wavelet decomposition structure of the original signal s. Soft or Hard Thresholding Hard and soft thresholding are examples of shrinkage rules. After you have determined your threshold, you have to decide how to apply that threshold to your data. The simplest scheme is hard thresholding. Let T denote the threshold and x your data. The hard thresholding is x | x | T ( x) 0 | x | T The soft thresholding is x T x T ( x) 0 | x | T x T x T You can apply your threshold using the hard or soft rule with wthresh. y = linspace(-1,1,100); thr = 0.4; ythard = wthresh(y,'h',thr); ytsoft = wthresh(y,'s',thr); subplot(131); plot(y); title('Original Data'); subplot(132); plot(ythard,'*'); title('Hard Thresholding'); subplot(133); plot(ytsoft,'*'); title('Soft Thresholding'); 5-7 5 Denoising, Nonparametric Function Estimation, and Compression Dealing with Unscaled Noise and Nonwhite Noise Usually in practice the basic model cannot be used directly. We examine here the options available to deal with model deviations in the main de-noising function wden. The simplest use of wden is sd = wden(s,tptr,sorh,scal,n,wav) which returns the denoised version sd of the original signal s obtained using the tptr threshold selection rule. Other parameters needed are sorh, scal, n, and wav. The parameter sorh specifies the thresholding of details coefficients of the decomposition at level n of s by the wavelet called wav. The remaining 5-8 Denoising and Nonparametric Function Estimation parameter scal is to be specified. It corresponds to threshold’s rescaling methods. Option Corresponding Model 'one' Basic model 'sln' Basic model with unscaled noise 'mln' Basic model with nonwhite noise • Option scal = 'one' corresponds to the basic model. • In general, you can ignore the noise level and it must be estimated. The detail coefficients cD1 (the finest scale) are essentially noise coefficients with standard deviation equal to σ. The median absolute deviation of the coefficients is a robust estimate of σ. The use of a robust estimate is crucial for two reasons. The first one is that if level 1 coefficients contain f details, then these details are concentrated in a few coefficients if the function f is sufficiently regular. The second reason is to avoid signal end effects, which are pure artifacts due to computations on the edges. Option scal = 'sln' handles threshold rescaling using a single estimation of level noise based on the first-level coefficients. • When you suspect a nonwhite noise e, thresholds must be rescaled by a level-dependent estimation of the level noise. The same kind of strategy as in the previous option is used by estimating σlev level by level. This estimation is implemented in the file wnoisest, directly handling the wavelet decomposition structure of the original signal s. Option scal = 'mln' handles threshold rescaling using a level-dependent estimation of the level noise. For a more general procedure, the wdencmp function performs wavelet coefficients thresholding for both de-noising and compression purposes, while directly handling one-dimensional and two-dimensional data. It allows you to define your own thresholding strategy selecting in xd = wdencmp(opt,x,wav,n,thr,sorh,keepapp); where 5-9 5 Denoising, Nonparametric Function Estimation, and Compression • opt = 'gbl' and thr is a positive real number for uniform threshold. • opt = 'lvd' and thr is a vector for level dependent threshold. • keepapp = 1 to keep approximation coefficients, as previously and • keepapp = 0 to allow approximation coefficients thresholding. • x is the signal to be denoised and wav, n, sorh are the same as above. Denoising in Action We begin with examples of one-dimensional de-noising methods with the first example credited to Donoho and Johnstone. You can use the following file to get the first test function using wnoise. % Set signal to noise ratio and set rand seed. sqrt_snr = 4; init = 2055615866; % Generate original signal xref and a noisy version x adding % a standard Gaussian white noise. [xref,x] = wnoise(1,11,sqrt_snr,init); % De-noise noisy signal using soft heuristic SURE thresholding % and scaled noise option, on detail coefficients obtained % from the decomposition of x, at level 3 by sym8 wavelet. xd = wden(x,'heursure','s','one',3,'sym8'); 5-10 Denoising and Nonparametric Function Estimation Blocks Signal Denoising Since only a small number of large coefficients characterize the original signal, the method performs very well. As a second example, let us try the method on the highly perturbed part of the electrical signal studied above. According to this previous analysis, let us use db3 wavelet and decompose at level 3. To deal with the composite noise nature, let us try a level-dependent noise size estimation. % Load electrical signal and select part of it. load leleccum; indx = 2000:3450; 5-11 5 Denoising, Nonparametric Function Estimation, and Compression x = leleccum(indx); % Find first value in order to avoid edge effects. deb = x(1); % De-noise signal using soft fixed form thresholding % and unknown noise option. xd = wden(x-deb,'sqtwolog','s','mln',3,'db3')+deb; Electrical Signal De-Noising The result is quite good in spite of the time heterogeneity of the nature of the noise after and before the beginning of the sensor failure around time 2450. Extension to Image De-Noising The de-noising method described for the one-dimensional case applies also to images and applies well to geometrical images. A direct translation of the one-dimensional model is 5-12 Denoising and Nonparametric Function Estimation s(i,j) = f(i,j) + σe(i,j) where e is a white Gaussian noise with unit variance. The two-dimensional de-noising procedure has the same three steps and uses two-dimensional wavelet tools instead of one-dimensional ones. For the threshold selection, prod(size(s)) is used instead of length(s) if the fixed form threshold is used. Note that except for the “automatic” one-dimensional de-noising case, de-noising and compression are performed using wdencmp. As an example, you can use the following file illustrating the de-noising of a real image. % Load original image. load woman % Generate noisy image. x = X + 15*randn(size(X)); % Find default values. In this case fixed form threshold % is used with estimation of level noise, thresholding % mode is soft and the approximation coefficients are % kept. [thr,sorh,keepapp] = ddencmp('den','wv',x); % thr is equal to estimated_sigma*sqrt(log(prod(size(X)))) % De-noise image using global thresholding option. xd = wdencmp('gbl',x,'sym4',2,thr,sorh,keepapp); % Plots. colormap(pink(255)), sm = size(map,1); subplot(221), image(wcodemat(X,sm)), title('Original Image') subplot(222), image(wcodemat(x,sm)), title('Noisy Image') subplot(223), image(wcodemat(xd,sm)), title('denoised Image') The result shown below is acceptable. 5-13 5 Denoising, Nonparametric Function Estimation, and Compression Image De-Noising One-Dimensional Wavelet Variance Adaptive Thresholding The idea is to define level by level time-dependent thresholds, and then increase the capability of the de-noising strategies to handle nonstationary variance noise models. More precisely, the model assumes (as previously) that the observation is equal to the interesting signal superimposed on a noise (see “Denoising and Nonparametric Function Estimation” on page 5-2). s(n) = f(n) + σe(n) But the noise variance can vary with time. There are several different variance values on several time intervals. The values as well as the intervals are unknown. Let us focus on the problem of estimating the change points or equivalently the intervals. The algorithm used is based on an original work of Marc Lavielle about detection of change points using dynamic programming (see [Lav99] in “References”). 5-14 Denoising and Nonparametric Function Estimation Let us generate a signal from a fixed-design regression model with two noise variance change points located at positions 200 and 600. % Generate blocks test signal. x = wnoise(1,10); % Generate noisy blocks with change points. bb = randn(1,length(x)); cp1 = 200; cp2 = 600; x = x + [bb(1:cp1),bb(cp1+1:cp2)/4,bb(cp2+1:end)]; The aim of this example is to recover the two change points from the signal x. In addition, this example illustrates how the GUI tools locate the change points for interval dependent thresholding. Step 1. Recover a noisy signal by suppressing an approximation. % Perform a single-level wavelet decomposition % of the signal using db3. wname = 'db3'; lev = 1; [c,l] = wavedec(x,lev,wname); % Reconstruct detail at level 1. det = wrcoef('d',c,l,wname,1); The reconstructed detail at level 1 recovered at this stage is almost signal free. It captures the main features of the noise from a change points detection viewpoint if the interesting part of the signal has a sparse wavelet representation. To remove almost all the signal, we replace the biggest values by the mean. Step 2. To remove almost all the signal, replace 2% of biggest values by the mean. x = sort(abs(det)); v2p100 = x(fix(length(x)*0.98)); ind = find(abs(det)>v2p100); det(ind) = mean(det); 5-15 5 Denoising, Nonparametric Function Estimation, and Compression Step 3. Use the wvarchg function to estimate the change points with the following parameters: • The minimum delay between two change points is d = 10. • The maximum number of change points is 5. [cp_est,kopt,t_est] = wvarchg(det,5) Two change points and three intervals are proposed. Since the three interval variances for the noise are very different the optimization program detects easily the correct structure. The estimated change points are close to the true change points: 200 and 600. Step 4. (Optional) Replace the estimated change points. For 2 ≤ i ≤ 6, t_est(i,1:i-1) contains the i-1 instants of the variance change points, and since kopt is the proposed number of change points; then cp_est = t_est(kopt+1,1:kopt); You can replace the estimated change points by computing % cp_New = t_est(knew+1,1:knew); % where 1 5-16 knew 5 One-Dimensional Adaptive Thresholding of Wavelet Coefficients One-Dimensional Adaptive Thresholding of Wavelet Coefficients This section takes you through the features of local thresholding of wavelet coefficients for one-dimensional signals or data. This capability is available through graphical interface tools throughout the Wavelet Toolbox software: • Wavelet De-noising 1-D • Wavelet Compression 1-D • SWT De-noising 1-D • Regression Estimation 1-D • Density Estimation 1-D This tool allows you to define, level by level, time-dependent (x-axis-dependent) thresholds, and then increase the capability of the de-noising strategies handling nonstationary variance noise. More precisely, the model assumes that the observation is equal to the interesting signal superimposed on noise. The noise variance can vary with time. There are several different variance values on several time intervals. The values as well as the intervals are unknown. This section will use one of the graphical interface tool (SWT De-noising 1-D) to illustrate this capability. The behavior of all the above-mentioned tools is similar. One-Dimensional Interactive Local Thresholding 1 From the MATLAB prompt, type wavemenu The Wavelet Toolbox Main Menu appears. 5-17 5 Denoising, Nonparametric Function Estimation, and Compression Click the SWT De-noising 1-D menu item. The discrete stationary wavelet transform de-noising tool for one-dimensional signals appears. 2 Load data. From the File menu, choose the Load Signal option. When the Load Signal dialog box appears, select the MAT-file nblocr1.mat, which should reside in the MATLAB folder toolbox/wavelet/wavedemo. Click the OK button. The noisy blocks signal with two change points in the noise variance located at positions 200 and 600, is loaded into the SWT De-noising 1-D tool. 3 Perform signal decomposition. 5-18 One-Dimensional Adaptive Thresholding of Wavelet Coefficients Select the db1 wavelet from the Wavelet menu and select 5 from the Level menu, and then click the Decompose Signal button. After a pause for computation, the tool displays the stationary wavelet approximation and detail coefficients of the decomposition. Accept the defaults of Fixed form soft thresholding and Unscaled white noise. Click the De-noise button. The result is quite satisfactory, but seems to be oversmoothed when the signal is irregular. Select hard for the thresholding mode instead of soft, and then click the De-noise button. 5-19 5 Denoising, Nonparametric Function Estimation, and Compression The result is not satisfactory. The denoised signal remains noisy before position 200 and after position 700. This illustrates the limits of the classical de-noising strategies. In addition, the residuals obtained during the last trials clearly suggest to try a local thresholding strategy. 4 Generate interval-dependent thresholds. Click the Int. dependent threshold Settings button located at the bottom of the thresholding method frame. A new window titled Int. Dependent Threshold Settings for figure ... appears. 5-20 One-Dimensional Adaptive Thresholding of Wavelet Coefficients Click the Generate button. After a pause for computation, the tool displays the default intervals associated with adapted thresholds. 5-21 5 Denoising, Nonparametric Function Estimation, and Compression Three intervals are proposed. Since the variances for the three intervals are very different, the optimization program easily detects the correct structure. Nevertheless, you can visualize the intervals proposed for a number of intervals from 1 to 6 using the Select Number of Intervals menu (which replaces the Generate button). Using the default intervals automatically propagates the interval delimiters and associated thresholds to all levels. Denoise with Interval-Dependent Thresholds. Click the Close button in the Int. Dependent Threshold Settings for ... window. When the Update thresholds dialog box appears, click Yes. The SWT De-noising 1-D main window is updated. The sliders located to the right of the window control the level and interval dependent thresholds. For a given interval, the threshold is indicated by yellow dotted lines running horizontally through the graphs on the left of the window. The red dotted lines running vertically through the graphs indicate the interval delimiters. Next click the De-noise button. 5-22 One-Dimensional Adaptive Thresholding of Wavelet Coefficients Modifying Interval Dependent Thresholds. The thresholds can be increased to keep only the highest values of the wavelet coefficients at each level. Do this by dragging the yellow lines directly on the graphs on the left of the window, or using the View Axes button (located at the bottom of the screen near the Close button), which allows you to see each axis in full size. Another way is to edit the thresholds by selecting the interval number located near the sliders and typing the desired value. 5-23 5 Denoising, Nonparametric Function Estimation, and Compression Note that you can also change the interval limits by holding down the left mouse button over the vertical dotted red lines, and dragging them. You can also define your own interval dependent strategy. Click the Int. dependent threshold settings button. The Int. Dependent Threshold Settings for ... window appears again. We shall explore this window for a little while. Click the Delete button, so that the interval delimiters disappear. Double click the left mouse button to define new interval delimiters; for example at positions 300 and 500 and adjust the thresholds manually. Each level must be considered separately using the Level menu for adjusting the thresholds. The current interval delimiters can be propagated to all levels by clicking the Propagate button. So click the Propagate button. Adjust the thresholds for each level, one by one. At the end, click the Close button of the Int. Dependent Threshold settings for ... window. When the Update thresholds dialog box appears, click Yes. Then click the denoise button. Note that • By double-clicking again on an interval delimiter with the left mouse button, you delete it. • You can move the interval delimiters (vertical red dotted lines) and the threshold levels (horizontal yellow dotted lines) by holding down the left mouse button over these lines and dragging them. • The maximum number of interval delimiters at each level is 10. 5-24 One-Dimensional Adaptive Thresholding of Wavelet Coefficients Examples of Denoising with Interval Dependent Thresholds. From the File menu, choose the Example Analysis > Noisy Signals Interval Dependent Noise Variance > option. From the drop down men, choose with haar at level 4 ---> Elec. consumption 3 intervals. The proposed items contain, in addition to the usual information, the “true” number of intervals. You can then experiment with various signals for which local thresholding is needed. Importing and Exporting Information from the Graphical Interface The tool lets you save the denoised signal to disk. The toolbox creates a MAT-file in the current folder with a name you choose. 5-25 5 Denoising, Nonparametric Function Estimation, and Compression To save the denoised signal from the present de-noising process, use the menu option File > Save denoised Signal. A dialog box appears that lets you specify a folder and filename for storing the signal. Type the name dnelec. After saving the signal data to the file dnelec.mat, load the variables into your workspace: load dnelec whos Name Size Bytes Class dnelec 1x2000 16000 double array thrParams 1x4 656 cell array wname 1x4 8 char array The denoised signal is given by dnelec. In addition, the parameters of the de-noising process are given by the wavelet name contained in wname: wname wname = haar and the level dependent thresholds contained in thrParams, which is a cell array of length 4 (the level of the decomposition). For i from 1 to 4, thrParams{i} is an array nbintx3 (where nbint is the number of intervals, here 3), and each row contains the lower and upper bounds of the interval of thresholding and the threshold value. For example, for level 1, thrParams{1} ans = 1.0e+03 * 0.0010 0.0980 0.0060 0.0980 1.1240 0.0204 1.1240 2.0000 0.0049 5-26 Multivariate Wavelet Denoising Multivariate Wavelet Denoising This section demonstrates the features of multivariate de-noising provided in the Wavelet Toolbox software. The toolbox includes the wmulden function and a graphical user interface (GUI) tool available from wavemenu. This section also describes the command-line and GUI methods and includes information about transferring signal and parameter information between the disk and the GUI. This multivariate wavelet de-noising problem deals with models of the form X(t) = F(t) + e(t), where the observation X is p-dimensional, F is the deterministic signal to be recovered, and e is a spatially correlated noise signal. This kind of model is well suited for situations for which such additive, spatially correlated noise is realistic. Multivariate Wavelet Denoising — Command Line This example uses noisy test signals. In this section, you will • Load a multivariate signal. • Display the original and observed signals. • Remove noise by a simple multivariate thresholding after a change of basis. • Display the original and denoised signals. • Improve the obtained result by retaining less principal components. • Display the number of retained principal components. • Display the estimated noise covariance matrix. 1 Load a multivariate signal by typing the following at the MATLAB prompt: load ex4mwden whos Name Size Bytes Class covar 4x4 128 double array x 1024x4 32768 double array x_orig 1024x4 32768 double array 5-27 5 Denoising, Nonparametric Function Estimation, and Compression Usually, only the matrix of data x is available. Here, we also have the true noise covariance matrix (covar) and the original signals (x_orig). These signals are noisy versions of simple combinations of the two original signals. The first one is “Blocks” which is irregular, and the second is “HeavySine,” which is regular except around time 750. The other two signals are the sum and the difference of the two original signals. Multivariate Gaussian white noise exhibiting strong spatial correlation is added to the resulting four signals, which leads to the observed data stored in x. 2 Display the original and observed signals by typing kp = 0; for i = 1:4 subplot(4,2,kp+1), plot(x_orig(:,i)); axis tight; title(['Original signal ',num2str(i)]) subplot(4,2,kp+2), plot(x(:,i)); axis tight; title(['Observed signal ',num2str(i)]) kp = kp + 2; end 5-28 Multivariate Wavelet Denoising The true noise covariance matrix is given by covar covar = 1.0000 0.8000 0.6000 0.7000 0.8000 1.0000 0.5000 0.6000 0.6000 0.5000 1.0000 0.7000 0.7000 0.6000 0.7000 1.0000 3 Remove noise by simple multivariate thresholding. The de-noising strategy combines univariate wavelet de-noising in the basis where the estimated noise covariance matrix is diagonal with noncentered 5-29 5 Denoising, Nonparametric Function Estimation, and Compression Principal Component Analysis (PCA) on approximations in the wavelet domain or with final PCA. First, perform univariate de-noising by typing the following to set the de-noising parameters: level wname tptr sorh = = = = 5; 'sym4'; 'sqtwolog'; 's'; Then, set the PCA parameters by retaining all the principal components: npc_app = 4; npc_fin = 4; Finally, perform multivariate de-noising by typing x_den = wmulden(x, level, wname, npc_app, npc_fin, tptr, sorh); 4 Display the original and denoised signals by typing kp = 0; for i = 1:4 subplot(4,3,kp+1), plot(x_orig(:,i)); set(gca,'xtick',[]); axis tight; title(['Original signal ',num2str(i)]) subplot(4,3,kp+2), plot(x(:,i)); set(gca,'xtick',[]); axis tight; title(['Observed signal ',num2str(i)]) subplot(4,3,kp+3), plot(x_den(:,i)); set(gca,'xtick',[]); axis tight; title(['denoised signal ',num2str(i)]) kp = kp + 3; end 5-30 Multivariate Wavelet Denoising 5 Improve the first result by retaining fewer principal components. The results are satisfactory. Focusing on the two first signals, note that they are correctly recovered, but the result can be improved by taking advantage of the relationships between the signals, leading to an additional de-noising effect. To automatically select the numbers of retained principal components by Kaiser’s rule (which keeps the components associated with eigenvalues exceeding the mean of all eigenvalues), type npc_app = 'kais'; npc_fin = 'kais'; Perform multivariate de-noising again by typing 5-31 5 Denoising, Nonparametric Function Estimation, and Compression [x_den, npc, nestco] = wmulden(x, level, wname, npc_app, ... npc_fin, tptr, sorh); 6 Display the number of retained principal components. The second output argument gives the numbers of retained principal components for PCA for approximations and for final PCA. npc npc = 2 2 As expected, since the signals are combinations of two initial ones, Kaiser’s rule automatically detects that only two principal components are of interest. 7 Display the estimated noise covariance matrix. The third output argument contains the estimated noise covariance matrix: nestco nestco = 1.0784 0.8333 0.6878 0.8141 0.8333 1.0025 0.5275 0.6814 0.6878 0.5275 1.0501 0.7734 0.8141 0.6814 0.7734 1.0967 As you can see by comparing with the true matrix covar given previously, the estimation is satisfactory. 8 Display the original and final denoised signals by typing kp = 0; for i = 1:4 subplot(4,3,kp+1), plot(x_orig(:,i)); set(gca,'xtick',[]); axis tight; title(['Original signal ',num2str(i)]); set(gca,'xtick',[]); axis tight; subplot(4,3,kp+2), plot(x(:,i)); set(gca,'xtick',[]); 5-32 Multivariate Wavelet Denoising axis tight; title(['Observed signal ',num2str(i)]) subplot(4,3,kp+3), plot(x_den(:,i)); set(gca,'xtick',[]); axis tight; title(['denoised signal ',num2str(i)]) kp = kp + 3; end The results are better than those previously obtained. The first signal, which is irregular, is still correctly recovered, while the second signal, which is more regular, is denoised better after this second stage of PCA. 5-33 5 Denoising, Nonparametric Function Estimation, and Compression Interactive Multivariate Wavelet Denoising This section explores a de-noising strategy for multivariate signals using the graphical interface tools. 1 Start the Multivariate De-noising Tool by first opening the Wavelet Toolbox Main Menu. wavemenu 2 Click Multivariate Denoising to open the Multivariate De-Noising GUI. 3 Load data. Select File > Load Signals. In the Select dialog box, select the MAT-file ex4mwden.mat from the MATLAB folder toolbox/wavelet/wmultsig1d. Click Open to load the noisy multivariate signal into the GUI. The signal is a matrix containing four columns, where each column is a signal to be denoised. 5-34 Multivariate Wavelet Denoising These signals are noisy versions from simple combinations of the two original signals. The first one is “Blocks” which is irregular and the second is “HeavySine” which is regular except around time 750. The other two signals are the sum and the difference between the original signals. Multivariate Gaussian white noise exhibiting strong spatial correlation is added to the resulting four signals. The following example illustrates the two different aspects of the proposed de-noising method. First, perform a convenient change of basis to cope with spatial correlation and denoise in the new basis. Then, use PCA to take advantage of the relationships between the signals, leading to an additional de-noising effect. 4 Perform a wavelet decomposition and diagonalize the noise covariance matrix. Use the displayed default values for the Wavelet, the DWT Extension Mode, and the decomposition Level, and then click Decompose and Diagonalize. The tool displays the wavelet approximation and detail coefficients of the decomposition of each signal in the original basis. 5-35 5 Denoising, Nonparametric Function Estimation, and Compression Select Noise Adapted Basis to display the signals and their coefficients in the noise-adapted basis. To see more information about this new basis, click More on Noise Adapted Basis. A new figure displays the robust noise covariance estimate matrix and the corresponding eigenvectors and eigenvalues. Eigenvectors define the change of basis, and eigenvalues are the variances of uncorrelated noises in the new basis. The multivariate de-noising method proposed below is interesting if the noise covariance matrix is far from diagonal exhibiting spatial correlation, which, in this example, is the case. 5 denoise the multivariate signal. A number of options are available for fine-tuning the de-noising algorithm. However, we will use the defaults: fixed form soft thresholding, scaled white noise model, and the proposed numbers of retained principal components. In this case, the default values for PCA lead to retaining all the components. Select Original Basis to return to the original basis and then click Denoise. 5-36 Multivariate Wavelet Denoising The results are satisfactory. Both of the two first signals are correctly recovered, but they can be improved by getting more information about the principal components. Click More on Principal Components. A new figure displays information to select the numbers of components to keep for the PCA of approximations and for the final PCA after getting back to the original basis. You can see the percentages of variability explained by each principal component and the corresponding cumulative plot. Here, it is clear that only two principal components are of interest. 5-37 5 Denoising, Nonparametric Function Estimation, and Compression Close the More on Principal Components window. Select 2 as the Nb. of PC for APP. Select 2 as the Nb. of PC for final PCA, and then click denoise. The results are better than those previously obtained. The first signal, which is irregular, is still correctly recovered. The second signal, which is more regular, is denoised better after this second stage of PCA. You can get more information by clicking Residuals. Importing and Exporting from the GUI The tool lets you save denoised signals to disk by creating a MAT-file in the current folder with a name of your choice. To save the signal denoised in the previous section, 1 Select File > Save denoised Signals. 2 Select Save denoised Signals and Parameters. A dialog box appears that lets you specify a folder and filename for storing the signal. 3 Type the name s_ex4mwden and click OK to save the data. 4 Load the variables into your workspace: load s_ex4mwdent whos 5-38 Multivariate Wavelet Denoising Name Size Bytes Class DEN_Params 1x1 430 struct array PCA_Params 1x1 1536 struct array x 1024x4 32768 struct array The denoised signals are in matrix x. The parameters (PCA_Params and DEN_Params) of the two-stage de-noising process are also available. • PCA_Params are the change of basis and PCA parameters: PCA_Params PCA_Params = NEST: {[4x4 double] APP: {[4x4 double] FIN: {[4x4 double] [4x1 double] [4x1 double] [4x1 double] [4x4 double]} [2]} [2]} PCA_Params.NEST{1} contains the change of basis matrix. PCA_Params.NEST{2} contains the eigenvalues, and PCA_Params.NEST{3} is the estimated noise covariance matrix. PCA_Params.APP{1} contains the change of basis matrix, PCA_Params.APP{2} contains the eigenvalues, and PCA_Params.APP{3} is the number of retained principal components for approximations. The same structure is used for PCA_Params.FIN for the final PCA. • DEN_Params are the de-noising parameters in the diagonal basis: DEN_Params DEN_Params = thrVAL: [4.8445 2.0024 1.1536 1.3957 0] thrMETH: 'sqtwolog' thrTYPE: 's' 5-39 5 Denoising, Nonparametric Function Estimation, and Compression The thresholds are encoded in thrVAL. For j from 1 to 5, thrVAL(j) contains the value used to threshold the detail coefficients at level j. The thresholding method is given by thrMETH and the thresholding mode is given by thrTYPE. 5-40 Multiscale Principal Components Analysis Multiscale Principal Components Analysis This section demonstrates the features of multiscale principal components analysis provided in the Wavelet Toolbox software. The toolbox includes the wmspca function and a graphical user interface (GUI) available from wavemenu. This section describes the command-line and GUI methods, and information about transferring signal and parameter information between the disk and the GUI. The aim of multiscale PCA is to reconstruct, starting from a multivariate signal and using a simple representation at each resolution level, a simplified multivariate signal. The multiscale principal components generalizes the normal PCA of a multivariate signal represented as a matrix by performing a PCA on the matrices of details of different levels simultaneously. A PCA is also performed on the coarser approximation coefficients matrix in the wavelet domain as well as on the final reconstructed matrix. By selecting the numbers of retained principal components, interesting simplified signals can be reconstructed. Since you can perform multiscale PCA either from the command line or using the GUI, this section has subsections covering each method. Multiscale Principal Components Analysis — Command Line This example uses noisy test signals. In this section, you will: • Load a multivariate signal. • Perform a simple multiscale PCA. • Display the original and simplified signals. • Improve the obtained result by retaining less principal components. 1 Load a multivariate signal by typing at the MATLAB prompt: load ex4mwden whos 5-41 5 Denoising, Nonparametric Function Estimation, and Compression Name Size Bytes Class covar 4x4 128 double array x 1024x4 32768 double array x_orig 1024x4 32768 double array The data stored in matrix x comes from two test signals, Blocks and HeavySine, and from their sum and difference, to which multivariate Gaussian white noise has been added. 2 Perform a simple multiscale PCA. The multiscale PCA combines noncentered PCA on approximations and details in the wavelet domain and a final PCA. At each level, the most significant principal components are selected. First, set the wavelet parameters: level= 5; wname = 'sym4'; Then, automatically select the number of retained principal components using Kaiser’s rule by typing npc = 'kais'; Finally, perform multiscale PCA: [x_sim, qual, npc] = wmspca(x ,level, wname, npc); 3 Display the original and simplified signals: kp = 0; for i = 1:4 subplot(4,2,kp+1), plot(x (:,i)); set(gca,'xtick',[]); axis tight; title(['Original signal ',num2str(i)]) subplot(4,2,kp+2), plot(x_sim(:,i)); set(gca,'xtick',[]); axis tight; title(['Simplified signal ',num2str(i)]) kp = kp + 2; 5-42 Multiscale Principal Components Analysis end The results from a compression perspective are good. The percentages reflecting the quality of column reconstructions given by the relative mean square errors are close to 100%. qual qual = 98.0545 93.2807 97.1172 98.8603 4 Improve the first result by retaining fewer principal components. The results can be improved by suppressing noise, because the details at levels 1 to 3 are composed essentially of noise with small contributions from the signal. Removing the noise leads to a crude, but large, denoising effect. The output argument npc contains the numbers of retained principal components selected by Kaiser’s rule: npc npc = 1 1 1 1 1 2 2 5-43 5 Denoising, Nonparametric Function Estimation, and Compression For d from 1 to 5, npc(d) is the number of retained noncentered principal components (PCs) for details at level d. The number of retained noncentered PCs for approximations at level 5 is npc(6), and npc(7) is the number of retained PCs for final PCA after wavelet reconstruction. As expected, the rule keeps two principal components, both for the PCA approximations and the final PCA, but one principal component is kept for details at each level. To suppress the details at levels 1 to 3, update the npc argument as follows: npc(1:3) = zeros(1,3); npc npc = 0 0 0 1 1 2 2 Then, perform multiscale PCA again: [x_sim, qual, npc] = wmspca(x, level, wname, npc); 5 Display the original and final simplified signals: kp = 0; for i = 1:4 subplot(4,2,kp+1), plot(x (:,i)); set(gca,'xtick',[]); axis tight; title(['Original signal ',num2str(i)]); set(gca,'xtick',[]); axis tight; subplot(4,2,kp+2), plot(x_sim(:,i)); set(gca,'xtick',[]); axis tight; title(['Simplified signal ',num2str(i)]) kp = kp + 2; end 5-44 Multiscale Principal Components Analysis As shown, the results are improved. Interactive Multiscale Principal Components Analysis This section explores multiscale PCA using the GUIs. 1 Start the Multiscale Princ. Comp. Analysis tool by first opening the Wavelet Toolbox Main Menu: wavemenu 5-45 5 Denoising, Nonparametric Function Estimation, and Compression 2 Click Multiscale Princ. Comp. Analysis to open the Multiscale Principal Components Analysis GUI. 3 Load data. Select File > Load Signals. In the Select dialog box, select the MAT-file ex4mwden.mat from the MATLAB folder toolbox/wavelet/wmultsig1d. Click Open to load the multivariate signal into the GUI. The signal is a matrix containing four columns, where each column is a signal to be simplified. 5-46 Multiscale Principal Components Analysis These signals are noisy versions from simple combinations of the two original signals, Blocks and HeavySine and their sum and difference, each with added multivariate Gaussian white noise. 4 Perform a wavelet decomposition and diagonalize each coefficients matrix. Use the default values for the Wavelet, the DWT Extension Mode, and the decomposition Level, and then click Decompose and Diagonalize. The tool displays the wavelet approximation and detail coefficients of the decomposition of each signal in the original basis. To get more information about the new bases allowed for performing a PCA for each scale, click More on Adapted Basis. A new figure displays the corresponding eigenvectors and eigenvalues for the matrix of the detail coefficients at level 1. 5-47 5 Denoising, Nonparametric Function Estimation, and Compression You can change the level or select the coarser approximations or the reconstructed matrix to investigate the different bases. When you finish, click Close. 5 Perform a simple multiscale PCA. The initial values for PCA lead to retaining all the components. Select Kaiser from the Provide default using drop-down list, and click Apply. The results are good from a compression perspective. 6 Improve the obtained result by retaining fewer principal components. 5-48 Multiscale Principal Components Analysis The results can be improved by suppressing the noise, because the details at levels 1 to 3 are composed essentially of noise with small contributions from the signal, as you can see by careful inspection of the detail coefficients. Removing the noise leads to a crude, but large, de-noising effect. For D1, D2 and D3, select 0 as the Nb. of non-centered PC and click Apply. The results are better than those previously obtained. The first signal, which is irregular, is still correctly recovered, while the second signal, which is more regular, is denoised better after this second stage of PCA. You can get more information by clicking Residuals. Importing and Exporting from the GUI The Multiscale Principal Components Analysis tool lets you save the simplified signals to disk. The toolbox creates a MAT-file in the current folder with a name of your choice. To save the simplified signals from the previous section: 1 Select File > Save Simplified Signals. 5-49 5 Denoising, Nonparametric Function Estimation, and Compression 2 Select Save Simplified Signals and Parameters. A dialog box appears that lets you specify a folder and file name for storing the signal. 3 Type the name s_ex4mwden and click OK to save the data. 4 Load the variables into your workspace: load s_ex4mwden whos Name Size Bytes Class PCA_Params 1x7 2628 struct array x 1024x4 32768 double array The simplified signals are in matrix x. The parameters of multiscale PCA are available in PCA_Params: PCA_Params PCA_Params = 1x7 struct array with fields: pc variances npc PCA_Params is a structure array of length d+2 (here, the maximum decomposition level d=5) such that PCA_Params(d).pc is the matrix of principal components. The columns are stored in descending order of the variances. PCA_Params(d).variances is the principal component variances vector, and PCA_Params(d).npc is the vector of selected numbers of retained principal components. 5-50 Data Compression Data Compression The compression features of a given wavelet basis are primarily linked to the relative scarceness of the wavelet domain representation for the signal. The notion behind compression is based on the concept that the regular signal component can be accurately approximated using the following elements: a small number of approximation coefficients (at a suitably chosen level) and some of the detail coefficients. Like de-noising, the compression procedure contains three steps: 1 Decompose Choose a wavelet, choose a level N. Compute the wavelet decomposition of the signal s at level N. 2 Threshold detail coefficients For each level from 1 to N, a threshold is selected and hard thresholding is applied to the detail coefficients. 3 Reconstruct Compute wavelet reconstruction using the original approximation coefficients of level N and the modified detail coefficients of levels from 1 to N. The difference of the de-noising procedure is found in step 2. There are two compression approaches available. The first consists of taking the wavelet expansion of the signal and keeping the largest absolute value coefficients. In this case, you can set a global threshold, a compression performance, or a relative square norm recovery performance. Thus, only a single parameter needs to be selected. The second approach consists of applying visually determined level-dependent thresholds. Let us examine two real-life examples of compression using global thresholding, for a given and unoptimized wavelet choice, to produce a nearly complete square norm recovery for a signal (see Signal Compression on page 5-52) and for an image (see Image Compression on page 5-53). 5-51 5 Denoising, Nonparametric Function Estimation, and Compression % Load electrical signal and select a part. load leleccum; indx = 2600:3100; x = leleccum(indx); % Perform wavelet decomposition of the signal. n = 3; w = 'db3'; [c,l] = wavedec(x,n,w); % Compress using a fixed threshold. thr = 35; keepapp = 1; [xd,cxd,lxd,perf0,perfl2] = ... wdencmp('gbl',c,l,w,n,thr,'h',keepapp); Signal Compression The result is quite satisfactory, not only because of the norm recovery criterion, but also on a visual perception point of view. The reconstruction uses only 15% of the coefficients. % Load original image. load woman; x = X(100:200,100:200); nbc = size(map,1); 5-52 Data Compression % Wavelet decomposition of x. n = 5; w = 'sym2'; [c,l] = wavedec2(x,n,w); % Wavelet coefficients thresholding. thr = 20; keepapp = 1; [xd,cxd,lxd,perf0,perfl2] = ... wdencmp('gbl',c,l,w,n,thr,'h',keepapp); Image Compression If the wavelet representation is too dense, similar strategies can be used in the wavelet packet framework to obtain a sparser representation. You can then determine the best decomposition with respect to a suitably selected entropy-like criterion, which corresponds to the selected purpose (de-noising or compression). Compression Scores When compressing using orthogonal wavelets, the Retained energy in percentage is defined by 5-53 5 Denoising, Nonparametric Function Estimation, and Compression 100 * ( vector-norm(coeffs of the current decomposition, 2) ) 2 ( vector-norm(original signal, 2) )2 When compressing using biorthogonal wavelets, the previous definition is not convenient. We use instead the Energy ratio in percentage defined by 100 * ( vector-norm(compressed signal, 2) ) 2 ( vector-norm(original signal, 2) )2 and as a tuning parameter the Norm cfs recovery defined by 100 * ( vector-norm(coeffs of the current decomposition, 2) ) ( vector-norm(coeffs of the original decomposition, 2) )2 The Number of zeros in percentage is defined by 100 * (number of zeros of the current decomposition) (number of coefficients) 5-54 2 True Compression for Images True Compression for Images In “Data Compression” on page 5-51, we addressed the aspects specifically related to compression using wavelets. However, in addition to the algorithms related to wavelets like DWT and IDWT, it is necessary to use other ingredients concerning the quantization mode and the coding type in order to deal with true compression. This more complex process can be represented by the following figure. Effects of Quantization Let us show the effects of quantization on the visualization of the fingerprint image. This indexed image corresponds to a matrix of integers ranging between 0 and 255. Through quantization we can decrease the number of colors which is here equal to 256. 5-55 5 Denoising, Nonparametric Function Estimation, and Compression The next figure illustrates how to decrease from 256 to 16 colors by working on the values of the original image. We can see on this figure: • At the top - On the left: the original image On the right: the corresponding histogram of values • At the bottom - On the left: the reconstructed image On the right: the corresponding histogram of quantized values This quantization leads to a compression of the image. Indeed, with a fixed length binary code, 8 bits per pixel are needed to code 256 colors and 4 bits per pixel to code 16 colors. We notice that the image obtained after quantization is of good quality. However, within the framework of true compression, quantization is not used on the original image, but on its wavelet decomposition. Let us decompose the fingerprint image at level 4 with the Haar wavelet. The histogram of wavelet coefficients and the quantized histogram are normalized 5-56 True Compression for Images so that the values vary between –1 and +1. The 15 intervals of quantization do not have the same length. The next figure illustrates how to decrease information by binning on the wavelet coefficient values of the original image. We can see on this figure: • At the top - On left: the original image On the right: the corresponding histogram (central part) of coefficient values • At the bottom - On the left: the reconstructed image On the right: the corresponding histogram (central part) of quantized coefficient values The key point is that the histogram of the quantized coefficients is massively concentrated in the class centered in 0. Let us note that yet again the image obtained is of good quality. 5-57 5 Denoising, Nonparametric Function Estimation, and Compression True Compression Methods The basic ideas presented above are used by three methods which cascade in a single step, coefficient thresholding (global or by level), and encoding by quantization. Fixed or Huffman coding can be used for the quantization depending on the method. The following table summarizes these methods, often called Coefficients Thresholding Methods (CTM), and gives the MATLAB name used by the true compression tools for each of them. MATLAB Name Compression Method Name 'gbl_mmc_f' Global thresholding of coefficients and fixed encoding 'gbl_mmc_h' Global thresholding of coefficients and Huffman encoding 'lvl_mmc' Subband thresholding of coefficients and Huffman encoding More sophisticated methods are available which combine wavelet decomposition and quantization. This is the basic principle of progressive methods. On one hand, progressivity makes it possible during decoding to obtain an image whose resolution increases gradually. In addition, it is possible to obtain a set of compression ratios based on the length of the preserved code. This compression usually involves a loss of information, but this kind of algorithm enables also lossless compression. Such methods are based on three ideas. The two first, already mentioned, are the use of wavelet decomposition to ensure sparsity (a large number of zero coefficients) and classical encoding methods. The third idea, decisive for the use of wavelets in image compression, is to exploit fundamentally the tree structure of the wavelet decomposition. Certain codes developed from 1993 to 2000 use this idea, in particular, the EZW coding algorithm introduced by Shapiro. See [Sha93] in “References”. EZW combines stepwise thresholding and progressive quantization, focusing on the more efficient way to encode the image coefficients, in order to minimize 5-58 True Compression for Images the compression ratio. Two variants SPIHT and STW (see the following table) are refined versions of the seminal EZW algorithm. Following a slightly different objective, WDR (and the refinement ASWDR) focuses on the fact that in general some portions of a given image require more refined coding leading to a better perceptual result even if there is generally a small price to pay in terms of compression ratio. A complete review of these progressive methods is in the Walker reference [Wal99] in “References”. The following table summarizes these methods, often called Progressive Coefficients Significance Methods (PCSM), and gives the MATLAB coded name used by the true compression tools for each of them. MATLAB Name Compression Method Name 'ezw' Embedded Zerotree Wavelet 'spiht' Set Partitioning In Hierarchical Trees 'stw' Spatial-orientation Tree Wavelet 'wdr' Wavelet Difference Reduction 'aswdr' Adaptively Scanned Wavelet Difference Reduction 'spiht_3d' Set Partitioning In Hierarchical Trees 3D for truecolor images Quantitative and Perceptual Quality Measures Let us close this section by defining two quantitative measures of the compression performance as well as two measures of the perceptual quality. Compression Performance Two quantitative measures giving equivalent information are commonly used as a performance indicator for the compression: • The compression ratio CR, which means that the compressed image is stored using only CR% of the initial storage size. 5-59 5 Denoising, Nonparametric Function Estimation, and Compression • The Bit-Per-Pixel ratio BPP, which gives the number of bits required to store one pixel of the image. Perceptual Quality Two measures are commonly used to evaluate the perceptual quality: • The Mean Square Error (MSE). It represents the mean squared error between the compressed and the original image and is given by: MSE = 1 m=1 n=1 2 X (i, j) − X c (i, j) ∑ ∑ mn i=0 j =0 The lower the value of MSE, the lower the error. • The Peak Signal to Noise Ratio (PSNR). It represents a measure of the peak error and is expressed in decibels. It is defined by: ⎛ 2552 ⎞ PSNR = 10 ⋅ log10 ⎜ ⎟ ⎝ MSE ⎠ The higher the PSNR, the better the quality of the compressed or reconstructed image. Typical values for lossy compression of an image are between 30 and 50 dB and when the PSNR is greater than 40 dB, then the two images are indistinguishable. More Information on True Compression You can find examples illustrating command-line mode and GUI tools for true compression in “True Compression for Images” on page 5-55 and the reference page for wcompress. More information on the true compression for images and more precisely on the compression methods is in [Wal99], [Sha93], [Sai96], [StrN96], and [Chr06]. See “References”.. 5-60 Two-Dimensional True Compression Two-Dimensional True Compression This section takes you through the features of two-dimensional true compression using the Wavelet Toolbox software. For more information on the compression methods see “True Compression for Images” on page 5-55 in the Wavelet Toolbox User’s Guide. For more information on the main function available when using command-line mode, see the wcompress reference pages. Starting from a given image, the goal of the true compression is to minimize the length of the sequence of bits needed to represent it, while preserving information of acceptable quality. Wavelets contribute to effective solutions for this problem. The complete chain of compression includes phases of quantization, coding and decoding in addition of the wavelet processing itself. The purpose of this section is to show how to compress and uncompress a grayscale or truecolor image using various compression methods. In this section, you’ll learn to • Compress using global thresholding and Huffman encoding • Uncompress • Compress using progressive methods • Handle truecolor images Two-Dimensional True Compression — Command Line Compression by Global Thresholding and Huffman Encoding First load and display the grayscale image mask. load mask; image(X) axis square; 5-61 5 Denoising, Nonparametric Function Estimation, and Compression colormap(pink(255)) title('Original Image: mask') A synthetic performance of the compression is given by the compression ratio and the Bit-Per-Pixel ratio which are equivalent. The compression ratio CR means that the compressed image is stored using only CR% of the initial storage size. The Bit-Per-Pixel ratio BPP gives the number of bits used to store one pixel of the image. For a grayscale image, the initial BPP is 8 while for a truecolor image the initial BPP is 24 because 8 bits are used to encode each of the three colors (RGB color space). The challenge of compression methods is to find the best compromise between a weak compression ratio and a good perceptual result. Let us begin with a simple method cascading global coefficients thresholding and Huffman encoding. We use the default wavelet bior4.4 and the default level which is the maximum possible level (see the wmaxlev function) divided by 2. 5-62 Two-Dimensional True Compression The desired Bit-Per-Pixel ratio BPP is set to 0.5 and the compressed image will be stored in the file named 'mask.wtc'. meth = 'gbl_mmc_h'; % Method name option = 'c'; % 'c' stands for compression [CR,BPP] = wcompress(option,X,'mask.wtc',meth,'bpp',0.5) CR = 6.6925 BPP = 0.5354 The achieved Bit-Per-Pixel ratio is actually about 0.53 (closed to the desired one) for a compression ratio of 6.7%. Uncompression Let us uncompress the image retrieved from the file 'mask.wtc' and compare it to the original image. option = 'u'; % 'u' stands for uncompression Xc = wcompress(option,'mask.wtc'); colormap(pink(255)) subplot(1,2,1); image(X); axis square; title('Original Image') subplot(1,2,2); image(Xc); axis square; title('Compressed Image') xlabel({['Compression Ratio: ' num2str(CR,'%1.2f %%')], ... ['BPP: ' num2str(BPP,'%3.2f')]}) 5-63 5 Denoising, Nonparametric Function Estimation, and Compression Compression by Progressive Methods Let us now illustrate the use of progressive methods starting with the well known EZW algorithm using the Haar wavelet. The key parameter is the number of loops. Increasing it, leads to better recovery but worse compression ratio. meth = 'ezw'; % Method name wname = 'haar'; % Wavelet name nbloop = 6; % Number of loops [CR,BPP] = wcompress('c',X,'mask.wtc',meth, ... 'maxloop',nbloop,'wname',wname); Xc = wcompress('u','mask.wtc'); colormap(pink(255)) subplot(1,2,1); image(X); axis square; title('Original Image') subplot(1,2,2); image(Xc); axis square; title('Compressed Image - 6 steps') xlabel({['Compression Ratio: ' num2str(CR,'%1.2f %%')], ... ['BPP: ' num2str(BPP,'%3.2f')]}) 5-64 Two-Dimensional True Compression A too small number of steps (here 6) produces a very coarse compressed image. So let us examine a little better result for 9 steps and a satisfactory result for 12 steps. [CR,BPP]= wcompress('c',X,'mask.wtc',meth,'maxloop',9, ... 'wname','haar'); Xc = wcompress('u','mask.wtc'); colormap(pink(255)) subplot(1,2,1); image(Xc); axis square; title('Compressed Image - 9 steps') xlabel({['Compression Ratio: ' num2str(CR,'%1.2f %%')],... ['BPP: ' num2str(BPP,'%3.2f')]}) [CR,BPP] = wcompress('c',X,'mask.wtc',meth,'maxloop',12, ... 'wname','haar'); Xc = wcompress('u','mask.wtc'); subplot(1,2,2); image(Xc); axis square; title('Compressed Image - 12 steps') xlabel({['Compression Ratio: ' num2str(CR,'%1.2f %%')],... ['BPP: ' num2str(BPP,'%3.2f')]}) 5-65 5 Denoising, Nonparametric Function Estimation, and Compression As can be seen, the reached BPP ratio is about 0.92 when using 12 steps. Let us try to improve it by using the wavelet bior4.4 instead of haar and looking at obtained results for steps 12 and 11. [CR,BPP] = wcompress('c',X,'mask.wtc','ezw','maxloop',12, ... 'wname','bior4.4'); Xc = wcompress('u','mask.wtc'); colormap(pink(255)) subplot(1,2,1); image(Xc); axis square; title('Compressed Image - 12 steps - bior4.4') xlabel({['Compression Ratio: ' num2str(CR,'%1.2f %%')], ... ['BPP: ' num2str(BPP,'%3.2f')]}) [CR,BPP] = wcompress('c',X,'mask.wtc','ezw','maxloop',11, ... 'wname','bior4.4'); Xc = wcompress('u','mask.wtc'); subplot(1,2,2); image(Xc); axis square; title('Compressed Image - 11 steps - bior4.4') xlabel({['Compression Ratio: ' num2str(CR,'%1.2f %%')], ... ['BPP: ' num2str(BPP,'%3.2f')]}) 5-66 Two-Dimensional True Compression Starting from the eleventh loop, the result can be considered satisfactory. The reached BPP ratio is now about 0.35. It can even be slightly improved by using a more recent method: SPIHT (Set Partitioning In Hierarchical Trees). [CR,BPP] = wcompress('c',X,'mask.wtc','spiht','maxloop',12, ... 'wname','bior4.4'); Xc = wcompress('u','mask.wtc'); colormap(pink(255)) subplot(1,2,1); image(X); axis square; title('Original Image') subplot(1,2,2); image(Xc); axis square; title('Compressed Image - 12 steps - bior4.4') xlabel({['Compression Ratio: ' num2str(CR,'%1.2f %%')], ... ['BPP: ' num2str(BPP,'%3.2f')]}) [psnr,mse] = psnr_mse_maxerr(X,Xc) delete('mask.wtc') 5-67 5 Denoising, Nonparametric Function Estimation, and Compression The final compression ratio (2.8%) and the Bit-Per-Pixel ratio (0.23) are very satisfactory. Let us recall that the first ratio means that the compressed image is stored using only 2.8% of the initial storage size. Handling Truecolor Images Finally, let us illustrate how to compress the wpeppers.jpg truecolor image. Truecolor images can be compressed along the same scheme as the grayscale images by applying the same strategies to each of the three color components. The progressive compression method used is SPIHT (Set Partitioning In Hierarchical Trees) and the number of encoding loops is set to 12. X = imread('wpeppers.jpg'); [CR,BPP] = wcompress('c',X,'wpeppers.wtc','spiht','maxloop',12); Xc = wcompress('u','wpeppers.wtc'); subplot(1,2,1); image(X); axis square; title('Original Image') subplot(1,2,2); image(Xc); axis square; title('Compressed Image - 12 steps - bior4.4') xlabel({['Compression Ratio: ' num2str(CR,'%1.2f %%')], ... ['BPP: ' num2str(BPP,'%3.2f')]}) delete('wpeppers.wtc') 5-68 Two-Dimensional True Compression The compression ratio (1.65%) and the Bit-Per-Pixel ratio (0.4) are very satisfactory while maintaining a good visual perception. Interactive Two-Dimensional True Compression In this section, we explore the different methods for two-dimensional true compression, using the graphical interface tools. 1 Start the True Compression 2-D Tool. From the MATLAB prompt, type wavemenu The Wavelet Toolbox Main Menu appears. Click the True Compression 2-D menu item. The true compression tool for images appears. 2 Load the image. From the File menu, choose the Load Image option and select the Matlab Supported Formats item. When the Load Image dialog box appears, select the MAT-file mask.mat, which should reside in the MATLAB folder toolbox/wavelet/wavedemo. 5-69 5 Denoising, Nonparametric Function Estimation, and Compression Click the OK button. A window appears asking you if you want to consider the loaded image as a truecolor one. Click the No button since it is a grayscale image. The mask image is loaded into the True Compression 2-D tool. It appears at the top left of the window together with the gray level histogram just below. 3 Perform a Wavelet Decomposition. Accept the default wavelet bior4.4 and select 4 from the Level menu which is the maximum possible level divided by 2 and then click the Decompose button. After a pause for computation, the tool displays the wavelet approximation and details coefficients of the decomposition for the three directions, together with the histogram of the original coefficients. 5-70 Two-Dimensional True Compression The peak of the bin containing zero illustrates the capability of wavelets to concentrate the image energy into a few nonzero coefficients. 4 Try a simple method. Begin with a simple method cascading global coefficients thresholding and Huffman encoding. Choose the GBL_MMC_H option from the menu Compression method located at the top right of the Compression Parameters frame. For more information on the compression methods, see “True Compression for Images” on page 5-55 in the Wavelet Toolbox User’s Guide. Set the desired Bit-Per-Pixel ratio to 0.5. 5-71 5 Denoising, Nonparametric Function Estimation, and Compression Values of the other parameters are automatically updated. Note that these values are only approximations of the true performances since the quantization step cannot be exactly predicted without performing it. Click the Compress button. Synthetic performance is given by the compression ratio and the computed Bit-Per-Pixel (BPP). This last one is actually about 0.53 (close to the desired one 0.5) for a compression ratio of 6.7%. 5-72 Two-Dimensional True Compression The compressed image, at the bottom left, can be compared with the original image. The result is satisfactory but a better compromise between compression ratio and visual quality can be obtained using more sophisticated true compression which combines the thresholding and quantization steps. 5 Compress using a first progressive method: EZW. Let us now illustrate the use of progressive methods starting with the well known EZW algorithm. Let us start by selecting the wavelet haar from the Wavelet menu and select 8 from the Level menu. Then click the Decompose button. Choose the EZW option from the menu Compression method. The key parameter is the number of loops: increasing it leads to a better recovery but worse compression ratio. From the Nb. Encoding Loops menu, set the number of encoding loops to 6, which is a small value. Click the Compress button. 6 Refine the result by increasing the number of loops. 5-73 5 Denoising, Nonparametric Function Estimation, and Compression Too few steps produce a very coarse compressed image. So let us examine a little better result for 9 steps. Set the number of encoding loops to 9 and click the Compress button. As can be seen, the result is better but not satisfactory, both by visual inspection and by calculating the Peak Signal to Noise Ratio (PSNR) which is less than 30. Now try a large enough number of steps to get a satisfactory result. Set the number of encoding loops to 12 and click the Compress button. 5-74 Two-Dimensional True Compression The result is now acceptable. But for 12 steps, we attain a Bit-Per-Pixel ratio about 0.92. 7 Get better compression performance by changing the wavelet and selecting the best adapted number of loops. Let us try to improve the compression performance by changing the wavelet: select bior4.4 instead of haar and then click the Decompose button. In order to select the number of loops, the GUI tool allows you to examine the successive results obtained by this kind of stepwise procedure. Set the number of encoding loops at a large value, for example 13, and click the Show Compression Steps button. Moreover you could execute the procedure stepwise by clicking the Stepwise button. 5-75 5 Denoising, Nonparametric Function Estimation, and Compression Then, click the Compress button. Thirteen progressively more finely compressed images appear, and you can then select visually a convenient value for the number of loops. A satisfactory result seems to be obtained after 11 loops. So, you can set the number of encoding loops to 11 and click the Compress button. The reached BPP ratio is now about 0.35 which is commonly considered a very satisfactory result. Nevertheless, it can be slightly improved by using a more recent method SPIHT (Set Partitioning In Hierarchical Trees). 8 Obtain a final compressed image by using a third method. Choose the SPIHT option from the menu Compression method, set the number of encoding loops to 12, and click the Compress button. 5-76 Two-Dimensional True Compression The final compression ratio and the Bit-Per-Pixel ratio are very satisfactory: 2.8% and 0.22. Let us recall that the first ratio means that the compressed image is stored using only 2.8% of the initial storage size. 9 Handle truecolor images. Finally, let us illustrate how to compress truecolor images. The truecolor images can be compressed along the same scheme by applying the same strategies to each of the three-color components. From the File menu, choose the Load Image option and select the Matlab Supported Formats item. When the Load Image dialog box appears, select the MAT-file wpeppers.jpg which should reside in the MATLAB folder toolbox/wavelet/wavedemo. Click the OK button. A window appears asking you if you want to consider the loaded image as a truecolor one. Click the Yes button. Accept the 5-77 5 Denoising, Nonparametric Function Estimation, and Compression defaults for wavelet and decomposition level menus and then click the Decompose button. Then, accept the default compression method SPIHT and set the number of encoding loops to 12. Click the Compress button. The compression ratio and Bit-Per-Pixel (BPP) ratio are very satisfactory: 1.65% and 0.4 together with a very good perceptual result. 10 Inspect the wavelet tree. For both grayscale and truecolor images, more insight on the multiresolution structure of the compressed image can be retrieved by clicking the Inspect Wavelet Trees button and then on the various active menus available from the displayed tree. 5-78 Two-Dimensional True Compression Importing and Exporting from the GUI You can save the compressed image to disk in the current folder with a name of your choice. The Wavelet Toolbox compression tools can create a file using either one of the Matlab Supported Format types or a specific format which can be recognized by the extension of the file: wtc (Wavelet Toolbox Compressed). To save the above compressed image, use the menu option File > Save Compressed Image > Wavelet Toolbox Compressed Image. A dialog box appears that lets you specify a folder and filename for storing the image. Of course, the use of the wtc format requires you to uncompress the stored image using the Wavelet Toolbox true compression tools. 5-79 5 Denoising, Nonparametric Function Estimation, and Compression One-Dimensional Wavelet Regression Estimation This section takes you through the features of one-dimensional wavelet regression estimation using one of the Wavelet Toolbox specialized tools. The toolbox provides a graphical interface tool to explore some de-noising schemes for equally or unequally sampled data. For the examples in this section, switch the extension mode to symmetric padding, using the command dwtmode('sym') Regression for Equally-Spaced Observations 1 Start the Regression Estimation 1-D Tool. From the MATLAB prompt, type wavemenu The Wavelet Toolbox Main Menu appears. 5-80 One-Dimensional Wavelet Regression Estimation Click the Regression Estimation 1-D menu item. The discrete wavelet analysis tool for one-dimensional regression estimation appears. 2 Load data. From the File menu, choose the Load Data for Fixed Design Regression option. When the Load data for Fixed Design Regression dialog box appears, select the MAT-file noisbloc.mat, which should reside in the MATLAB folder toolbox/wavelet/wavedemo. 5-81 5 Denoising, Nonparametric Function Estimation, and Compression Click the OK button. The noisy blocks data is loaded into the Regression Estimation 1-D - Fixed Design tool. The loaded data denoted (X,Y) and the processed data obtained after a binning, are displayed. 3 Choose the processed data. The default value for the number of bins is 256 for this example. Enter 64 in the Nb bins (number of bins) edit box, or use the slider to adjust the value. The new binned data to be processed appears. The binned data appears to be very smoothed. Select 1000 from the Nb bins edit and press Enter or use the slider. The new data to be processed appears. The binned data appears to be very close to the initial data, since noisbloc is of length 1024. 4 Perform a Wavelet Decomposition of the processed data. Select the haar wavelet from the Wavelet menu and select 5 from the Level menu, and then click the Decompose button. After a pause for computation, the tool displays the detail coefficients of the decomposition. 5-82 One-Dimensional Wavelet Regression Estimation 5 Perform a regression estimation. While a number of options are available for fine-tuning the estimation algorithm, we’ll accept the defaults of fixed form soft thresholding and unscaled white noise. The sliders located to the right of the window control the level dependent thresholds, indicated by yellow dotted lines running horizontally through the graphs on the left part of the window. Continue by clicking the Estimate button. 5-83 5 Denoising, Nonparametric Function Estimation, and Compression You can see that the process removed the noise and that the blocks are well reconstructed. The regression estimate (in yellow) is the sum of the signals located below it: the approximation a5 and the reconstructed details after coefficient thresholding. You can experiment with the various predefined thresholding strategies by selecting the appropriate options from the menu located on the right part of the window or directly by dragging the yellow horizontal lines with the left mouse button. Let us now illustrate the regression estimation using the graphical interface for randomly or irregularly spaced observations, focusing on the differences from the previous situation. Regression for Randomly-Spaced Observations 1 From the File menu, choose the Load > Data for Stochastic Design Regression option. When the Load data for Stochastic Design Regression dialog box appears, select the MAT-file ex1nsto.mat, which should reside in the MATLAB folder toolbox/wavelet/wavedemo. Click the OK button. This short set of data (of size 500) is loaded into the Regression Estimation 1-D -- Stochastic Design tool. The loaded data denoted (X,Y), the histogram of X, and the processed data obtained after a binning are displayed. The histogram is interesting, because the values of X are randomly distributed. The binning step is essential since it transforms a problem of regression estimation for irregularly spaced X data into a classical fixed design scheme for which fast wavelet transform can be used. 2 Select the sym4 wavelet from the Wavelet menu, select 5 from the Level menu, and enter 125 in the Nb bins edit box. Click the Decompose button. The tool displays the detail coefficients of the decomposition. 3 From the Select thresholding method menu, select the item Penalize low and click the Estimate button. 5-84 One-Dimensional Wavelet Regression Estimation 4 Check Overlay Estimated Function to validate the fit of the original data. Importing and Exporting Information from the Graphical Interface Saving Function This tool lets you save the estimated function to disk. The toolbox creates a MAT-file in the current folder with a name you choose. To save the estimated function from the present estimation, use the menu option File > Save Estimated Function. A dialog box appears that lets you specify a folder and filename for storing the function. Type the name fex1nsto. After saving the function data to the file fex1nsto.mat, load the variables into your workspace: load fex1nsto whos 5-85 5 Denoising, Nonparametric Function Estimation, and Compression Name Size Bytes Class thrParams 1x5 580 cell array wname 1x4 8 char array xdata 1x125 1000 double array ydata 1x125 1000 double array The estimated function is given by xdata and ydata. The length of these vectors is equal to the number of bins you choose in step 2. In addition, the parameters of the estimation process are given by the wavelet name contained in wname: wname wname = sym4 and the level dependent thresholds contained in thrParams, which is a cell array of length 5 (the level of the decomposition). For i from 1 to 5, thrParams{i} contains the lower and upper bounds of the interval of thresholding and the threshold value (since interval dependent thresholds are allowed). For more information, see “One-Dimensional Adaptive Thresholding of Wavelet Coefficients” on page 5-17 in the Wavelet Toolbox User’s Guide. For example, for level 1, thrParams{1} ans = -0.4987 0.4997 1.0395 Loading Data To load data for regression estimation, your file must contain at least one vector. If your file contains only one vector, this vector is considered as ydata and an xdata vector is automatically generated. 5-86 One-Dimensional Wavelet Regression Estimation If your file contains at least two vectors, they must be called xdata and ydata or x and y. These vectors must be the same length. For example, load the file containing the data considered in the previous example: clear load ex1nsto whos Name Size Bytes Class x 1x500 4000 double array y 1x500 4000 double array At the end of this section, turn back the extension mode to zero padding using the command dwtmode('zpd') 5-87 5 5-88 Denoising, Nonparametric Function Estimation, and Compression 6 Matching Pursuit • “Sparse Representation in Redundant Dictionaries” on page 6-2 • “Matching Pursuit Algorithms” on page 6-4 • “Matching Pursuit — Command Line” on page 6-9 • “Matching Pursuit — Interactive Analysis” on page 6-22 6 Matching Pursuit Sparse Representation in Redundant Dictionaries In this section... “Redundant Dictionaries and Sparsity” on page 6-2 “Nonlinear Approximation in Dictionaries” on page 6-3 Redundant Dictionaries and Sparsity Representing a signal in a particular basis involves finding the signal’s unique set of expansion coefficients in that basis. While there are many advantages to signal representation in a basis, particularly an orthogonal basis, there are also disadvantages. The ability of a basis to provide a sparse representation of a signal depends on how well the signal characteristics match the characteristics of the basis vectors. For example, smooth continuous signals are sparsely represented in a Fourier basis, while impulses are not. On the other hand, a smooth signal with isolated discontinuities is sparsely represented in a wavelet basis, while a wavelet basis is not efficient at representing a signal whose Fourier transform has narrow high frequency support. Real-world signals often contain features that prohibit sparse representation in any single basis. For these signals, you want the ability to choose vectors from a set not limited to a single basis. Because you want to ensure that you can represent every vector in the space, the dictionary of vectors you choose from must span the space. However, because the set is not limited to a single basis, the dictionary is not linearly independent. Because the vectors in the dictionary are not a linearly independent set, the signal representation in the dictionary is not unique. However, by creating a redundant dictionary, you can expand your signal in a set of vectors that adapt to the time-frequency or time-scale characteristics of your signal. You are free to create a dictionary consisting of the union of several bases. For example, you can form a basis for the space of square-integrable functions consisting of a wavelet packet basis and a local cosine basis. A wavelet packet basis is well adapted to signals with different behavior in different frequency intervals. A local cosine basis is well adapted to signals with different behavior in different time intervals. The ability to choose vectors from each 6-2 Sparse Representation in Redundant Dictionaries of these bases greatly increases your ability to sparsely represent signals with varying characteristics. Nonlinear Approximation in Dictionaries Define a dictionary as a collection of unit-norm elementary building blocks for your signal space. These unit-norm vectors are called atoms. If the atoms of the dictionary span the entire signal space, the dictionary is complete. If the dictionary atoms form a linearly-dependent set, the dictionary is redundant. In most applications of matching pursuit, the dictionary is complete and redundant. Let {φk} denote the atoms of a dictionary. Assume the dictionary is complete and redundant. There is no unique way to represent a signal from the space as a linear combination of the atoms. x kk k An important question is whether there exists a best way. An intuitively satisfying way to choose the best representation is to select the φk yielding the largest inner products (in absolute value) with the signal. For example, the best single φk is max | x, k | k which for a unit-norm atom is the magnitude of the scalar projection onto the subspace spanned by φk. The central problem in matching pursuit is how you choose the optimal M-term expansion of your signal in a dictionary. 6-3 6 Matching Pursuit Matching Pursuit Algorithms In this section... “Basic Matching Pursuit” on page 6-4 “Orthogonal Matching Pursuit” on page 6-7 “Weak Orthogonal Matching Pursuit” on page 6-8 Basic Matching Pursuit Let Φ denote the dictionary of atoms as a N-by-M matrix with M>N. If the complete, redundant dictionary forms a frame for the signal space, you can obtain the minimum L2 norm expansion coefficient vector by using the frame operator. † * ( * )1 However, the coefficient vector returned by the frame operator does not perserve sparsity. If the signal is sparse in the dictionary, the expansion coefficients obtained with the canonical frame operator generally do not reflect that sparsity. Sparsity of your signal in the dictionary is a trait that you typically want to preserve. Matching pursuit addresses sparsity preservation directly. Matching pursuit is a greedy algorithm that computes the best nonlinear approximation to a signal in a complete, redundant dictionary. Matching pursuit builds a sequence of sparse approximations to the signal stepwise. Let Φ= {φk} denote a dictionary of unit-norm atoms. Let f be your signal. 1 Start by defining R0f = f 2 Begin the matching pursuit by selecting the atom from the dictionary that maximizes the absolute value of the inner product with R0f = f. Denote that atom by φp. 3 Form the residual R1f by subtracting the orthogonal projection of R0f onto the space spanned by φp. 6-4 Matching Pursuit Algorithms R1 f R0 f R0 f , p p 4 Iterate by repeating steps 2 and 3 on the residual. Rm1 f Rm f Rm f , k k 5 Stop the algorithm when you reach some specified stopping criterion. In nonorthogonal (or basic) matching pursuit, the dictionary atoms are not mutually orthogonal vectors. Therefore, subtracting subsequent residuals from the previous one can reintroduce components that are not orthogonal to the span of the previously included atoms. To illustrate this, consider the following example. The example is not intended to present a working matching pursuit algorithm. Consider the following dictionary for Euclidean 2-space. This dictionary is an equal-norm frame. 1 1 / 2 1 / 2 { , } , 0 3 / 2 1 / 2 Assume you have the following signal. 1 1 / 2 The following figure illustrates this example. The dictionary atoms are in red. The signal vector is in blue. 6-5 6 Matching Pursuit Construct this dictionary and signal in MATLAB. dictionary = [1 0; 1/2 sqrt(3)/2; -1/sqrt(2) -1/sqrt(2)]'; x = [1 1/2]'; Compute the inner (scalar) products between the signal and the dictionary atoms. scalarproducts = dictionary'*x; The largest scalar product in absolute value occurs between the signal and [-1/sqrt(2); -1/sqrt(2)]. This is clear because the angle between the two vectors is almost π radians. 6-6 Matching Pursuit Algorithms Form the residual by subtracting the orthogonal projection of the signal onto [-1/sqrt(2); -1/sqrt(2)] from the signal. Next, compute the inner products of the residual (new signal) with the remaining dictionary atoms. It is not necessary to include [-1/sqrt(2); -1/sqrt(2)] because the residual is orthogonal to that vector by construction. residual = x-scalarproducts(3)*dictionary(:,3); scalarproducts = dictionary(:,1:2)'*residual; The largest scalar product in absolute value is obtained with [1;0]. The best two atoms in the dictionary from two iterations are [-1/sqrt(2); -1/sqrt(2)] and [1;0]. If you iterate on the residual, you see that the output is no longer orthogonal to the first atom chosen. This means that a previously selected atom may be selected again. This fact and the associated complications with convergence argues in favor of “Orthogonal Matching Pursuit” on page 6-7 (OMP). Orthogonal Matching Pursuit In orthogonal matching pursuit (OMP), the residual is always orthogonal to the atoms already selected. This means that the same atom can never be selected twice and results in convergence for a d-dimensional vector after at most d steps. Conceptually, you can do this by using Gram-Schmidt to create an orthonormal set of atoms. With an orthonormal set of atoms, you see that for a d-dimensional vector, you can find at most d orthogonal directions. The OMP algorithm is: 1 Denote your signal by f. Initialize the residual R0f = f. 2 Select the atom that maximizes the absolute value of the inner product with R0f = f. Denote that atom by φp. 3 Form a matrix, Φ, with previously selected atoms as the columns. Define the orthogonal projection operator onto the span of the columns of Φ. P ( * )1 * 6-7 6 Matching Pursuit 4 Apply the orthogonal projection operator to the residual. 5 Update the residual. Rm1 f ( I P) Rm f I is the identity matrix. Orthogonal matching pursuit ensures that the previously selected atoms are not chosen again in subsequent steps. Weak Orthogonal Matching Pursuit It can be computationally efficient to relax the criterion that the selected atom maximizes the absolute value of the inner product to a less strict one. | x, p | max | x, k | (0, 1] k This is known as weak matching pursuit. 6-8 Matching Pursuit — Command Line Matching Pursuit — Command Line In this section... “Creating Dictionaries” on page 6-9 “Matching Pursuit With Dictionaries” on page 6-11 “Matching Pursuit — Electricity Consumption Data” on page 6-12 Creating Dictionaries To create dictionaries for matching pursuit at the command line, use wmpdictionary. This example shows you how to create a dictionary consisting of the discrete cosine transform (DCT-II) and the shifted Kronecker delta bases for a signal of length 1000. Define the dictionary as a cell array. dict = {'dct','RnIdent'}; Create the dictionary. mpdict = wmpdictionary(1000,'LstCpt',dict); This example shows you how to create a dictionary consisting of the Daubechies extremal-phase wavelet with 2 vanishing moments and a basis for polynomials of degree at most 19. Specifying a wavelet family short name without the optional level input uses wavelets at level 5 by default. dict = {'db2','poly'}; mpdict = wmpdictionary(1000,'LstCpt',dict); This example shows you how to create a dictionary with wavelet packets and the discrete cosine transform basis (DCT-II). Construct the wavelet packets from the Daubechies least-asymmetric wavelet with 4 vanishing moments at level 3. 6-9 6 Matching Pursuit For a wavelet packet, prepend the wavelet family short name with wp and provide the wavelet packet name and level in a cell array. dict = {{'wpsym4',3},'dct'}; mpdict = wmpdictionary(1000,'LstCpt',dict); This example shows you how to visualize the dictionary constructed from the Haar wavelet down to level 2. The Haar wavelet dictionary at level 2 consists of translated scaling functions at level 2 and translated wavelet functions at levels 2 and 1. Create the dictionary. [mpdict,~,~,longs] = wmpdictionary(100,'lstcpt',{{'haar',2}}); Use the longs output argument to divide the wavelet dictionary by level and type of function (scaling or wavelet). Step through a plot of the translated scaling functions and wavelets by level. for nn = 1:size(mpdict,2) if (nn <= longs{1}(1)) plot(mpdict(:,nn),'k','linewidth',2); grid on; xlabel('Translation'); title('Haar Scaling Function - Level 2'); elseif (nn>longs{1}(1) & nn<= longs{1}(1)+longs{1}(2)) plot(mpdict(:,nn),'r','linewidth',2); grid on; xlabel('Translation'); title('Haar Wavelet - Level 2'); else title('Haar Wavelet - Level 1'); plot(mpdict(:,nn),'b','linewidth',2); grid on; title('Haar Wavelet - Level 1'); xlabel('Translation'); end pause(0.2); end 6-10 Matching Pursuit — Command Line Matching Pursuit With Dictionaries Use wmpalg to compute the matching pursuit of a 1-D input signal in a dictionary. This example shows you how to obtain the matching pursuit of the cuspamax signal in a dictionary constructed from the discrete cosine transform (DCT-II) basis and sym4 wavelets. Load the cuspamax signal. Create a dictionary consisting of Daubechies least-asymmetric wavelets with 4 vanishing moments at level 2 and level 5 along with the DCT-II basis. load cuspamax; dict = {{'sym4',2},'sym4','dct'}; mpdict = wmpdictionary(length(cuspamax),'LstCpt',dict); Obtain the basic (nonorthogonal) matching pursuit in your dictionary with 25 iterations and plot the approximation. [YFIT1,R1,COEFF1,IOPT1,QUAL1] = wmpalg('BMP',cuspamax,mpdict); plot(YFIT1); axis tight; hold on; plot(cuspamax,'k'); legend('BMP','Original Signal','Location','NorthWest'); 6-11 6 Matching Pursuit Use orthogonal matching pursuit with the identical dictionary. [YFIT2,R2,COEFF2,IOPT2,QUAL2] = wmpalg('OMP',cuspamax,mpdict); Matching Pursuit — Electricity Consumption Data This example compares matching pursuit with a nonlinear approximation in the discrete Fourier transform basis. The data is electricity consumption data collected over a 24-hour period. This example shows you that by selecting atoms from a dictionary, matching pursuit is often able to approximate a vector more efficiently with M vectors than any single basis. 6-12 Matching Pursuit — Command Line Use a signal measuring electric consumption sampled at one minute intervals over 24 hours. This signal is also used in the wavelet multisignal analysis example. See Multisignal 1-D Wavelet Analysis. Load the dataset. The dataset contains 35 days of electric consumption. Choose day 32 for further analysis. The data is centered and scaled. Accordingly, the actual units of usage are not relevant. Plot the data. load elec35_nor; x = signals(32,:); plot(x); xlabel('Minutes'); ylabel('Usage'); set(gca,'xlim',[1 1440]); 6-13 6 Matching Pursuit The electricity consumption data contains smooth oscillations punctuated by abrupt increases and decreases in usage. Zoom in on a time interval from 500 minutes to 1200 minutes. plot(x); xlabel('Minutes'); ylabel('Usage'); set(gca,'xlim',[500 1200]); grid on; set(gca,'xtick',[500:50:1200]); In the preceding plot, you can clearly see the abrupt changes in the slowly-varying signal at approximately 650, 760, and 1120 minutes. 6-14 Matching Pursuit — Command Line In many real-world signals like these data, the interesting and important information is contained in the transients. Accordingly, it is important to adequately model these transient phenomena. Construct a signal approximation using 35 vectors chosen from a dictionary with orthogonal matching pursuit. Construct a dictionary consisting of the Daubechies extremal phase wavelet and scaling vectors at level 2, the discrete cosine transform basis, a sine basis, the Kronecker delta basis, and the Daubechies least asymmetric phase wavelet and scaling vectors with 4 vanishing moments at levels 1 and 4. dictionary = {{'db4',2},'dct','sin',{'sym4',1},{'sym4',4}}; [mpdict,nbvect] = wmpdictionary(length(x),'lstcpt',dictionary); Use orthogonal matching pursuit to find the best 35-term greedy approximation of the electric consumption data. Plot the result. [y,r,coef,iopt,qual] = wmpalg('OMP',x,mpdict,'itermax',35); plot(x); hold on; plot(y,'r'); xlabel('Minutes'); ylabel('Usage'); legend('Original Signal','OMP','Location','NorthEast'); set(gca,'xlim',[1 1440]); 6-15 6 Matching Pursuit You can see in the preceding plot that with 35 coefficients, orthogonal matching pursuit approximates both the smoothly-oscillating part of the signal as well as the abrupt changes in electricity usage. Determine how many vectors the OMP algorithm has selected from each of the subdictionaries. basez = cumsum(nbvect); k = 1; for nn = 1:length(basez) if (nn == 1) basezind{nn} = 1:basez(nn); else basezind{nn} = basez(nn-1)+1:basez(nn); 6-16 Matching Pursuit — Command Line end end dictvectors = cellfun(@(x) intersect(iopt,x),basezind,... 'UniformOutput',false) The following table summarizes how many vectors the OMP algorithm selected from each of the subdictionaries. Subdictionary Number of Vectors Selected Daubechies wavelet (db4) level 2 3 Discrete cosine transform 16 Sine 5 Daubechies least-asymmetric wavelet (sym4) level 1 2 Daubechies least-asymmetric wavelet (sym4) level 4 9 You see that the majority of the vectors come from the DCT or sine basis (60%). This is not surprising given the overall slowly-varying nature of the electricity consumption data. However, the addition of the 14 vectors from the wavelet subdictionaries enables you to accurately capture the abrupt signal changes. Compare matching pursuit with a DCT-sine dictionary to the full dictionary. To demonstrate that the addition of the wavelet subdictionaries has improved the signal approximation, repeat the OMP with just the DCT and sine subdictionaries. Do not restrict the approximation to just the 21 vectors chosen in the preceding example. Enable the OMP to select the 35 best vectors from the DCT-sine dictionary. Construct the dictionary and perform the OMP. dictionary2 = {'dct','sin'}; [mpdict2,nbvect2] = wmpdictionary(length(x),'lstcpt',dictionary2); [y2,r2,coef2,iopt2,qual2] = wmpalg('OMP',x,mpdict2,'itermax',35); Compare the OMP with the DCT-sine dictionary to the OMP with the addition of the wavelet subdictionaries. 6-17 6 Matching Pursuit plot(x); hold on; plot(y,'r','linewidth',2); title('Full Dictionary'); xlabel('Minutes'); ylabel('Usage'); set(gca,'xlim',[500 1200]); set(gca,'xtick',[500:50:1200]); figure; plot(x); hold on; plot(y2,'r','linewidth',2); title('DCT and Sine Dictionary'); xlabel('Minutes'); ylabel('Usage'); set(gca,'xlim',[500 1200]); set(gca,'xtick',[500:50:1200]); The results for the DCT-sine dictionary are shown in the following figure. 6-18 Matching Pursuit — Command Line The results including the wavelet subdictionaries is shown in the following figure. The addition of the wavelet subdictionaries enables you to more accurately model the abrupt changes in electricity usage in the data. The advantage of including the wavelet bases is especially clear in approximating the upward and downward spikes in usage at approximately 650 and 1120 minutes. This example shows you how to obtain the best 35-term nonlinear approximation of the signal in the discrete Fourier basis. 6-19 6 Matching Pursuit Obtain the DFT of the data. Sort the DFT coefficients and select the 35 largest coefficients. Because the DFT of a real-valued signal is conjugate symmetric, only consider frequencies from 0 (DC) to the Nyquist (1/2 cycles/minute). xdft = fft(x); [~,I] = sort(xdft(1:length(x)/2+1),'descend'); ind = I(1:35); Examine the vector ind. None of the indices correspond to 0 or the Nyquist. Therefore, you must add the corresponding complex conjugate in order to obtain the nonlinear approximation in the DFT basis. indconj = length(xdft)-ind+2; ind = [ind indconj]; xdftapp = zeros(size(xdft)); xdftapp(ind) = xdft(ind); xrec = ifft(xdftapp); Plot the approximation along with the original signal. plot(x); hold on; plot(xrec,'r'); xlabel('Minutes'); ylabel('Usage'); legend('Original Signal','Nonlinear DFT Approximation',... 'Location','NorthEast'); set(gca,'xlim',[1 1440]); Similar to the DCT-sine dictionary, the nonlinear DFT approximation does well at matching the smooth oscillations in electricity consumption data. Zoom in on the interval of the data containing the abrupt changes in consumption. plot(x); hold on; plot(xrec,'r','linewidth',2); xlabel('Minutes'); ylabel('Usage'); legend('Original Signal','Nonlinear DFT Approximation',... 'Location','NorthEast'); set(gca,'xlim',[500 1200]); set(gca,'xtick',[500:50:1200]); 6-20 Matching Pursuit — Command Line You can see in the above figure that the nonlinear DFT approximation is not able to accurately approximate the abrupt changes in consumption. 6-21 6 Matching Pursuit Matching Pursuit — Interactive Analysis In this section... “Matching Pursuit 1-D Interactive Tool” on page 6-22 “Interactive Matching Pursuit of Electricity Consumption Data” on page 6-38 Matching Pursuit 1-D Interactive Tool You can perform basic, orthogonal, and weak orthogonal matching pursuit using the interactive tool wavemenu. To access the Matching Pursuit 1-D interactive tool, enter wavemenu at the MATLAB command prompt. 6-22 Matching Pursuit — Interactive Analysis Click Matching Pursuit 1-D. 6-23 6 Matching Pursuit To demonstrate the Matching Pursuit 1-D tool, select File —> Example —> Cuspamax. 6-24 Matching Pursuit — Interactive Analysis In the upper left corner, you see the plot of the signal with the matching pursuit approximation superimposed. 6-25 6 Matching Pursuit Underneath the plot, you see the relative errors using the L1, L2, and L-infinity norms. The maximum relative error in a given norm is || R || 100 , || Y || where || || denotes the specified norm, R is the residual vector at each iteration in the matching pursuit algorithm, and Y is the signal. In the middle panel on the left is the plot of the final residual vector after the matching pursuit algorithm terminates. 6-26 Matching Pursuit — Interactive Analysis The bottom left panel displays the percentage of retained signal energy (L2 norm) and the relative error percentages for the L1, L2, and L-infinity norms over the algorithm iterations. 6-27 6 Matching Pursuit In the top middle panel of the Matching Pursuit 1-D tool, you see the indices of the selected coefficients from the subdictionaries. 6-28 Matching Pursuit — Interactive Analysis The left vertical axis shows the name of the subdictionary. The right vertical axis gives the ratio of selected vectors to the total number of vectors in the subdictionary. The location of the vertical bars along the horizontal axis gives the relative positions of the selected vectors in the subdictionaries. More detailed information on selected components is available by clicking More on Components in the bottom right panel. The bottom middle panel displays the superposition of selected vectors from the subdictionaries. 6-29 6 Matching Pursuit This plot enables you to assess the relative contribution of the subdictionaries to the signal approximation. In this example, you can see that the cosine and DCT subdictionaries contribute significantly to the approximation of the slowly-varying portions of the signal. The Daubechies least asymmetric wavelet with 4 vanishing moments (sym4) enables the matching pursuit to sparsely represent the cusp around index 700. In the top right panel of the Matching Pursuit 1-D tool, you see the dictionary used in the analysis. 6-30 Matching Pursuit — Interactive Analysis You have the ability to add or delete subdictionaries with Add Component and Del Component. The next panel contains the algorithm stopping rules. • Max. Iterations — This controls the number of iterations of the greedy matching pursuit algorithm. The value is equal to the number of expansion coefficients (vectors) used in the approximation. The utility of matching pursuit is that you can approximate many real-world signals efficiently with far fewer vectors than needed to span the signal space. • Max Relative Error — Specifies the stopping criterion based on the maximum relative error. Choose one of None, L2 norm, L1 norm, or Linf norm. The maximum relative error in a given norm is 6-31 6 Matching Pursuit || R || 100 , || Y || where || || denotes the specified norm, R is the residual vector at each iteration in the matching pursuit algorithm, and Y is the signal. In the next panel you select the algorithm used in the matching pursuit. Choose one of Basic MP for basic matching pursuit, Orthogonal MP for orthogonal matching pursuit, and Weak MP for weak orthogonal matching pursuit. See “Matching Pursuit Algorithms” on page 6-4 for a brief description of these algorithms. In the Display Parameters panel, you can control how the progress of the matching pursuit is displayed. Select one of • Final Plot — Plots the result of matching pursuit only after the algorithm terminates. • Stepwise — Updates the result every N iterations where N is a positive integer. If you select Stepwise, the Display every iterations item becomes visible. Select the number of iterations from the drop down menu. You are prompted to step through the algorithm with the Next or Final Plot. • Movie — Updates the result every N iterations where N is a positive integer in a continous manner. If you select Movie, the Display every iterations item becomes visible. Select the number of iterations from the drop down menu. Click Continue to step through the algorithm as a movie, which continues until the algorithm terminates. Click Pause to pause the algorithm, or Final Plot to update only at the termination of the algorithm. 6-32 Matching Pursuit — Interactive Analysis After you obtain a matching pursuit of a signal, use to obtain detailed interactive plots and information on the selected dictionary atoms and the final residual vector. Click More on Components. 6-33 6 Matching Pursuit From the above figure, you can see that while the DCT and cosine subdictionaries contribute energy across the extent of the signal, the wavelet and wavelet packet contributions are localized at the cusp around sample 700. This result is expected because wavelets and wavelet packets excel at sparsely representing abrupt changes in a signal or image. Change the Display to the Coefficients view. The Selection of Coefficients panel enables you to selectively sort and display contributions to the signal approximation by the various subdictionaries. Under Selection parameters, choose By Family and sym4 Select 6-34 lev5. Click Matching Pursuit — Interactive Analysis From the preceding operation, you see that the wavelet packet contributes to the approximation of the cusp, but does not contribute significantly to the global approximation. Choose dct and click Select. 6-35 6 Matching Pursuit The DCT basis contributes significantly to the global approximation of the signal but the smooth DCT basis vectors are not able to sparsely represent the cusp. Selecting More on Residuals allows you to examine the residual vector, a histogram of the residuals, a cumulative histogram, the estimated autocorrelation sequence, and the magnitude-squared discrete Fourier transform. 6-36 Matching Pursuit — Interactive Analysis You can control which plots are displayed and the appearance of the histogram by the options in the right panel. 6-37 6 Matching Pursuit Interactive Matching Pursuit of Electricity Consumption Data This example shows you how to perform an interactive matching pursuit of electricity consumption data collected over a 24-hour period. You can see the command-line equivalent of this example in “Matching Pursuit — Electricity Consumption Data” on page 6-12. Use a signal measuring electric consumption sampled at one minute intervals over 24 hours. This signal is also used in the wavelet multisignal analysis example, see Multisignal 1-D Wavelet Analysis. Load the electricity consumption signals in the workspace. Select the data for the 32nd day for further matching pursuit. load elec35_nor; x = signals(32,:); Start the interactive tool, wavemenu wavemenu Click the Matching Pursuit 1-D tool. 6-38 Matching Pursuit — Interactive Analysis Select File —> Import Signal from Workspace Load x. 6-39 6 Matching Pursuit Construct the following matching pursuit dictionary. In the Algorithm Stopping Rules panel, set Max. Iterations to 30. Select Orthogonal MP to use orthogonal matching pursuit. Click Approximate. 6-40 Matching Pursuit — Interactive Analysis 6-41 6 6-42 Matching Pursuit 7 Generating MATLAB Code from Wavelet Toolbox GUI • “Generating MATLAB Code for 1-D Decimated Wavelet Denoising and Compression” on page 7-2 • “Generating MATLAB Code for 2-D Decimated Wavelet Denoising and Compression” on page 7-13 • “Generating MATLAB Code for 1-D Stationary Wavelet Denoising” on page 7-20 • “Generating MATLAB Code for 2-D Stationary Wavelet Denoising” on page 7-27 • “Generating MATLAB Code for 1-D Wavelet Packet Denoising and Compression” on page 7-31 • “Generating MATLAB Code for 2-D Wavelet Packet Denoising and Compression” on page 7-35 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI Generating MATLAB Code for 1-D Decimated Wavelet Denoising and Compression Wavelet 1-D Denoising You can generate MATLAB code to reproduce GUI-based 1-D wavelet denoising at the command line. You must perform this operation in the Wavelet 1-D - - De-noising tool. You must first denoise your signal before you can enable the File > Generate Matlab Code (Denoising Process) operation. The generated MATLAB code does not include the calculation of the thresholds using thselect or wbmpen. Denoise Doppler Signal 1 Enter wavemenu at the MATLAB command prompt. 2 Select Wavelet 1-D in the Wavelet Toolbox Main Menu. 7-2 Generating MATLAB® Code for 1-D Decimated Wavelet Denoising and Compression 3 Load the noisy Doppler example analysis. Select File > Example Analysis > Noisy Signals - Constant Noise Variance > with sym4 at level 5 - -> Noisy Doppler. 4 Click De-noise. 7-3 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI 5 In the Select thresholding method drop-down menu, select the default Fixed form threshold. Use the default soft option. Set the thresholds by level as follows: • level 5 — 3.5 • level 4 — 3.72 • level 3 — 3.0 • level 2 — 2.0 • level 1 — 3.0 7-4 Generating MATLAB® Code for 1-D Decimated Wavelet Denoising and Compression Click De-noise. 6 Generate the MATLAB code by selecting File > Generate Matlab Code (Denoising Process). The operation generates the following MATLAB code. function sigDEN = func_denoise_dw1d(SIG) % FUNC_DENOISE_DW1-D Saved Denoising Process. % SIG: vector of data % ------------------% sigDEN: vector of denoised data % Analysis parameters. %--------------------wname = 'sym4'; level = 5; % Denoising parameters. %---------------------- 7-5 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI % meth = 'sqtwolog'; % scal_or_alfa = one; sorh = 's'; % Specified soft or hard thresholding thrParams = [... 3.00000000 ; ... 2.00000000 ; ... 3.00000000 ; ... 3.72000000 ; ... 3.50000000 ... ]; % Denoise using CMDDENOISE. %-------------------------sigDEN = cmddenoise(SIG,wname,level,sorh,NaN,thrParams); 7 Save func_denoise_dw1d.m in a folder on the MATLAB search path. Execute the following code. load noisdopp; SIG = noisdopp; % func_denoise_dw1d.m is generated code sigDEN = func_denoise_dw1d(SIG); 8 Export the denoised signal from the GUI by selecting File > Save > De-noised Signal. 7-6 Generating MATLAB® Code for 1-D Decimated Wavelet Denoising and Compression Save the denoised signal as denoiseddoppler.mat in a folder on the MATLAB search path. Load denoiseddoppler.mat in the MATLAB workspace. Compare denoiseddoppler with your command line result. load denoiseddoppler; plot(sigDEN,'k'); axis tight; hold on; plot(denoiseddoppler,'r'); legend('Command Line','GUI','Location','SouthEast'); 7-7 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI Interval Dependent 1-D Wavelet Denoising 1 Enter wavemenu at the MATLAB command prompt. 2 Select Wavelet 1-D. 3 Select File > Load > Signal, and load leleccum.mat from the matlab/toolbox/wavelet/wavedemo folder. 4 Select the sym4 wavelet, and set Level equal to 3. Click Analyze. 7-8 Generating MATLAB® Code for 1-D Decimated Wavelet Denoising and Compression When you inspect the original signal and the finest-scale wavelet coefficients, you see that the noise variance is not constant. In this situation, interval-dependent thresholding is useful. To implement interval-dependent denoising: 1 Click De-noise. 2 Under Select thresholding method, select Rigorous SURE. 3 Select Int. dependent threshold settings. 4 In the Interval Dependent Threshold Settings for Wavelet 1-D tool, choose Generate Default Intervals. Three intervals are created. Click Propagate to propagate the intervals to all levels. 5 Click Close, and answer Yes to Update Thresholds?. 6 Select De-noise. 7 Generate the MATLAB code by selecting File > Generate Matlab Code (Denoising Process). 7-9 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI The operation generates the following MATLAB code. function sigDEN = func_denoise_dw1d(SIG) % FUNC_DENOISE_DW1D Saved Denoising Process. % SIG: vector of data % ------------------- % sigDEN: vector of denoised data % Analysis parameters. %--------------------wname = 'sym4'; level = 3; % Denoising parameters. %---------------------% meth = 'rigrsure'; % scal_or_alfa = one; sorh = 's'; % Specified soft or hard thresholding thrSettings = {... [... 1.000000000000000 2410.000000000000000 5.659608351110114; ... 2410.000000000000000 3425.000000000000000 19.721391195242880; ... 3425.000000000000000 4320.000000000000000 4.907947952868359; ... ]; ... [... 1.000000000000000 2410.000000000000000 5.659608351110114; ... 2410.000000000000000 3425.000000000000000 5.659608351110114; ... 3425.000000000000000 4320.000000000000000 5.659608351110114; ... ]; ... [... 1.000000000000000 2410.000000000000000 5.659608351110114; ... 2410.000000000000000 3425.000000000000000 5.659608351110114; ... 3425.000000000000000 4320.000000000000000 5.659608351110114; ... ]; ... }; % Denoise using CMDDENOISE. %-------------------------sigDEN = cmddenoise(SIG,wname,level,sorh,NaN,thrSettings); 7-10 Generating MATLAB® Code for 1-D Decimated Wavelet Denoising and Compression 8 To avoid confusion with the MATLAB code generated in “Denoise Doppler Signal ” on page 7-2, change the function definition line. Change the function definition to: function sigDEN = func_IDdenoise_dw1d(SIG) Save the MATLAB program as func_IDdenoise_dw1d.m in a folder on the MATLAB search path. 9 Save the denoised signal as denoisedleleccum.mat with File > Save > De-noised Signal in a folder on the MATLAB search path. Execute the following code. load leleccum; load denoisedleleccum; sigDEN = func_IDdenoise_dw1d(leleccum); plot(sigDEN,'k'); hold on; plot(denoisedleleccum,'r'); legend('Command Line','GUI'); norm(sigDEN-denoisedleleccum,2) 7-11 7 7-12 Generating MATLAB® Code from Wavelet Toolbox™ GUI Generating MATLAB® Code for 2-D Decimated Wavelet Denoising and Compression Generating MATLAB Code for 2-D Decimated Wavelet Denoising and Compression In this section... “2-D Decimated Discrete Wavelet Transform Denoising” on page 7-13 “2-D Decimated Discrete Wavelet Transform Compression” on page 7-17 2-D Decimated Discrete Wavelet Transform Denoising You can generate MATLAB code to reproduce GUI-based 2-D decimated wavelet denoising at the command line. You must perform this operation in the Wavelet 2-D – –De-noising tool. You must first denoise your image before you can enable the File > Generate Matlab Code (Denoising Process) operation. 1 Enter wavemenu at the MATLAB command prompt. 2 Select Wavelet 2-D. 3 Load the Noisy SinSin example indexed image. Using the default biorthogonal wavelet and level 3 decomposition, click De-noise. 4 In the Select thresholding method drop-down menu, select the default Fixed form threshold and soft options. Use the default Unscaled white noise. Set the thresholds by level for the horizontal, diagonal, and vertical coefficients as follows: • Level 3 — 4 • Level 2 — 4 • Level 1 — 8 Enter these thresholds for the horizontal, diagonal, and vertical coefficients. 5 Select De-noise. 6 Generate the MATLAB code with File > Generate Matlab Code (Denoising Process). 7-13 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI The operation generates the following MATLAB code. function [XDEN,cfsDEN,dimCFS] = func_denoise_dw2d(X) % FUNC_DENOISE_DW2-D Saved Denoising Process. % X: matrix of data % ----------------- % XDEN: matrix of denoised data % cfsDEN: decomposition vector (see WAVEDEC2) % dimCFS: corresponding bookkeeping matrix % Analysis parameters. %--------------------wname = 'bior6.8'; level = 3; % Denoising parameters. %----------------------% meth = 'sqtwolog'; % scal_OR_alfa = one; 7-14 Generating MATLAB® Code for 2-D Decimated Wavelet Denoising and Compression sorh = 's'; thrParams = % Specified soft or hard thresholding [... 8.00000000 4.00000000 4.00000000 ; ... 8.00000000 4.00000000 4.00000000 ; ... 8.00000000 4.00000000 4.00000000 ... ]; roundFLAG = true; % Denoise using CMDDENOISE. %-------------------------[coefs,sizes] = wavedec2(X,level,wname); [XDEN,cfsDEN,dimCFS] = wdencmp('lvd',coefs,sizes, ... wname,level,thrParams,sorh); if roundFLAG , XDEN = round(XDEN); end if isequal(class(X),'uint8') , XDEN = uint8(XDEN); end 7 Save func_denoise_dw2d.m in a folder on the MATLAB search path, and execute the following code. load noissi2d.mat; noissi2d = X; [XDEN,cfsDEN,dimCFS] = func_denoise_dw2d(noissi2d); 8 Save your denoised image in a folder on the MATLAB search path as denoisedsin.mat. 7-15 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI Load the denoised image in the MATLAB workspace. Compare the result with your generated code. load denoisedsin.mat; % denoised image loaded in variable X subplot(121); imagesc(X); title('Image denoised in the GUI'); subplot(122); imagesc(XDEN); title('Image denoised with generated code'); % Norm of the difference is zero norm(XDEN-X,2) 7-16 Generating MATLAB® Code for 2-D Decimated Wavelet Denoising and Compression 2-D Decimated Discrete Wavelet Transform Compression You can generate MATLAB code to reproduce GUI-based 2-D decimated wavelet compression at the command line. You must perform this operation in the Wavelet 2-D --Compression tool. You must first compress your image before you can enable the File > Generate Matlab Code (Compression Process) operation. 1 Enter wavemenu at the MATLAB command prompt. 2 Select Wavelet 2-D. 3 Select File > Load > Image and load the detfingr.mat indexed image from the matlab/toolbox/wavelet/wavedemo folder. When the Loading an Image dialog appears, select No to load the grayscale image. 7-17 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI 4 Select the bior3.5 wavelet, and set Level to 3. 5 Click Analyze, then click Compress. 6 Using the default Global thresholding, set Select thresholding method to Bal.sparsity-norm (sqrt). 7 Click Compress. 8 File > Generate Code (Compression Process) generates the following code. function [XCMP,cfsCMP,dimCFS] = func_compress_dw2d(X) % FUNC_COMPRESS_DW2D Saved Compression Process. % X: matrix of data % ----------------% XCMP: matrix of compressed data % cfsCMP: decomposition vector (see WAVEDEC2) % dimCFS: corresponding bookkeeping matrix % Analysis parameters. %--------------------wname = 'bior3.5'; level = 3; % Compression parameters. %-----------------------% meth = 'sqrtbal_sn'; sorh = 'h'; % Specified soft or hard thresholding thrSettings = 10.064453124999996; roundFLAG = true; % Compression using WDENCMP. %-------------------------[coefs,sizes] = wavedec2(X,level,wname); [XCMP,cfsCMP,dimCFS] = wdencmp('gbl',coefs,sizes, ... wname,level,thrSettings,sorh,1); if roundFLAG , XCMP = round(XCMP); end if isequal(class(X),'uint8') , XCMP = uint8(XCMP); end 7-18 Generating MATLAB® Code for 2-D Decimated Wavelet Denoising and Compression 9 Save the MATLAB program, func_compress_dw2d.m, in a folder on the MATLAB search path. Execute the following code at the command line. load detfingr.mat; % Image data is in X [XCMP,cfsCMP,dimCFS] = func_compress_dw2d(X); 10 Save the compressed image from the Wavelet 2-D - - Compression tool in a folder on the MATLAB search path. Use File > Save > Compressed Image, and name the file compressed_fingerprint.mat. Execute the following code. load compressed_fingerprint.mat; % Image data is in X norm(XCMP-X,2) 7-19 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI Generating MATLAB Code for 1-D Stationary Wavelet Denoising You can generate MATLAB code to reproduce GUI-based 1-D nondecimated (stationary) wavelet denoising at the command line. You must perform this operation in the Stationary Wavelet Transform Denoising 1-D tool. You must first denoise your signal before you can enable the File > Generate Matlab Code (Denoising Process) operation. 1-D Stationary Wavelet Transform Denoising 1 Enter wavemenu at the MATLAB command prompt. 2 Select SWT Denoising 1-D. 3 Load the Noisy bumps example. Select File > Example Analysis > Noisy Signals > with sym4 at level 5 - - -> Noisy bumps 7-20 Generating MATLAB® Code for 1-D Stationary Wavelet Denoising 4 Set the thresholds as follows: • Level 1 — 3.5 • Level 2 — 3.4 • Level 3 — 2.3 • Level 4 — 5.3 • Level 5 — 2.2 Click De-noise. 7-21 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI 5 Generate the MATLAB code with File > Generate Matlab Code (Denoising Process). 7-22 Generating MATLAB® Code for 1-D Stationary Wavelet Denoising The operation generates the following MATLAB code. function [sigDEN,wDEC] = func_denoise_sw1d(SIG) % FUNC_DENOISE_SW1-D Saved Denoising Process. % SIG: vector of data % ------------------- % sigDEN: vector of denoised data % wDEC: stationary wavelet decomposition % Analysis parameters. %--------------------wname = 'sym4'; level = 5; % Denoising parameters. %---------------------% meth = 'sqtwolog'; % scal_OR_alfa = one; sorh = 's'; thrParams = % Specified soft or hard thresholding {... 7-23 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI [... 1.00000000 1024.00000000 3.50000000; ... 1024.00000000 3.40000000; ... 1024.00000000 2.30000000; ... 1024.00000000 5.29965570; ... 1024.00000000 2.20000000; ... ]; ... [... 1.00000000 ]; ... [... 1.00000000 ]; ... [... 1.00000000 ]; ... [... 1.00000000 ]; ... }; % Decompose using SWT. %--------------------wDEC = swt(SIG,level,wname); % Denoise. %--------len = length(SIG); for k = 1:level thr_par = thrParams{k}; if ~isempty(thr_par) NB_int = size(thr_par,1); x = [thr_par(:,1) ; thr_par(NB_int,2)]; x = round(x); x(x<1) = 1; x(x>len) = len; thr = thr_par(:,3); for j = 1:NB_int if j==1 , d_beg = 0; else d_beg = 1; end j_beg = x(j)+d_beg; j_end = x(j+1); j_ind = (j_beg:j_end); wDEC(k,j_ind) = wthresh(wDEC(k,j_ind),sorh,thr(j)); end 7-24 Generating MATLAB® Code for 1-D Stationary Wavelet Denoising end end % Reconstruct the denoise signal using ISWT. %------------------------------------------sigDEN = iswt(wDEC,wname); 6 Save func_denoise_sw1d.m in a folder on the MATLAB search path. Execute the following code. load noisbump.mat; [sigDEN,wDEC] = func_denoise_sw1d(noisbump); 7 Select File > Save De-noised Signal, and save the denoised signal as denoisedbumps.mat in a folder on the MATLAB search path. Execute the following code. load denoisedbump.mat; plot(sigDEN,'k'); axis tight; hold on; 7-25 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI plot(denoisedbump,'r'); % norm of the difference norm(sigDEN-denoisedbump,2) 7-26 Generating MATLAB® Code for 2-D Stationary Wavelet Denoising Generating MATLAB Code for 2-D Stationary Wavelet Denoising You can generate MATLAB code to reproduce GUI-based 2-D stationary wavelet denoising at the command line. You can generate code to denoise both indexed and truecolor images. You must perform this operation in the SWT Denoising 2-D tool. You must first denoise your image before you can enable the File > Generate Matlab Code (Denoising Process) operation. 2-D Stationary Wavelet Transform Denoising 1 Enter wavemenu at the MATLAB command prompt. 2 Select SWT Denoising 2-D. 3 Select File > Load Image, and load noiswom.mat from the matlab/toolbox/wavelet/wavedemo folder. Choose No when prompted to use the grayscale image. 7-27 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI 4 Select the db4 wavelet, and set the Level to 5. 5 Click Decompose Image. 6 Use the default soft thresholding method with Fixed form threshold and Unscaled white noise for Select noise structure. 7 Set the following thresholds for the horizontal, diagonal, and vertical details. Ensure that you set the thresholds for the three detail coefficient types. • Level 1 — 5 • Level 2 — 4 • Level 3 — 3 • Level 4 — 2 • Level 5 — 1 8 Click De-noise. 9 Select File > Generate Matlab Code (Denoising Process). The operation generates the following MATLAB code. function [XDEN,wDEC] = func_denoise_sw2d(X) % FUNC_DENOISE_SW2D Saved Denoising Process. % X: matrix of data % ----------------% XDEN: matrix of denoised data % wDEC: stationary wavelet decomposition % Analysis parameters. %--------------------wname = 'db2'; level = 5; % Denoising parameters. %----------------------% meth = 'sqtwolog'; % scal_OR_alfa = one; sorh = 's'; % Specified soft or hard thresholding 7-28 Generating MATLAB® Code for 2-D Stationary Wavelet Denoising thrSettings = [... 1.0000 2.0000 1.0000 2.0000 1.0000 2.0000 ]; roundFLAG = false; 3.0000 3.0000 3.0000 4.0000 4.0000 4.0000 5.0000 ; ... 5.0000 ; ... 5.0000 ... % Decompose using SWT2. %--------------------wDEC = swt2(X,level,wname); % Denoise. %--------permDir = [1 3 2]; for j = 1:level for kk=1:3 ind = (permDir(kk)-1)*level+j; thr = thrSettings(kk,j); wDEC(:,:,ind) = wthresh(wDEC(:,:,ind),sorh,thr); end end % Reconstruct the denoise signal using ISWT2. %------------------------------------------XDEN = iswt2(wDEC,wname); if roundFLAG , XDEN = round(XDEN); end 10 Save this MATLAB program as func_denoise_sw2d.m in a folder on the MATLAB search path. Execute the following code. load noiswom [XDEN,wDEC] = func_denoise_sw2d(X); 11 Save the denoised image as denoisedwom.mat in a folder on the MATLAB search path. 7-29 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI 12 Execute the following code. load denoisedwom % Compare the GUI and command line results imagesc(X); title('GUI Operation'); colormap(gray); figure; imagesc(XDEN); title('Command Line Operation'); colormap(gray); norm(XDEN-X,2) 7-30 Generating MATLAB® Code for 1-D Wavelet Packet Denoising and Compression Generating MATLAB Code for 1-D Wavelet Packet Denoising and Compression 1-D Wavelet Packet Denoising You can generate MATLAB code to reproduce GUI–based 1-D wavelet packet denoising at the command line. You must perform this operation in the Wavelet Packet 1-D - - De-noising tool. You must first denoise your signal before you can enable the File > Generate Matlab Code (Denoising Process) operation. 1 Enter wavemenu at the MATLAB command prompt. 2 Select Wavelet Packet 1-D. 3 Select File > Load Signal and load noisbump.mat from the matlab/toolbox/wavelet/wavedemo folder. 4 Select the db4 wavelet, and set the Level to 4. Accept the default value Shannon for Entropy. 7-31 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI 5 Click Analyze. 6 Click De-noise. 7 Under Select thresholding method, accept the default Fixed form thr. (unscaled wn) with the soft radio button enabled. Set Select Global Threshold to 2.75. 8 Click De-noise. 9 Select File > Generate Matlab Code (Denoising Process) The operation generates the following MATLAB code. function [sigDEN,wptDEN] = func_denoise_wp1d(SIG) % FUNC_DENOISE_WP1D Saved Denoising Process. % SIG: vector of data % ------------------- % sigDEN: vector of denoised data % wptDEN: wavelet packet decomposition (wptree object) % Analysis parameters. %--------------------Wav_Nam = 'db4'; Lev_Anal = 4; Ent_Nam = 'shannon'; Ent_Par = 0; % Denoising parameters. %---------------------% meth = 'sqtwologuwn'; sorh = 's'; % Specified soft or hard thresholding thrSettings = {sorh,'nobest',2.750000000000000,1}; % Decompose using WPDEC. %---------------------wpt = wpdec(SIG,Lev_Anal,Wav_Nam,Ent_Nam,Ent_Par); % Nodes to merge. 7-32 Generating MATLAB® Code for 1-D Wavelet Packet Denoising and Compression %----------------n2m = []; for j = 1:length(n2m) wpt = wpjoin(wpt,n2m(j)); end % Denoise using WPDENCMP. %-----------------------[sigDEN,wptDEN] = wpdencmp(wpt,thrSettings{:}); Save func_denoise_wp1d.m in a folder on the MATLAB search path. Save the denoised signal from the Wavelet Packet 1-D - - De-noising tool as wp_denoisedbump.mat in a folder on the MATLAB search path. Execute the following code. load noisbump; [sigDEN,wptDEN] = func_denoise_wp1d(noisbump); 7-33 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI load wp_denoisedbump; plot(sigDEN); title('Denoised Signal'); axis([1 1024 min(sigDEN)-1 max(sigDEN+1)]); norm(sigDEN-wp_denoisedbump,2) 7-34 Generating MATLAB® Code for 2-D Wavelet Packet Denoising and Compression Generating MATLAB Code for 2-D Wavelet Packet Denoising and Compression 2-D Wavelet Packet Compression You can generate MATLAB code to reproduce GUI–based 2-D wavelet packet compression at the command line. You must perform this operation in the Wavelet 2-D - - Compression tool. You must first compress your image before you can enable the File > Generate Matlab Code (Compression Process) operation. 1 Enter wavemenu at the MATLAB command prompt. 2 Select Wavelet Packet 2-D. 3 Select File > Load > Example Analysis > Indexed Images, and load the tire. 4 Using the default parameter settings, click Best Tree. 7-35 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI 5 Click Compress. 6 Set Select thresholding method to Bal.sparsity-norm (sqrt). 7 Click Compress. 7-36 Generating MATLAB® Code for 2-D Wavelet Packet Denoising and Compression 8 File > Generate Code (Compression Process) generates the following code. function [XCMP,wptCMP] = func_compress_wp2d(X) % FUNC_COMPRESS_WP2D Saved Compression Process. % X: matrix of data % ----------------- % XCMP: matrix of compressed data % wptCMP: wavelet packet decomposition (wptree object) % Analysis parameters. 7-37 7 Generating MATLAB® Code from Wavelet Toolbox™ GUI %--------------------Wav_Nam = 'haar'; Lev_Anal = 2; Ent_Nam = 'shannon'; Ent_Par = 0; % Compression parameters. %----------------------% meth = 'sqrtbal_sn'; sorh = 'h'; % Specified soft or hard thresholding thrSettings = {sorh,'nobest',16.499999999999886,1}; roundFLAG = true; % Decompose using WPDEC2. %----------------------wpt = wpdec2(X,Lev_Anal,Wav_Nam,Ent_Nam,Ent_Par); % Nodes to merge. %----------------n2m = [2 3]; for j = 1:length(n2m) wpt = wpjoin(wpt,n2m(j)); end % Compression using WPDENCMP. %---------------------------[XCMP,wptCMP] = wpdencmp(wpt,thrSettings{:}); if roundFLAG , XCMP = round(XCMP); end if isequal(class(X),'uint8') , XCMP = uint8(XCMP); end 9 Save the generated MATLAB code as func_compress_wp2d.m in a folder on the MATLAB search path, and execute the following code. load tire; [XCMP,wptCMP] = func_compress_wp2d(X); 10 Save the compressed image from the Wavelet 2-D -- Compression tool as compressed_tire.mat in a folder on the MATLAB search path. Use File > Save > Compressed Image to save the compressed image. 11 Execute the following code to compare the command line and GUI result. 7-38 Generating MATLAB® Code for 2-D Wavelet Packet Denoising and Compression load compressed_tire.mat; norm(XCMP-X,2) 7-39 7 7-40 Generating MATLAB® Code from Wavelet Toolbox™ GUI A GUI Reference This appendix explains some of the features of the Wavelet Toolbox graphical user interface (GUI). • “General Features” on page A-2 • “Continuous Wavelet Tool Features” on page A-12 • “Wavelet 2-D Tool Features” on page A-13 • “Wavelet Packet Tool Features (1-D and 2-D)” on page A-14 • “Wavelet Display Tool” on page A-16 • “Wavelet Packet Display Tool” on page A-17 A GUI Reference General Features Some features of the Wavelet Toolbox graphical user interface are • Color coding • Connectedness of plots • Using the mouse • Controlling the colormap • Controlling the number of colors • Controlling the coloration mode • Customizing graphical objects • Zooming in on plots • Using menus • Using View Axes button • Using Interval Dependent Threshold Settings tool Note In this appendix, axis (or axes) refers to the MATLAB graphic object. Color Coding In all the graphical tools, signals and analysis components are color coded as follows. Signal Shown In Original Red Reconstructed or synthesized Yellow Approximations Variegated shades of blue (high level = darker) Details Variegated shades of green (high level = darker) A-2 General Features Connection of Plots Plots containing related information and graphed on the same abscissa are connected in the sense that manipulations performed on one plot affect all others in the same way. For images, the connection holds in both abscissa and ordinate. You can manipulate all plots along an individual axis (X or Y) or you can manipulate all plots along both axes at the same time (XY). For example, the approximations and details shown in the separate mode view of a decomposition all respond together when any of the plots is magnified or zoomed. Click and drag your mouse over the region you want to zoom. Clicking XY+ results in the zoom being applied to all the plots. • Zoom in on relevant detail. One advantage of using the graphical interface tools is that you can zoom in easily on any part of the signal and examine it in greater detail. Drag a rubber band box (by holding down the left mouse button) over the portion of the signal you want to magnify. Here, we’ve selected the noisy part of the original signal. A-3 A GUI Reference Click the X+ button (located at the bottom of the screen) to zoom horizontally. The Wavelet 1-D tool zooms all the displayed signals. The other zoom controls do more or less what you’d expect them to. The X- button, for example, zooms out horizontally. The history function keeps A-4 General Features track of all your views of the signal. Return to a previous zoom level by clicking the left arrow button. • Zooming in on Detail Drag a rubber band box (by holding down the left mouse button) over the portion of the image you want to magnify. Click the XY+ button (located at the bottom of the screen) to zoom horizontally and vertically. • The History pane enables you to remember how you zoom the axes so that you can toggle back and forth between views. Using the Mouse Wavelet Toolbox software uses three types of mouse control. Left Mouse Button Middle Mouse Button Right Mouse Button Make selections. Activate controls. Display cross-hairs to show position-dependent information. Translate plots up and down, and left and right. A-5 A GUI Reference Note The functionality of the middle mouse button and the right mouse button can be inverted depending on the platform. Making Selections and Activating Controls Most of your work with Wavelet Toolbox graphical tools involves making selections and activating controls. You do this using the left (or only) mouse button. Translating Plots By holding down the right mouse button (or its equivalent on a one- or two-button mouse), you can move the mouse to draw a rectangle in either a horizontal or vertical orientation. Releasing the middle mouse button then causes the plot to shift horizontally (or vertically) by an amount proportional to the width (or height) of the rectangle. Displaying Position-Dependent Information When you hold down the middle mouse button (or its equivalent on a oneor two-button mouse), a cross-hair cursor appears over the graph or plot. Position-dependent information also appears in the Info box located at the A-6 General Features bottom center of the tool. The type of information that appears depends on what tool you are using and the plot in which your cursor is located.. Controlling the Colormap The Colormap selection box, located at the lower right of the window, allows you to adjust the colormap that is used to plot images or coefficients (wavelet or wavelet packet). This is more than an aesthetic adjustment because you are likely to see different features depending on your colormap selection. Controlling the Number of Colors The Nb. Colors slider, located at the bottom right of the window, allows you to adjust how many colors the tool uses to plot images or coefficients (wavelet or wavelet packet). You can also use the edit control to adjust the number of colors. Adjusting the number of colors can highlight different features of the plot. Consider the coefficients plot of the Koch curve generated in the Continuous Wavelet tool, shown here using 129 colors. A-7 A GUI Reference and here using 68 colors. Controlling the Coloration Mode In the Continuous Wavelet tools, the coloration of coefficients can be done in several different ways. Coloration mode — Three parameters are used color the coefficients. • init or current — When you select init, coloration is performed with all the coefficient values. When you select current, only the coefficients displayed in the current axis limits are used. • by scales or all scales — When you select by scale, the coloration is done separately for each scale. When you select all scales, all scales are used. • abs — When you select abs, the absolute values of the coefficients are used. In the Wavelet 1-D tool, you access coefficients coloration with the More Display Options button, and then select the desired Coloration Mode option. A-8 General Features The More Display Options button appears only when the Display mode is one of the following — Show and Scroll, Show and Scroll (Stem Cfs), Superimposed, and Separate). In this case, scales are replaced by levels in all options of the Coloration Mode menu. Using Menus General Menu Bar At the top of most windows you find the same kind of structure. The menu bar of each figure in Wavelet Toolbox software is very similar to the menu bar of the default MATLAB figures. You can use many of the tools that are offered in the menus and associated toolbar of the standard MATLAB figures. One of the main differences is the View menu, which depends on the current tool used. View Dynamical Visualization Tool Option. The View > Dynamical Visualization Tool option lets you enable or disable the Dynamical Visualization Tool located at the bottom of each window. Enabling the Dynamical Visualization Tool activates the zoom, center, history, and axes options at the bottom of the interactive tool. Before using Zoom In, Zoom Out, or Rotate 3D options (or the equivalent icons from the toolbar), you must disable the Dynamical Visualization Tool to avoid possible conflicts. Default Display Mode Option. The Default Display Mode option is specific to the Wavelet 1-D tool and lets you set a default Display Mode for all the different analyses you perform inside the same tool. A-9 A GUI Reference Using the View Axes Button The Dynamical Visualization Tool is located at the bottom of most of the windows in the Wavelet Toolbox software. In this tool, the View Axes toggle button lets you magnify the axis that you choose. A-10 General Features The toggle buttons in the View Axes figure are positioned so that you can understand which axis is correlated with a button. When you click the same toggle button again, you restore the original view. Clicking the View Axes toggle button again closes the View Axes figure. A-11 A GUI Reference Continuous Wavelet Tool Features Here is an example of an option that allows you to perform analysis using different scale modes. Scale Settings • Step by Step Mode — Specify the initial scale, the step size, and the maximum scale. • Power 2 Mode — The scales are 20, 21, up to power you select in the Power drop down menu. These are the same scales used for discrete analysis. • Manual Mode — Enter a vector of scales. A-12 Wavelet 2-D Tool Features Wavelet 2-D Tool Features The Wavelet 2-D tool is described in “Two-Dimensional Discrete Wavelet Analysis” on page 3-137. Here is an example of an option that allows you to view a selected part of the window at a full window resolution. In the Full Size menu on the right side of the interactive tool choose the image you want to view as full size. Click your selection again to restore the original view. A-13 A GUI Reference Wavelet Packet Tool Features (1-D and 2-D) Coefficients Coloration NAT or FRQ is for Natural or Frequency order. By level or Global is for a coloration made level by level or taking all detail levels. abs is used to take the absolute values of coefficients. Node Action When you select a node in the tree, the selected option is performed. A complete description of options is provided in the following sections. Node Label The node labels can be changed using the pop-up menu. For example, the Type option labels the nodes with (a) for approximation and (d) for detail. Node Action Functionality The available options in the Node Action menu are • Visualize: When you select a node in the wavelet packet tree the corresponding signal appears. • Split/Merge: If a terminal node is selected, it is split, growing the wavelet packet tree. Selecting other nodes has the behavior of merging all the nodes below it in the wavelet packet tree. • Recons.: When you select a node in the wavelet packet tree, the corresponding reconstructed signal appears. • Select On/Off: When On, you can select many nodes in the wavelet packet tree. Then you can reconstruct a synthesized signal from the selected nodes using the Reconstruct button on the main window. Use the Off selection to deselect all the previous selected nodes. • Statistics: When you select a node in the wavelet packet tree, the Statistics tool appears using the signal corresponding to the selected node. A-14 Wavelet Packet Tool Features (1-D and 2-D) • View Col. Cfs.: When active, this option removes all the colored coefficients displayed, and lets you redraw only the corresponding coefficients by selecting a node in the wavelet packet tree. A-15 A GUI Reference Wavelet Display Tool The Wavelet Display tool is mentioned in the section “Introduction to Wavelet Families” in the Wavelet Toolbox Getting Started Guide. The Refinement drop down menu allows you choose the number of points that the wavelet and scaling functions are computed over. The number of points are in powers of 2. In the following figure, the db2 scaling and wavelet functions are computed over a grid of 28 points. The Information on: selections allow you to obtain more detailed information on the current wavelet family, or all supported families. A-16 Wavelet Packet Display Tool Wavelet Packet Display Tool The Refinement drop down menu allows you choose the number of points that the wavelet packets are computed over. The number of points are in powers of 2. In the following figure, the db2 wavelet packets are computed over a grid of 28 points. The Wav. Pack. from 0 to: allows you to choose the number of wavelet packets to display. The Information on: selections allow you to obtain more detailed information on the current wavelet family, Daubechies Family (DB), or wavelet packets in general, W Systems. A-17 A A-18 GUI Reference Index A B adding a new wavelet 1-26 algorithms decomposition 3-40 discrete wavelet transform (DWT) 3-37 fast wavelet transform (FWT) 3-37 filters 3-37 for biorthogonal 3-45 Mallat 3-37 polyphase 1-50 rationale 3-45 reconstruction 3-47 stationary wavelet transform (SWT) 3-62 analysis continuous features 2-2 continuous complex 2-21 continuous real 2-2 discrete 3-2 one-dimensional discrete wavelet 3-2 one-dimensional wavelet packet 4-7 orthogonal algorithm 3-45 filters 3-37 translation invariant 3-62 two-dimensional discrete wavelet 3-137 two-dimensional wavelet packet 4-15 wavelet packet 4-2 approximations coefficients discrete wavelet transform 3-40 extracting one-dimensional 3-6 extracting two-dimensional 3-143 reconstruction example code 3-6 axes view A-10 bases. See analysis,wavelet packets besttree function 4-51 binning processed data 5-84 biorthogonal wavelets definition 1-10 border distortion boundary value replication 3-52 periodic extension 3-52 periodic padding 3-53 periodized wavelet transform 3-62 smooth padding 3-53 symmetric extension 3-52 symmetrization 3-52 zero-padding 3-52 breakdowns variance 5-14 Index C calculating a default global threshold 3-77 coefficients approximation extracting one-dimensional 3-6 fast wavelet transform 3-40 coloration A-14 complex continuous wavelet 2-23 continuous wavelet 2-5 detail fast wavelet transform 3-40 one-dimensional 3-6 two-dimensional 3-143 discrete wavelet Wavelet 1-D tool 3-35 Wavelet 2-D tool 3-162 line 2-10 load. See importing to the GUI save. See exporting from the GUI coiflets Index-1 Index definition 1-9 Coloration Mode color coding A-2 controlling A-8 controlling the colormap A-7 Coloration Mode menu 2-18 complex frequency B-spline wavelets 1-19 complex Gaussian wavelets 1-18 complex Morlet wavelets 1-19 complex Shannon wavelets 1-20 compressing images true compression 5-55 using graphical interface 3-153 compression ddencmp function 4-4 difference with de-noising 5-51 energy ratio 5-54 norm recovery 5-54 number of zeros 5-54 procedure wavelet packets 4-5 wavelets 5-51 retained energy 5-53 true 5-55 using wavelet packets 4-18 continuous wavelet transform (CWT) 2-18 See also analysis, transforms CWT. See continuous wavelet transform (CWT) D ddencmp command 3-77 de-noising basic model two-dimensional 5-12 default values 4-4 fixed form threshold 5-5 minimax performance 5-5 noise size estimate 5-8 nonwhite noise 5-8 Index-2 procedure wavelet packets 4-5 SURE estimate 5-5 thresholding 3-10 using SWT 1-D using command line 3-77 1-D using graphical interface 3-80 2-D using graphical interface 3-175 using thresholding 3-173 variance adaptive 5-14 de-noising images stationary wavelet transform 3-178 two-dimensional procedure 5-12 de-noising signals process 3-8 decomposition best-level 4-47 choosing optimal 4-41 displaying results of multilevel 3-7 entropy-based criteria 4-42 load. See importing to the GUI performing multilevel using command line 3-5 using graphical interface 3-18 save. See exporting from the GUI single-level 3-15 structure one-dimensional 3-35 two-dimensional 3-162 details coefficients extracting one-dimensional 3-6 extracting two-dimensional 3-143 decomposition 4-30 orientation 3-43 reconstruction from command line 3-6 dilation equation twin-scale relation 3-37 discrete wavelet transform (DWT) 3-2 Index See also analysis, transforms display mode 2-D tree 3-151 downsampling one-dimensional 3-41 two-dimensional 3-43 DWT. See discrete wavelet transform, transforms E edge effects. See border distortion entropy definitions 4-42 exporting from the GUI complex continuous wavelet 2-29 continuous wavelet 2-18 discrete stationary wavelet 1-D 3-84 discrete stationary wavelet 2-D 3-179 discrete wavelet 2-D 3-156 discrete wavelet 3-D 3-189 variance adaptive thresholding 5-25 wavelet 1-D 3-28 wavelet packets 4-22 wavelet regression estimation 1-D 5-85 F fast wavelet transform (FWT). See transforms filters FIR MATLAB file used for construction 1-29 high-pass 3-40 low-pass 3-40 quadrature mirror construction example 3-39 scaling 3-38 frequencies parameter 4-39 frequency B-spline wavelets 1-19 Full Decomposition Mode 3-19 FWT. See transforms G Gaussian wavelets 1-14 GUI complex continuous wavelet 2-24 continuous wavelet 2-8 full window resolution A-13 image de-noising using SWT 3-175 local variance adaptive thresholding 5-17 multiscale principal components analysis 5-45 multisignal analysis 3-95 multivariate wavelet de-noising 5-34 regression estimation 5-80 signal de-noising using SWT 3-80 true compression 5-61 using menus A-9 using the mouse A-5 wavelet display A-16 wavelet one-dimensional 3-13 wavelet packet 4-7 wavelet packet display A-17 wavelet three-dimensional 3-182 wavelet two-dimensional 3-147 I IDWT. See inverse discrete wavelet transform, transforms images indexed 3-161 importing to the GUI complex continuous wavelet 2-29 continuous wavelet 2-18 discrete stationary wavelet 1-D 3-84 discrete stationary wavelet 2-D 3-179 discrete wavelet 2-D 3-156 discrete wavelet 3-D 3-189 Index-3 Index variance adaptive thresholding 5-25 wavelet 1-D 3-28 wavelet packets 4-22 wavelet regression estimation 1-D 5-85 inverse lifting wavelet transform (ILWT) 1-50 inverse stationary wavelet transform (ISWT) 3-67 L lifting 1-42 lifting wavelet transform (LWT) 1-50 load. See importing in the GUI Load data for Stochastic Design Regression dialog box 5-84 Load Signal dialog box wavelet packets 4-8 wavelets 3-15 local maxima lines 2-10 M MATLAB files for wavelet families 1-29 merge. See wavelet packets Mexican hat wavelet definition 1-15 Meyer wavelet definition 1-12 minimax 5-5 More Display Options button 3-19 More on Residuals for Wavelet 1-D Compression window 3-24 Morlet wavelet definition 1-16 multiresolution 3-45 N node action A-14 Index-4 noise suppressing 3-3 See also de-noising unscaled 5-8 nondecimated DWT. See transforms (stationary wavelet) O objects 4-53 P padding. See border distortion periodic-padding signal extension 3-53 periodized wavelet transform. See border distortion predefined wavelet families type 1 1-29 type 2 1-30 type 3 1-30 type 4 1-31 type 5 1-31 Q quadrature mirror filters (QMF) orthfilt function 3-39 R reconstruction MATLAB files 3-49 one step 3-44 one-dimensional IDWT 3-42 two-dimensional IDWT 3-43 regression estimation one-dimensional wavelet 5-80 regularity wavelet families 1-22 Index residuals display 1-D discrete wavelet compression 3-24 1-D stationary wavelet decomposition 3-83 2-D discrete wavelet compression 3-156 2-D stationary wavelet decomposition 3-178 reverse biorthogonal wavelets 1-12 S save. See exporting from the GUI scale choosing using command line 2-6 choosing using graphical interface 2-24 to frequency display 2-13 scale mode A-12 scaling filters definition 3-38 Separate Mode 3-19 Shannon wavelets 1-20 Show and Scroll Mode 3-19 Show and Scroll Mode (Stem Cfs) 3-19 shrink. See thresholding signal extensions border distortion 3-52 signal-end effects. See border distortion smooth padding signal extension 3-53 splines filter lengths 3-45 split. See wavelet packets stationary wavelet transform (SWT) 3-62 Superimpose Mode 3-19 support. See wavelet families symlets definition 1-8 symmetrization signal extension 3-53 symmetry. See wavelet families T thresholding for optimal de-noising 3-10 hard 5-7 interval dependent 5-15 rules tptr options 5-4 soft 5-7 thselect MATLAB file 5-4 transforms fast wavelet (FWT) 3-37 inverse lifting wavelet transform (ILWT) 1-50 inverse stationary wavelet (ISWT) 3-67 lifting wavelet (LWT) 1-50 stationary wavelet (SWT) 3-62 translation invariant 3-62 translation using the mouse A-6 translation invariance 3-62 Tree Mode definition 3-19 features 3-151 trees best 4-10 best-level 4-47 mode using 3-151 objects 4-53 Tree Mode 3-19 wavelet two-dimensional 3-44 wavelet packet notation 4-40 subtrees 4-46 true compression for images 5-55 twin-scale relations definition 3-37 Index-5 Index U upsampling two-dimensional IDWT 3-44 V vanishing moments wavelet families 1-22 variance adaptive thresholding 5-14 view axes A-10 W wavelet families adding new 1-26 criteria 1-2 full name 1-27 properties (Part 1) 1-22 properties (Part 2) 1-24 regularity advantage 1-2 short name 1-27 support 1-2 symmetry 1-2 vanishing moments 1-2 Wavelet Packet 1-D Compression window 4-11 Wavelet Packet 1-D menu item 4-7 Wavelet Packet 1-D tool starting 4-7 Wavelet Packet 2-D Compression window 4-18 wavelet packets and wavelet analysis differences 4-31 atoms 4-38 bases 4-40 best level decomposition 4-47 best tree 4-10 besttree function 4-4 building 4-35 compression 4-51 Index-6 de-noising ideas 4-51 using SURE 4-14 decomposition 4-50 decomposition tree subtrees 4-46 definition 4-30 finding best level 4-4 frequency order 4-39 from wavelets to 4-30 merge 4-41 natural order 4-39 objects 4-53 organization 4-40 selecting threshold for compression 4-11 split 4-41 tree notation 4-40 wavelets adding new 1-26 biorthogonal definition 1-10 coiflets definition 1-9 complex frequency B-spline 1-19 complex Gaussian 1-18 complex Morlet 1-19 complex Shannon 1-20 Daubechies definition 1-7 defining order 1-28 determining type 1-27 Gaussian 1-14 Haar definition 1-8 Mexican hat definition 1-15 Meyer definition 1-12 Morlet Index definition 1-16 one-dimensional capabilities objects 3-49 table 3-2 reverse biorthogonal 1-12 symlets definition 1-8 tree two-dimensional 3-44 two-dimensional capabilities objects 3-50 table 3-137 vanishing moments number of 1-2 wavelets.asc file 1-40 wavelets.inf file 1-40 wavelets.prv file 1-40 wthresh command 3-77 Z zero-padding signal extension 3-52 zoom 2-15 Index-7
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