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Direct-current-based image reconstruction
versus direct-current included or excluded
frequency-domain reconstruction
in diffuse optical tomography
Guan Xu,1 Daqing Piao,1,* Charles F. Bunting,1 and Hamid Dehghani2
1

School of Electrical and Computer Engineering, Oklahoma State University, Stillwater, Oklahoma, USA 74078
2

University of Birmingham, Birmingham B15 2TT, UK
*Corresponding author: daqing.piao@okstate.edu

Received 25 September 2009; revised 16 March 2010; accepted 29 March 2010;
posted 8 April 2010 (Doc. ID 117721); published 25 May 2010

We study the level of image artifacts in optical tomography associated with measurement uncertainty
under three reconstruction configurations, namely, by using only direct-current (DC), DC-excluded frequency-domain, and DC-included frequency-domain data. Analytic and synthetic studies demonstrate
that, at the same level of measurement uncertainty typical to optical tomography, the ratio of the standard deviation of μa over μa reconstructed by DC only is at least 1.4 times lower than that by frequencydomain methods. The ratio of standard deviations of D (or μ0s ) over D (or μ0s ) reconstructed by DC only are
slightly lower than those by frequency-domain methods. Frequency-domain reconstruction including DC
generally outperforms that excluding DC, but as the amount of measurements increases, the difference
between the two diminishes. Under the condition of a priori structural information, the performances of
three reconstruction configurations are seemingly equivalent. © 2010 Optical Society of America
OCIS codes: 170.3880, 170.3010, 170.6960, 170.5270.

1. Introduction

Diffuse optical tomography (DOT) based on measurement of near-infrared (NIR) light diffused through
thick biological tissue aims to quantify the heterogeneities of NIR-absorbing chromophors and scattering
particles [1]. There are generally three categories
of DOT measurements: (1) continuous wave (CW),
wherein only steady-state or direct-current (DC)
detection is carried out, (2) time domain, wherein
the attenuation and pulse-width broadening of the
excitation light are the measurands [2–5], and (3)
frequency domain, which is mathematically the Fourier-transform equivalent of the time-domain method
[6–17] but is considerably less complicated in instrumentation. Frequency-domain detection ideally
0003-6935/10/163059-12$15.00/0
© 2010 Optical Society of America

renders three types of information: the DC attenuation, the modulation intensity change (AC), and the
modulation phase shift (PHS). Some frequencydomain DOT works, however, have utilized AC and
PHS [6–12,14,15], rather than the complete measurands of DC, AC, and PHS. Excluding the DC in frequency-domain DOT reconstruction implied that
the DC information was considered unlikely to improve the outcome of reconstruction when the AC
and PHS are available. Such consideration could have
been prompted if the DC information had been redundant in frequency-domain reconstruction, but indeed
it has not been either justified or negated.
On the other hand, many works in DOT have relied
on only the DC measurements [18–26]. Although
lacking phase information will certainly reduce the
accuracy or confidence of quantitative reconstruction,
almost all these studies have demonstrated that the
absorption and reduced scattering characteristics can
1 June 2010 / Vol. 49, No. 16 / APPLIED OPTICS

3059

be separately and absolutely reconstructed by use of
DC information only. But all these works lack a direct
comparison of the outcome of DC-based reconstruction with that of frequency-domain reconstruction,
which is needed to provide a basis to assess the
compromise for reconstruction based solely upon
DC information. Out of these DC-based DOT reconstructions, there also exists a common but not widely
stated feature in the images—the recovered background is usually more homogeneous than the general level of background artifacts seen in images
reconstructed in the frequency domain. Fewer image
artifacts in the background may be beneficial for identifying the target of interest over a relatively heterogeneous background, but what contributes to fewer
image artifacts in the background has not been well
understood.
This work studies the level of artifacts associated
with measurement uncertainties in three modes of
image reconstruction, namely DC, AC þ PHS, and
DC þ AC þ PHS. The studies are conducted both
analytically and by synthetic measurements, to address why the DC-based reconstruction results in
fewer background artifacts and to demonstrate that
including DC information in frequency domain generally improves the reconstruction outcome. Clearly,
the analysis of this study shall be based upon the propagation of measurement noises to the image. Contributing to the image artifacts are a number of noise
sources, among which is an error due to coupling loss,
as studied by Schweiger et al. [27]. That study treated coupling errors as coupling coefficients appended
to the solution space, and demonstrated reconstruction of frequency-domain data contaminated with
synthetic coupling errors. Similar studies are necessary to understanding reconstruction with contaminated DC data.
The level of artifacts is a critical indicator of the
capability of reliably recovering the optical heterogeneity. Ntziachristos et al. [28] demonstrated that the
reconstruction of localized lesions deteriorated as a
function of background heterogeneity. They also
found that increasing the dataset size, specifically
the number of detectors used, improves the reconstruction of the lesion structure, but does not remove
the artifacts. Those results, performed on frequencydomain synthetic and experimental data, indicate
that certain artifacts are inherent to image formation and, thereby, cannot be removed completely. The
cause of such artifacts must also be inherent to DCbased reconstruction, wherein the outcome relative
to frequency-domain reconstruction is unknown.
The analytic approach of this study is based primarily upon a method introduced by Fantini et al.
[29] to model the accuracies or, equivalently, the errors associated with a two-distance measurement
technique for quantifying the optical properties of
a bulk homogeneous medium. Reconstructing optical
properties in a homogeneous medium is essentially a
process of fitting the slopes of measurements with
respect to different source–detector distances, for
3060

APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

which Fantini et al. introduced their models of the
“relative error” of absorption and reduced scattering
coefficients using the intensity exponential factor,
the AC exponential factor, and the phase factor between the measurements made at two different
source–detector distances. The tomography of optical
heterogeneity relies on multiple measurements
among spatially resolved sources and detectors, and
image reconstruction is a process of optimizing the
local optical properties to minimize the difference
of model prediction for these source–detector pairs
with respect to the measured values. The accuracy
of reconstruction is thereby dependent upon the capability of distinguishing the signal variations for a
single source–detector pair due to all types of measurement fluctuations, as well as local changes of
tissue optical properties, such variations among different source–detector pairs, and mapping such variations to the image space. Hence, the “relative error”
initially discussed in [29] equally applies to tomography of optical heterogeneity, because the “relative error” of measurement determines the upper limit of
reconstruction accuracy; in other words, it sets the
“parameter-recovery-uncertainty level” (PRUL) in
the tomography images.
This study analyzes the PRULs of the absorption
coefficient, the reduced scattering coefficient, and the
diffusion coefficient, for the measurement conditions
of DC, AC + PHS, and DC + AC + PHS and examines
their representations as image artifacts in synthetic
models. Much of the analytic approach of this study
is based upon the method established in [29]; however, there are substantial differences in the measurement configurations investigated, and also, in
this novel study, the analytic results partially suggested by [29] are quantitatively evaluated to compare the PRULs among these configurations. It is
also noted that [29] considered the measurement
configurations of DC þ AC, AC þ PHS, and DCþ
PHS. When frequency-domain (FD) information is
available, it is straightforward to apply AC þ PHS,
as employed by many works [6–12,14,15], to image
reconstruction. The utilization of DC þ AC and DC þ
PHS are mathematically valid; however, those configurations have seldom been used for image reconstruction. This study investigates the level of
artifacts in the DC, AC þ PHS, and DC þ AC þ PHS
configurations, as they are the most likely implemented approaches toward image reconstruction. Therefore, among the results previously stated in [29], only
those related to AC þ PHS have been included in this
study when appropriate. The AC þ PHS result for
the absorption coefficient in [29] is cited directly,
but the AC þ PHS result in [29] for reduced scattering is revised to a more generalized form that is consistent with the result for the absorption coefficient.
Table 1 in Subsection 2.A is introduced to make clear
these distinctions. This study also investigates reconstruction of the diffusion coefficient, because,
not only are the absorption and reduced scattering
coefficients coupled, but also generally the diffusion

Table 1.

Comparison of the Analytic Derivations in This Work with That in [29]

Measurements

Δμa σ μa
μa ð μa Þ
Δμ0s σ μ0s
μ0s ð μ0s Þ
ΔD σ D
D ðDÞ
a

DC

DC þ AC

AC þ PHS

DC þ PHS

DC þ AC þ PHS

This study

[29]

[29]

[29]

This study

This study

[29]

This study

[29]

This study

This study
This study

The derivation was revised to a more generalized form.

coefficient is involved in the reconstruction process
prior to formulating the reduced scattering coefficient. The diffusion coefficient image may provide
new insights to the study even though its artifacts
are expected to be close to those seen in reduced scattering image.
The rest of the paper is organized in the following
sections. Section 2 analyses the PRUL for three categories: (1) D.C. only, (2) AC þ PHS, and (3) DCþ
AC þ PHS. Tissue and measurement parameters
typical to optical tomography applications are implemented to evaluate quantitatively the PRULs expected in the images. Section 3 uses synthetic data
to examine the uncertainty of the parameters recovered for homogeneous medium, single inclusion with
different types of optical contrast, and multiple inclusions with specific optical contrasts. These synthetic models are also evaluated selectively for the
condition of having spatial a priori information in
the image reconstruction. Section 4 discusses the implications of the results.
2. Theory

The reconstruction accuracy of optical tomography is
determined by many factors, including the accuracy
of the forward model, the determinacy of inverse formulation, and the characteristics of instrument noise
[30]. An analytic approach has been introduced in
[29] to demonstrate that the uncertainty (or error)
in the measurement maps to the uncertainty of recovering the assembled optical properties of bulk
tissue. The same uncertainty (or error) of the measurement, when involved in tomographic reconstruction to recover spatially resolved tissue optical
properties, will translate to spatially varying artifacts that reduce the contrast-to-noise ratio (CNR)
of the target of interest. This effect may seem obvious; however, the extent of it is not well understood.
This work closes this gap of knowledge in three conditions of DOT measurements, namely DC, ACþ
PHS, and complete frequency-domain information
by DC þ AC þ PHS.
A.

[29]

a

Parameter-Recovery-Uncertainty Level

The variation of the recovered optical properties is
modeled as PRUL, which for AC þ PHS has been
derived in [29] in terms of the attenuation of the
AC amplitude and phase shift versus a change of
source–detector distances. We implement the approach in [29], but extend it to DC-only and DC þ
AC þ PHS configurations, and apply it to diffusion

coefficients in addition to absorption and reduced
scattering coefficients.
The frequency-domain measurement of photon density consists of a steady state and time-varying components as U FD ð~
r; ωÞ ¼ U DC ð~
rÞ þ U AC ð~
r; ωÞ, where~
r is
the position vector and ω is the angular modulation
frequency of the light source. The U FD ð~
r; ωÞ satisfies
the photon diffusion equation of




μa ð~
rÞ
iω
−
þ
U FD ð~
r; ωÞ þ ∇2 U FD ð~
r; ωÞ
Dð~
rÞ vDð~
rÞ
¼−

Sð~
r; ωÞ
;
Dð~
rÞ

ð1Þ

where v is the speed of light in the medium, μa is the
absorption coefficient, D ¼ ½3ðμa þ μ0s Þ
−1 is the diffusion coefficient, μ0s is the reduced scattering coefficient,
and the source term Sð~
r; ωÞ has a DC component
SDC ð~
rÞ and a time-varying component SAC ð~
r; ωÞ. For
a homogeneous infinite medium with a detector at ~
r
and a source at~
r0 , thereby a source–detector distance
of d ¼ j~
r0 −~
rj, we have
U FD ðr; ωÞ ¼ U DC ðrÞ þ jU AC ðr; ωÞj expðiΦAC Þ
¼

SDC ðr0 Þ
expð−kDC dÞ
4πDd
S ðr0 ; ωÞ
expð−kAC dÞ · expðikPHS dÞ;
þ AC
4πDd
ð2Þ

where
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
u 
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffi

u
2
μa
ω
t μa
1þ 2 2þ1 ;
kDC ¼
;
kAC ¼
D
2D
v μa
ð3Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
u 
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

uμ
2
ω
t a
1þ 2 2−1 :
kPHS ¼
2D
v μa
It is noted that kAC > kDC and kAC is correlated with,
but not linearly dependent upon, kDC . The attenuation
of the DC component of the photon density is thus not
equal to or linearly dependent upon that of the AC
component, which is an indication that the DC information would not be a duplication of any of AC or PHS.
Denoting d2 > d1 and ρ ¼ jd1 − d2 j as the difference of source–detector distance between two mea1 June 2010 / Vol. 49, No. 16 / APPLIED OPTICS

3061

surements corresponding to the same source, one has
[29] (reproduced here for convenience)

and a PRUL of [29]

ð4Þ


 2
δ
¼D·
:
ρ

ð5Þ

References [31,32] suggest that, for steady-state surface measurements, μa and D collectively determine
the diffuse reflectance, denoted as R∞ , by the relationship ½μa · D
 ¼ KðR∞ Þ. It is noted that the diffuse
reflectance is not U DC ð~
rÞ, which implies treating
KðR∞ Þ as not significantly dependent upon U DC ð~
rÞ,
thereby Eq. (5) may be converted to
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
KðR∞ Þ
· δ;
¼
ρ

ð6Þ

and estimating the PRUL of μa for DC by



σ μa 
1 ∂μa
σ
σδ ¼ δ
¼

μa DC μa ∂δ
δ


 2 1=2
σδ
:
δ2

or

μa jACþPHS ¼

ð7Þ

σμ
¼ η · ðξÞ1=2 ;
μ

(7)

DC

(9)

AC þ PHS

(11)

DC þ AC þ PHS

3062

Expression


α2 þϕ2
α2 −ϕ2



ð12Þ

which contains apmultiplication
factor η and a
ffiffiffi
square-root term ξ. The relative levels of these
σ2

σ2

PRULs become comparable as ϕϕ2 , ασα2 , and δ2δ are practically the same [29]. It is indicated in Table 2 that
the PRUL of μa will be the lowest in DC-based reconstruction, but whether the PRUL of μa is lower in
AC þ PHS or in DC þ AC þ PHS depends upon the
difference in α and ϕ.
Because the image reconstruction recovers D to
formulate μ0s , it is imperative to analyze the PRUL
of D. For the case of DC, similar to the derivation
for μa, we have
2


 
ρ
;
δ

ð13Þ

ð8Þ

Comparison on PRUL of μa ðσ μa =μa Þ

pffiffiffi
ξ

η
Condition

ð10Þ

The PRULs in Eqs. (7), (9), and (11) all have the
shape of



ω ϕ α
−
;
2v α ϕ

ω δ2
·
v 2αϕ


σ μa 
μa DCþACþPHS




 




1
∂μa 2 2
∂μa 2 2
∂μa 2 2 1=2
¼
σδ þ
σα þ
σϕ
∂δ
∂α
∂ϕ
μa

 2

σ
σ 2 σ 2ϕ 1=2
¼ 4 2δ þ α2 þ 2
:
ð11Þ
δ
α
ϕ

DjDC ¼ KðR∞ Þ ·

Table 2.

Eq.

ð9Þ

and, accordingly, a PRUL of

We have, for AC þ PHS [29],



1
μa

μa jDCþACþPHS ¼ −

Table 1 lists the PRUL of five different measurement
configurations, among which three were investigated
in [29]. As stated previously, the configuration of
DC þ AC and DC þ PHS were seldom used for image
reconstruction, therefore, only the AC þ PHS results
of [29] are cited for this comparative study.
In CW measurement, we have

μa jDC

¼

For DC þ AC þ PHS measurement, we have

ϕ ¼ Φðd2 Þ − Φðd1 Þ ¼ ρ · kPHS
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
u 
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

uμ
2
ω
t a
1þ 2 2−1 :
¼ρ·
2D
v μa

μa jDC




 


∂μa 2 2
∂μa 2 2 1=2
σα þ
σϕ
∂α
∂ϕ



α2 þ ϕ2 σ 2α σ 2ϕ 1=2
¼ 2
þ
:
α − ϕ2 α2 ϕ2

σ μa
j
μa ACþPHS

rffiffiffiffiffi



d2 U DC ðd2 Þ
μa
δ ¼ ln
¼ −ρ · kDC ¼ −ρ ·
;
d1 U DC ðd1 Þ
D



d U ðd Þ
α ¼ ln 2 AC 2 ¼ −ρ · kAC
d1 U AC ðd1 Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
u 
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

uμ
2
ω
t a
1þ 2 2 þ1 ;
¼ −ρ ·
2D
v μa

Value
1
>1
1

APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

Expression
 2 1=2


σδ
δ2

σ 2ϕ
ϕ2

1=2

σα
þ
α2
 2

2
σ 2 1=2
σδ
4 δ2 þ ασα2 þ ϕϕ2
2

Normalized value
1

Normalized η ·
1

1.41

>1:41

2.45

2.45

pffiffiffi
ξ

Table 3.

Eq.
(14)
(16)

Comparison on PRUL of D

Condition

Expression
 2 1=2

DC



AC þ PHS & DC
þAC þ PHS

σ 2α
α2

σδ
δ2

þ

σ 2ϕ
ϕ2

1=2

Normalized
Value
1
∼1:41

ð14Þ

For AC þ PHS and DC þ AC þ PHS, the expressions
are the same:
ωρ2 1
·
:
¼−
2v αϕ

ð15Þ

Therefore,
σD
σ
jACþPHS ¼ D jDCþACþPHS
D
D

 2

 2 1=2
1
∂D
∂D
2
¼
σα þ
σ 2ϕ
D
∂α
∂ϕ

 2

σ α σ 2ϕ 1=2
¼
þ
:
α2 ϕ2

ð16Þ


 0 2 1=2

 0 2
σ μ0s
1
∂μs
∂μs
2
σD þ
σ 2μa
0 ¼ 0
μs
∂D
∂μa
μs
−1 
 2 
 2

 2 1=2

σ μa
1
1
σD
− μa
¼
·
þ ðμa Þ2
;
D
μa
3D
3D
ð17Þ
so the PRUL of μ0s for DC is

−1 
 2 
 2 


 2 1=2
σ μ0s 
1
1
σδ
σδ
2
þ
μ
−
μ
¼
·
·
:
a
a
μ0s DC
3D
3D
δ2
δ2
ð18Þ
For AC þ PHS, it is [29,33]
Table 4.

Eq.

Condition
DC

(19)

AC þ PHS

(20)

DC þ AC þ PHS

−1

σ μ0s
1
− μa
j
¼
μ0s DCþACþPhs
3D


 2 
 2
1
σ α σ 2ϕ
þ μ2a
·
þ
3D
α2 ϕ 2

 2

σ δ σ 2α σ 2ϕ 1=2
· 4 2þ 2þ 2
:
δ
α
ϕ

ð20Þ

Based on the estimation leading to Table 2, the
PRULs in Eqs. (19) and (20) can be normalized with
respect to Eq. (18). The results are given in Table 4.
Again, the PRUL of μ0s will be the lowest for DC.
Whether the PRUL of μ0s is lower in AC þ PHS or
in DC þ AC þ PHS depends also upon the difference
in α and ϕ as for the PRUL of μa, but, because of the
dominance of 1=3D over μa, the difference between
AC þ PHS and DC þ AC þ PHS will be less than that
observed for PRUL of μa in Table 2.
B. Summary of the PRUL Analyses

The PRULs of D in Eqs. (14) and (16) are compared in
Table 3. Apparently, when AC and phase are employed, the DC component is redundant for the recovery of D.
The PRUL of μ0s is derived by

(18)

ð19Þ

and for DC þ AC þ PHS, it is




 2 1=2

σ D 
1 ∂D
σδ
σ
σ δ ¼ or 2δ
¼
:

D DC D ∂δ
δ
δ

DjACþPHS ¼ DjDCþACþPHS


−1 
 2 
 2


σ μ0s 
1
1
σ α σ 2ϕ
− μa
¼
·
þ
μ0s ACþPhs
3D
3D
α2 ϕ 2
 
 2


 2
α þ ϕ2 2
σ α σ 2ϕ 1=2
·
þ
;
þ μ2a ·
α2 − ϕ2
α2 ϕ 2

The DC-only reconstruction seems to give the least
level of relative uncertainty of the parameter in
the reconstruction. The AC þ PHS configuration
seems to be equivalent to DC þ AC þ PHS in the level of PRULs of reduced scattering and diffusion coefficient, but it is unclear for the absorption coefficient.
These analyses have been conducted for an infinite
homogeneous medium, but the results will be readily
translatable to a medium with boundaries and with
inclusions.
3. Synthetic Studies

Simulations are carried out to study the practical
issues of PRUL, such as background noise, the accuracy of optical property recovery, and the interparameter cross coupling, of the three measurements
setups.
A. Synthetic Model

The forward model is carried out by the finiteelement method (FEM) solution of Eq. (1) using the
Robin-type boundary condition [34]:

Comparison on PRUL of μ0s

Expression
 2 i1=2
h 2  2 
σδ
σ
1
þ μ2a · δ2δ
3D
δ2
h 2  2 σ 2 
 2 2 2  2 σ2 i1=2
σα
1
þ ϕϕ2 þ μ2a · αα2þϕ
· σαα2 þ ϕϕ2
3D
α2
−ϕ2
h 2  2 σ2 
i1=2
 2
2
σ2
σ
σα
1
þ ϕϕ2 þ μ2a · 4 δ2δ þ σαα2 þ ϕϕ2
3D
α2

Normalized as
1
>1:41
>1:41

1 June 2010 / Vol. 49, No. 16 / APPLIED OPTICS

3063

performed to facilitate stable convergence [35] and is
set at 0.5 in this study when included. For pixelwise
reconstructions using NIRFAST [36,37], α is set to 1.
B. Simulation Results

Synthetic data are generated for a homogeneous
medium, a medium with a single inclusion, and a
medium with multiple inclusions with mixed types
of optical heterogeneities.
1.

Fig. 1. (Color online) Imaging geometry for a homogeneous
medium.

Uð~
r0 ; ωÞ − 2DA^
n0 · ∇Uð~
r0 ; ωÞ ¼ 0;

ð21Þ

where A is related to refractive index mismatch and
^ 0 is an outgoing normal vector. The Jacobian is
n
structured to the form of
2

2

3

DC
J ¼ 4 AC 5 ¼
PHS

3

∂ ln U DC
6 ∂ ln∂μjUa j
AC
6
4 ∂μa
∂ΦAC
∂μa

∂ ln U DC
∂D
7
∂ ln jU AC j 7
∂D
5;
∂ΦAC
∂D

ð22Þ

where the indices of each block of the Jacobian could
be node based for pixelwise reconstruction or region
based for prior-guided regionwise reconstruction.
Utilizing only the first row leads to CW, utilizing
the second and third rows renders AC þ PHS, and
utilizing all three rows gives DC þ AC þ PHS.
The inverse solver implements the Levernberg–
Marquardt algorithm as
xkþ1 ¼ xk þ α · ½J T ðxk ÞJðxk Þ þ λI
−1 J T ðxk ÞΔvðxk Þ;
ð23Þ
where x is the array of unknown parameters, Δν is
the forward projection error and λ is a penalty or regularization term. The value of λ is initially set as 100,
and is reduced to its fourth root with each continued
iteration. The damping factor, α, in the range of (0, 1),
is introduced when only regionwise reconstruction is
Table 5.

DC
AC þ PHS
DC þ AC þ PHS

0.01
0.01
0.01

A cylinder-applicator geometry [38] of 60 mm in
height and 86 mm in diameter with 16 optodes is
adopted, like the one shown in Fig. 1. The optodes are
turned on sequentially for the measurements being
taken by all other optodes, generating a total of
240 measurements for each dataset.
The volume is discretized into a FEM mesh of
12,695 nodes for forward computation, while a smaller FEM mesh of 600 nodes is used in the reconstruction. Because this synthetic study specifically
investigates the level of artifacts reconstructed to
the same level of recovered parameters in an otherwise homogeneous medium, the same optical properties of μa ¼ 0:01 mm−1 and μ0s ¼ 0:01 mm−1 are used
for both forward computation and as the initial
values of the inverse routine, with 1% noise added
to the forward simulation data to maintain the same
measurement error. In addition, all controlling
parameters of the inverse model are maintained
the same for DC, AC þ PHS, and DC þ AC þ PHS
configurations.
Table 5 demonstrates that the variations recovered to the parameters of a homogeneous medium
are lowest in DC, as expected from the analytic analysis. The DC þ AC þ PHS slightly outperforms AC þ
PHS in μa recovery, but AC þ PHS slightly outperforms DC þ AC þ PHS in μ0s =D recovery.
The normalized numbers (1:45–1:64) for μ0s =D
recovery are considerably close to those in the analytical derivation—with the same average optical properties, the background standard deviation of the
images reconstructed by FD system measurements
is at least 1.41 times larger than those reconstructed
by the CW system. However, in μa reconstruction, the
variations in FD configurations are about twice those
predicted in Table 2. It is noted that the analytic
results in this study are based upon perturbation
analysis. It is well known that DOT is a nonlinear
process, wherein the absorption perturbation is more

Mean Value and Standard Deviation Reconstructed for Homogeneous Medium`

σ μa ðmm−1 Þ
μ

a

PRULs in a Homogeneous Medium

Abs.
10−6

0:69 ×
3:13 × 10−6
2:98 × 10−6

σ μ0s ðmm−1 Þ

Norm.
1
4.50
4.29

μ

0s
1.00
1.00
1.00

Abs.
10−4

0:80 ×
1:18 × 10−4
1:31 × 10−4

σ D ðmmÞ

Norm.
1
1.47
1.64



D
0.33
0.33
0.33

Abs.
10−3

2:64 ×
3:83 × 10−3
4:24 × 10−3

Norm.
1
1.45
1.60

a
“Abs.” denotes the absolute value of the standard deviation. “Norm.” denotes the standard deviation normalized with respect to the
standard deviation of DC. The same notations apply to Tables 6 and 8.

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APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

Fig. 2. (Color online) Contrast-to-noise-ratio (CNR) with respect to the measurement noise levels. (a), (b), (c) μa =μ0s =D distribution in the
z ¼ 0 plane of forward model; (d) μa CNR comparison; (e) μ0s CNR comparison; (f) D CNR comparison.

pronounced than scattering perturbation. In this
specific model of homogeneous medium, the signal
perturbation is evenly distributed to the entire volume of the homogeneous medium instead of mostly
confined to smaller lesions with higher optical property contrast, as in the later examinations. Therefore, the perturbations from AC and PHS could
have been coupled to and nonlinearly amplified as
the variation of absorptions.
2. Contrast-to-Noise Ratio Analysis for Single
Target
The results in Subsection 3.B.1 indicate that, for 1%
noise in the measurement of homogeneous medium,
DC-only reconstruction clearly maintains a lower
artifact level compared to DC þ AC þ PHS and
AC þ PHS. This study examines the contrast of a target inclusion in an otherwise homogeneous medium

at different measurement noise levels when reconstructed by DC, AC þ PHS, and DC þ AC þ PHS configurations. The synthetic model is similar to that in
Subsection 3.B.1, but with a spherical heterogeneity
added at (x ¼ 0 mm. y ¼ −20 mm, z ¼ 0 mm), with
μa ¼ 0:025 mm−1 and μ0s ¼ 1:75 mm−1 . The reconstruction basis of 2760 nodes is larger than the
one used for Subsection 3.B.1. Varying noise levels,
of 0% to 10%, are integrated into the forward data
to examine the CNR of the target (CNR ¼ ½max
ðtarget-region-valueÞ − meanðbackground-valueÞ
=
background-standard-deviation) with respect to
the background artifacts. The background deviation
is calculated by excluding the areas within a distance
of 1.5 times the target radius away from its center
[39]. The calculated CNRs are given in Fig. 2 for
the three types of target contrast. It is observed in
Fig. 2 that the CNR levels of μa and D look similar
when compared to that of μ0s, which supports the

Fig. 3. (Color online) Simulation studies for reconstructing multiple targets in a three-dimensional cylindrical geometry with the optodes
and targets located on one plane.
1 June 2010 / Vol. 49, No. 16 / APPLIED OPTICS

3065

Table 6.

Standard Deviation of Background Optical Properties in Fig. 3

σ μa ðmm−1 Þ
DC
AC þ PHS
DC þ AC þ PHS

σ μ0s ðmm−1 Þ
Norm.

Abs.

Norm.

Abs.

Norm.

1:92 × 10−4
3:63 × 10−4
3:45 × 10−4

1
1.89
1.79

2:46 × 10−2
2:88 × 10−2
2:49 × 10−2

1
1.17
1.01

7:25 × 10−3
9:01 × 10−3
7:75 × 10−3

1
1.24
1.07

assumptions made for deriving PRULs of μa and D in
Eqs. (7) and (14). In Fig. 2, the CNR levels of μa are
found to be lower than that of μ0s, which may be due to
underestimation of μa and overestimation of μ0s in
such a pixelwise image reconstruction [24]. Despite
this, several features can be observed in Fig. 2. (1)
At a zero noise level, the three methods are comparable in the CNR. (2) When the noise becomes higher,
the D.C. clearly outperforms the other two in CNR,
while DC þ AC þ PHS slightly outperforms ACþ
PHS. (3) At a 10% noise level, the CNRs of all methods are similar for μ0s and D recovery, but DC still
outperforms the other two in μa reconstruction.
3.

Multiple Target Case

The geometry for having multiple inclusions is shown
in Fig. 3, where three spherical targets with radii of
7:5 mm are located in the longitudinal middle plane
(z ¼ 0) of the cylindrical imaging volume and are all
20 mm away from the center of the circular cross section, ensuring the same spatial sensitivity at their positions. Target 1, at the upper left (x ¼ −14:14 mm,
y ¼ 14:14 mm, z ¼ 0 mm), has only absorption contrast (μa ¼ 0:025 mm−1 , μ0s ¼ 1 mm−1 ), target 2, at upper right (x ¼ 14:14 mm, y ¼ 14:14 mm, z ¼ 0 mm),
has only scattering contrast (μa ¼ 0:01 mm−1 , μ0s ¼
1:75 mm−1 ), and target 3, at lower side (x ¼ 0 mm,
y ¼ −20 mm, z ¼ 0 mm), has contrasts of both absorpTable 7.

tion and reduced scattering (μa ¼ 0:025 mm−1 ,
μ0s ¼ 1:75 mm−1 ). The dashed line in the figure marks
the position of the target when it presents no contrast
in that category. Table 6 lists the deviation of the
background optical property in the reconstructed
images. Standard deviation values in Table 6 are normalized along each column versus those of DC-only
reconstruction.
For background homogeneity, comparison in
Table 6 indicates that DC only demonstrates the lowest artifact level in the image background, while the
background artifact levels of DC þ AC þ PHS and
AC þ PHS are approximately 1 to 2 times higher.
Although the numerical simulative result does not
exactly match the values in Tables 2–4, it qualitatively agrees with the analytical derivations. The
analytical derivations given in Tables 2–4 indicate
that DC þ AC þ PHS and AC þ PHS produce similar
background homogeneities, but the simulation results all indicated a slightly lower background artifact level in DC þ AC þ PHS reconstruction. For
target accuracy, the reconstructed images in Fig. 3
and the data comparison in Table 7 are seen with
DC þ AC þ PHS as superior to AC þ PHS, which,
along with the comparison on the background
homogeneity, indicates that including DC generally
improves the FD reconstruction. In terms of interparameter cross coupling, DC has more coupling
than FD, which is well known. The cross coupling

Comparison of the Accuracy of Recovered Optical Properties in Fig. 3

μa1 ðmm−1 Þ
Value
Set
DC
AC þ PHS
DC þ AC þ PHS

0.025
0.0125
0.0146
0.0149

μ0s1 ðmm−1 Þ
Error

Value

−50:16%
−41:62%
−40:30%

1
1.398
1.293
1.201

μa2 ðmm−1 Þ

Set
DC
AC þ PHS
DC þ AC þ PHS

3066

D1 ðmmÞ
Error

Value

Error

39.84%
29.27%
20.06%

0.325
0.236
0.255
0.274

−27:35%
−21:59%
−15:67%

μ0s2 ðmm−1 Þ

D2 ðmmÞ

Value

Error

Value

Error

Value

Error

0.01
0.0114
0.0107
0.0104

13.68%
6.95%
3.81%

1.75
1.238
1.250
1.375

−29:25%
−28:56%
−21:45%

1.75
1.639
1.619
1.635

−6:34%
−7:47%
−6:55%

μa3 ðmm−1 Þ

Set
DC
AC þ PHS
DC þ AC þ PHS

σ D ðmmÞ

Abs.

μ0s3 ðmm−1 Þ

D3 ðmmÞ

Value

Error

Value

Error

Value

0.025
0.0141
0.0139
0.0137

−43:48%
−44:31%
−45:24%

1.75
1.639
1.619
1.635

−6:34%
−7:47%
−6:55%

0.188
0.2012
0.204
0.202

APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

Error
7.37%
8.70%
7.64%

Table 8.

Standard Deviation of Background Optical Properties in Fig. 4

σ μa ðmm−1 Þ
DC
AC þ PHS
DC þ AC þ PHS

σ μ0s ðmm−1 Þ

σ D ðmmÞ

Abs.

Norm.

Abs.

Norm.

Abs.

Norm.

2:26 × 10−4
4:07 × 10−4
3:95 × 10−4

1
1.80
1.75

3:00 × 10−2
3:26 × 10−2
3:18 × 10−2

1
1.09
1.06

8:47 × 10−3
9:78 × 10−3
9:51 × 10−3

1
1.15
1.12

in DC þ AC þ PHS is slightly less severe than that
in AC þ PHS.
A similar study is conducted for the same targets
in a three-ring setup [38] in Fig. 4, which has three
identical rings of optodes at the azimuthal planes of
z ¼ −10 mm, z ¼ 0 mm, and z ¼ 10 mm. Each set of
data contains a total of 2256 measurements by
turning on one source and detecting at all other optodes. The key values are compared in Tables 8 and 9.
Most features of the three aspects discussed for the
single-ring case can be reconfirmed, except that the
target contours recovered by FD reconstructions are
more accurately defined, but, nonetheless, the difference between DC þ AC þ PHS and AC þ PHS is
insignificant.
Prior-guided region-based reconstructions are also
performed on both of the imaging geometries of
Figs. 3 and 4 to examine if including accurate a priori
structural information of the target affects the outcome of the three reconstruction configurations. As
is shown in Figs. 5 and 6, with forward models the
same as those in Figs. 3 and 4, the inverse model
has integrated spatial a priori information by assuming a homogeneous target of the accurate size in a
homogeneous background. Results of both cases indicate that, with the structural a priori information,
the performances of the three configurations are essentially equivalent.
4. Discussions

Using only the DC information to simultaneously recover the absorption and diffusion (or the reduced
scattering) distributions has been controversial.

The nonuniqueness that may be inherent to DC-only
measurements was described in a seminal study [40].
However, despite the negative predictions in [40]
that there could be an infinite number of diffusion
and absorption pairs leading to the same surface
measurements, Harrach [41] proved that, at most,
one of them consists of a piecewise constant diffusion
and piecewise analytic absorption, and if the true
medium has these properties, as in virtually any
practical condition, a reconstruction algorithm favoring these properties will pick the right combination
of profiles. Harrach’s study theoretically justified the
experiences in many works that the absorption and
scattering distributions have been separately and
uniquely recovered by surface measurement of DC
only [18–26].
The primary aim of this work is to understand the
expectation for DC-based reconstruction in a more
systematic approach, thereby establishing a certain
level of confidence for the recovered information
when only DC information can be relied upon. This
work, conveyed by a side-by-side comparison of
the reconstructions based on DC, AC þ PHS, and
DC þ AC þ PHS, does provide direct evidence that
DC-based reconstruction is much less accurate in recovering the absolute optical properties of the target
of interest when no additional spatial information is
available to confine the reconstruction, as having
been universally recognized by the DOT community.
However, apart from these well-expected shortcomings, it seems that DC-based reconstruction may not
be completely unfavorable. This study generalized
the analytical approach initially proposed in [29]

Fig. 4. (Color online) Simulation studies for reconstructing multiple targets in a three-dimensional cylindrical geometry with the optodes
located on three different planes and targets located on the middle plane.
1 June 2010 / Vol. 49, No. 16 / APPLIED OPTICS

3067

Table 9.

Comparison of the Accuracy of Recovered Optical Properties in Fig. 4

μa1 ðmm−1 Þ

Set
DC
AC þ PHS
DC þ AC þ PHS

μ0s1 ðmm−1 Þ
Error

Value

Error

Value

Error

0.025
0.0133
0.0169
0.0171

−46:93%
−32:39%
−31:42%

1
1.528
1.288
1.292

52.81%
28.77%
29.21%

0.325
0.216
0.256
0.255

−33:49%
−21:43%
−21:70%

μa2 ðmm−1 Þ

Set
DC
AC þ PHS
DC þ AC þ PHS

μ0s2 ðmm−1 Þ

D2 ðmmÞ

Value

Error

Value

Error

Value

Error

0.01
0.0117
0.0104
0.0103

16.95%
3.73%
3.09%

1.75
1.319
1.427
1.441

−24:63%
−18:45%
−17:66%

0.189
0.251
0.232
0.230

32.26%
22.46%
21.31%

μa3 ðmm−1 Þ

Set
DC
AC þ PHS
DC þ AC þ PHS

D1 ðmmÞ

Value

μ0s3 ðmm−1 Þ

D3 ðmmÞ

Value

Error

Value

Error

Value

Error

0.025
0.0156
0.0163
0.0163

−37:57%
−35:02%
−34:77%

1.75
1.847
1.731
1.726

5.57%
−1:10%
−1:38%

0.188
0.178
0.191
0.191

−4:73%
1.61%
1.88%

to quantify the level of image artifacts that is expressed by the standard deviation of a parameter
over the parameter itself. Parameters representative
of tissue measurements are used to evaluate the analytic results and conduct the synthetic studies, in
both of which the DC reconstruction produced a lower level of relative variation in the optical parameters recovered, and some advantages in the CNR.
It may be argued that DC flattens images, leading to
a lower standard deviation in the background and,
because the background standard deviation is the denominator of CNR, the CNR of DC could become better. But, if there were flattening of the image, then
the numerator of the CNR would also be flattened,
and perhaps flattened more strongly owing to the
nonlinearity of DOT and, thereby, underestimated
at a higher level, which collectively might reduce
the CNR rather than increase the CNR. The slight
but notable CNR advantage of DC-based over FDbased reconstruction demonstrated in this study
strongly suggests some inherent advantages of DC,
but it could be just because DC has lower information
content, similar to what one could expect by reducing
the amount of data available or increasing the regularization in FD-based reconstructions.

It is worthwhile to note that this study (as well as
most other synthetic studies) assumes a step change
of the optical properties of the target of interest with
respect to the background. This is not a faithful
representation of actual tissue-imaging applications, wherein the target of interest frequently has
a tapered or smooth change of contrast over the background. The stronger cross talk between absorption
and scattering seen for DC-only reconstruction in
this study, as well as many other studies, could have
been the outcome of the nonuniqueness, revealed by
[40], which is pronounced when the target of interest
has a step contrast over the background. In fact, the
DC-based reconstruction of in vivo measurements
has encountered notably different absorption and
scattering patterns of a target of interest [42], which
may indicate a weaker cross talk for smoother contrast of the target of interest. It is also noted that this
study, as well as most other synthetic studies, assumes a globally homogenous yet locally heterogeneous background. An actual tissue environment
could be locally homogenous but globally strongly
heterogeneous, such as is found in the prostate [26].
In such conditions, a balance or trade-off may exist
between the ability of suppressing the background
heterogeneity and the likelihood of identifying a

Fig. 5. (Color online) Region-based reconstruction for multiple targets in a three-dimensional cylindrical geometry with the optodes and
targets located on one plane. (a) Imaging geometry and the regions of interest; (b) comparison of the results for DC, AC þ PHS, and
DC þ AC þ PHS.
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APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

Fig. 6. (Color online) Region-based reconstruction for multiple targets in a three-dimensional cylindrical geometry with the optodes
located on three different planes and targets located on the middle plane. (a) Imaging geometry and the regions of interest; (b) comparison
of the results for DC, AC þ PHS, and DC þ AC þ PHS.

target of interest in which the contrast is strong
locally but weak globally.
This study has also indicated that including DC information in FD reconstruction can sometimes lead
to better images than those obtained by ignoring
it. The expressions of δ and α in Eq. (4) demonstrate
that the DC attenuation is not linearly dependent
upon the AC attenuation, and the difference between
the two attenuation values increases as the modulation frequency increases. The necessity of including
DC in order to optimize the FD reconstruction is
made evident by the results in Subsections 3.B.2
and 3.B.3, wherein the DC þ AC þ PHS results have
always been slightly better than the AC þ PHS
results on the background artifacts, the target properties, and the cross coupling between μa and μ0s =D.
However, the slightly better performance of DC þ
AC þ PHS over AC þ PHS diminishes as the total
number of measurements goes up, as is shown in
the three-ring case in Subsection 3.B.3. When fewer
measurements are available in application situations, including the DC information in the limited
FD measurements likely will improve the overall reconstruction outcome.
This study is carried out for the measurements at a
single wavelength. Investigating the PRUL issues in
the context of multiband FD measurements will be a
natural and more practical extension of this work because most optical tomography measurements are
conducted with some kind of spectral information.
Besides, similar approaches may be extended to
other applications wherein the measurement data
contains multiple aspects of information, from which
the data usage may be optimized for the specific system configuration.
5. Conclusions

The level of variations of recovered optical properties
in optical tomography associated with the measurement uncertainty under three reconstruction configurations of DC-only, the DC-excluded FD, and the
DC-included FD is studied by analytic and synthetic
means. It is demonstrated that, at the same level of
measurement uncertainty typical to optical tomography and under pixelwise reconstruction without spatial a priori information, the standard deviations of
μa over μa reconstructed by DC only are at least 1.4
times lower than those obtained by FD methods. The
standard deviations of D (or μ0s ) over D (or μ0s ) reconstructed by DC only are slightly lower than those by

FD methods. Frequency-domain reconstruction including DC generally outperforms reconstruction excluding DC, but the difference between the two
becomes less significant when the total amount of
measurements becomes larger. For FD reconstruction with no spatial a priori information and a smaller number of measurements, including DC is
recommended. When a priori structural information
is available, the three reconstruction configurations
investigated in this study perform equally well.
This work has been supported in part by the Prostate Cancer Research Program of the U.S. Army Medical Research Acquisition Activity (USAMRAA)
through grant W81XWH-07-1-0247, and the Health
Research Program of Oklahoma Center for the Advancement of Science and Technology (OCAST)
through grant HR06-171. We are grateful to the
anonymous reviewers for their constructive comments that enriched the discussions of this work.
References and Notes
1. S. R. Arridge, “Optical tomography in medical imaging,”
Inverse Probl. 15, R41–R93 (1999).
2. F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “Optical tomographic mapping of cerebral haemodynamics by means of
time-domain detection: methodology and phantom validation,” Phys. Med. Biol. 49, 1055–1078 (2004).
3. H. Rinneberg, D. Grosenick, K. T. Moesta, J. Mucke,
B. Gebauer, C. Stroszczynski, H. Wabnitz, M. Moeller,
B. Wassermann, and P. M. Schlag, “Scanning time-domain optical mammography: detection and characterization of breast
tumors in vivo,” Technol. Cancer Res. Treat. 4, 483–496 (2005).
4. T. Tanifuji and M. Hijikata, “Finite difference time domain
(FDTD) analysis of optical pulse responses in biological tissues for spectroscopic diffused optical tomography,” IEEE
Trans. Med. Imaging 21, 181–184 (2002).
5. W. Mo and N. Chen, “Fast time-domain diffuse optical tomography using pseudorandom bit sequences,” Opt. Express 16,
13643–13650 (2008).
6. B. W. Pogue, M. S. Patterson, H. Jiang, and K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. 40, 1709–1729
(1995).
7. B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and
K. D. Paulsen, “Spatially variant regularization improves
diffuse optical tomography,” Appl. Opt. 38, 2950–2961 (1999).
8. J. Wang, S. Jiang, K. D. Paulsen, and B. W. Pogue, “Broadband
frequency-domain near-infrared spectral tomography using a
mode-locked Ti:sapphire laser,” Appl. Opt. 48, D198–D207
(2009).
9. D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang,
J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos,
1 June 2010 / Vol. 49, No. 16 / APPLIED OPTICS

3069

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

and B. Chance, “Non-invasive and quantitative near-infrared
hemoglobin spectrometry in the piglet brain during hypoxic
stress, using a frequency-domain multidistance instrument,”
Phys. Med. Biol. 46, 41–62 (2001).
M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida,
E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag,
and M. Kaschke, “Frequency-domain techniques enhance optical mammography: initial clinical results,” Proc. Natl. Acad.
Sci. USA 94, 6468–6473 (1997).
N. G. Chen, M. Huang, H. Xia, D. Piao, E. Cronin, and
Q. Zhu, “Portable near-infrared diffusive light imager for
breast cancer detection,” J. Biomed. Opt. 9, 504–510 (2004).
M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin,
D. Kidney, J. Butler, B. Chance, and A. G. Yodh, “Threedimensional diffuse optical mammography with ultrasound
localization in a human subject,” J. Biomed. Opt. 5, 237–247
(2000).
B. J. Tromberg, B. W. Pogue, K. D. Paulsen, A. G. Yodh,
D. A. Boas, and A. E. Cerussi, “Assessing the future of diffuse
optical imaging technologies for breast cancer management,”
Med. Phys. 35, 2443–2451 (2008).
E. S. Papazoglou, M. S. Weingarten, L. Zubkov, L. Zhu,
S. Tyagi, and K. Pourezaei, “Near infrared diffuse optical tomography: improving the quality of care in chronic wounds of patients
with diabetes,” Biomed. Instrum. Technol. 41, 83–87 (2007).
G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu,
“Combined diffuse optical tomography (DOT) and MRI system
for cancer imaging in small animals,” Technol. Cancer Res.
Treat. 5, 351–363 (2006).
J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran,
A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Threedimensional diffuse optical tomography in the parallel plane
transmission geometry: evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,”
Med. Phys. 30, 235–247 (2003).
H. K. Kim, U. J. Netz, J. Beuthan, and A. H. Hielscher, “Optimal source-modulation frequencies for transport-theorybased optical tomography of small-tissue volumes,” Opt.
Express 16, 18082–18101 (2008).
N. Iftimia and H. Jiang, “Quantitative optical image reconstruction of turbid media by use of direct-current measurements,” Appl. Opt. 39, 5256–5261 (2000).
A. M. Siegel, J. J. A. Marota, and D. A. Boas, “Design and
evaluation of a continuous-wave diffuse optical tomography
system,” Opt. Express 4, 287–298 (1999).
Y. Xu, X. Gu, T. Khan, and H. Jiang, “Absorption and scattering images of heterogeneous scattering media can be simultaneously reconstructed by use of dc data,” Appl. Opt. 41,
5427–5437 (2002).
Z. Yuan and H. Jiang, “Image reconstruction scheme that combines modified Newton method and efficient initial guess estimation for optical tomography of finger joints,” Appl. Opt.
46, 2757–2768 (2007).
H. Jiang, K. D. Paulsen, U. L. Osterbergy, and M. S. Patterson,
“Improved continuous light diffusion imaging in single- and
multi-target tissue-like phantoms,” Phys. Med. Biol. 43,
675–693 (1998).
Y. Pei, H. L. Graber, and R. L. Barbour, “Normalizedconstraint algorithm for minimizing inter-parameter crosstalk in DC optical tomography,” Opt. Express 9, 97–109 (2001).
Z. Jiang, D. Piao, G. Xu, J. W. Ritchey, G. R. Holyoak,
K. E. Bartels, C. F. Bunting, G. Slobodov, and J. S. Krasinski,
“Trans-rectal ultrasound-coupled near-infrared optical tomography of the prostate Part II: Experimental demonstration,”
Opt. Express 16, 17505–17520 (2008).
A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson,
M. Backhaus, U. Netz, and J. Beuthan, “ Sagittal laser optical

3070

APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.
42.

tomography for imaging of rheumatoid finger joints,” Phys.
Med. Biol. 49, 1147–1163 (2004).
K. K. Wang and T. C. Zhu, “Reconstruction of in-vivo optical
properties for human prostate using interstitial diffuse optical
tomography,” Opt. Express 17, 11665–11672 (2009).
M. Schweiger, I. Nissilä, D. A. Boas, and S. R. Arridge, “Image
reconstruction in optical tomography in the presence of coupling errors,” Appl. Opt. 46, 2743–2756 (2007).
V. Ntziachristos, A. H. Hielscher, A. G. Yodh, and B. Chance,
“Diffuse optical tomography of highly heterogeneous media,”
IEEE Trans. Med. Imaging 20, 470–478 (2001).
S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieriand,
and E. Gratton, “Quantitative determination of the absorption
spectra of chromophores in strongly scattering media: a lightemitting-diode based technique,” Appl. Opt. 33, 5204–5213
(1994).
D. Boas, T. Gaudette, and S. R. Arridge, “Simultaneous
imaging and optode calibration with diffuse optical tomography,” Opt. Express 8, 263–270 (2001).
S. L. Jacques, “Reflectance spectroscopy with optical fiber devices and transcutaneous bilirubinometers,” in Biomedical
Optical Instrumentation and Laser-Assisted Biotechnology,
A. M. Verga Scheggi, S. Martellucci, A. N. Chester, and R.
Pratesi, eds. (Kluwer Academic, 1996), pp. 83–94.
F. Fabbri, M. A. Franceschini, and S. Fantini, “Characterization of spatial and temporal variations in the optical properties of tissuelike media with diffuse reflectance imaging,”
Appl. Opt. 42, 3063–3072 (2003).
The original derivation in [29] for σ μ0s =μ0s has ρ in the equation,
which is inconsistent with that obtained for σ μa =μa. Equation
(19) corrected this inconsistency.
B. W. Pogue, S. Geimer, T. O. McBride, S. Jiang, U. L. Osterberg,
and K. D. Paulsen, “Three-dimensional simulation of nearinfrared diffusion in tissue: boundary condition and geometry
analysis for finite-element image reconstruction,” Appl. Opt.
40, 588–600 (2001).
G. Xu, D. Piao, C. H. Musgrove, C. F. Bunting, and H. Dehghani,
“Trans-rectal ultrasound-coupled near-infrared optical tomography of the prostate, part I: simulation,” Opt. Express 16,
17484–17504 (2008).
H. Dehghani, B. W. Pogue, S. Jiang, B. Brooksby, and
K. D. Paulsen, “Three-dimensional optical tomography: resolution in small-object imaging,” Appl. Opt. 42, 3117–3128
(2003).
H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis,
S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: algorithms for numerical model and image reconstruction
algorithms,” Commun. Numer. Meth. Eng. 25, 711–732 (2009).
P. K. Yalavarthy, H. Dehghani, B. W. Pogue, and K. D. Paulsen,
“Critical computational aspects of near infrared circular tomographic imaging: Analysis of measurement number, mesh
resolution and reconstruction basis,” Opt. Express 14,
6113–6127 (2006).
M. Huang and Q. Zhu, “A dual-mesh optical tomography
reconstruction method with depth correction using a priori
ultrasound information,” Appl. Opt. 43, 1654–1662 (2004).
S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23, 882–884
(1998).
B. Harrach, “On uniqueness in diffuse optical tomography,”
Inverse Probl. 25, 055010 (2009).
Z. Jiang, G. R. Holyoak, K. E. Bartels, J. W. Ritchey, G. Xu,
C. F. Bunting, G. Slobodov, and D. Piao, “In vivo trans-rectal
ultrasound coupled near-infrared optical tomography of a
transmissible venereal tumor model in the canine pelvic canal,” J. Biomed. Opt. 14, 030506 (2009).



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