Page 1
Implementation of a computationally efficient least-squares algorithm
for highly under-determined three-dimensional diffuse optical
tomography problems
Phaneendra K. Yalavarthy,a?Daniel R. Lynch, and Brian W. Pogueb?
Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire 03755
Hamid Dehghani
Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire 03755
and School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, United Kingdom
Keith D. Paulsen
Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire 03755
?Received 7 August 2007; revised 29 January 2008; accepted for publication 9 February 2008;
published 8 April 2008?
Three-dimensional ?3D? diffuse optical tomography is known to be a nonlinear, ill-posed and
sometimes under-determined problem, where regularization is added to the minimization to allow
convergence to a unique solution. In this work, a generalized least-squares ?GLS? minimization
method was implemented, which employs weight matrices for both data-model misfit and optical
properties to include their variances and covariances, using a computationally efficient scheme.
This allows inversion of a matrix that is of a dimension dictated by the number of measurements,
instead of by the number of imaging parameters. This increases the computation speed up to four
times per iteration in most of the under-determined 3D imaging problems. An analytic derivation,
using the Sherman–Morrison–Woodbury identity, is shown for this efficient alternative form and it
is proven to be equivalent, not only analytically, but also numerically. Equivalent alternative forms
for other minimization methods, like Levenberg–Marquardt ?LM? and Tikhonov, are also derived.
Three-dimensional reconstruction results indicate that the poor recovery of quantitatively accurate
values in 3D optical images can also be a characteristic of the reconstruction algorithm, along with
the target size. Interestingly, usage of GLS reconstruction methods reduces error in the periphery of
the image, as expected, and improves by 20% the ability to quantify local interior regions in terms
of the recovered optical contrast, as compared to LM methods. Characterization of detector photo-
multiplier tubes noise has enabled the use of the GLS method for reconstructing experimental data
and showed a promise for better quantification of target in 3D optical imaging. Use of these new
alternative forms becomes effective when the ratio of the number of imaging property parameters
exceeds the number of measurements by a factor greater than 2. © 2008 American Association of
Physicists in Medicine. ?DOI: 10.1118/1.2889778?
Key words: near infrared, diffuse optical tomography, three-dimensional imaging, image
reconstruction, inverse problems, least-squares minimization
I. INTRODUCTION
Diffuse optical tomography ?DOT? uses near infrared wave-
lengths ?600–1000 nm? to obtain optical absorption and
scattering images for characterizing functional properties of
the tissue under investigation.1–4The most important step in
forming these images is solving the inverse problem, i.e.,
estimating the optical properties by matching the experimen-
tal data with modeled results in the least-squares sense.3,5,6
This problem is typically ill posed and ill determined de-
pending on the noise in the data, the number of measure-
ments, and the dimensions of the parameter space.6Even
though light travels in three dimensions ?3D?, most of the
numerical models reported in the literature have been two
dimensional ?2D? because of computational considerations.
Moreover, the 3D DOT imaging problem is more under-
determined relative to the 2D case and has been found to
generate poor quantitative estimates of the optical properties
when compared to 2D results.7–15Several methods have ap-
pearedin the literature
computations,11,14,16but no unified approach to the problem
has been discussed. Recently, a generalized least-squares
?GLS? minimization scheme was presented for 2D DOT im-
age reconstruction.17This article reports a computationally
efficient approach for implementing GLS minimization in
describingefficient3D
1682 1682 Med. Phys. 35 „5…, May 2008 0094-2405/2008/35„5…/1682/16/$23.00© 2008 Am. Assoc. Phys. Med.
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3D which shows an improvement in the quantification of
optical properties relative to earlier studies. Even though the
focus is on GLS implementation, equivalent forms for other
methods are also presented in light of the GLS framework.
The inverse problem in DOT is solved by minimizing the
objective function ??? over the range of optical properties
???. Methods based on gradient optimization, which do not
require an explicit inversion of the Hessian matrix ?here,
Hessian approximates the second derivative; this can be
some form equivalent to JTJ or JJT, in general some form of
JTJ, J being the Jacobian? are known to be computationally
efficient.18,19But these methods require an optimization
scheme, which can be thought of as an inner iteration, for
choosing the step size, and are not as straightforward as a
direct inversion of the matrix. Alternatively, the full-Newton
methods require calculation of the Jacobian ?J?, the forward
data, and inversion of the dense Hessian matrix at each itera-
tion. Because full-Newton methods are relatively easy to
implement, they are widely used for DOT image reconstruc-
tion even though they require large matrix inversions at ev-
ery iteration. Thus, while the full-Newton method is ideal for
small problems, it rapidly becomes intractable for larger do-
mains, such as those encountered in 3D imaging problems.
This manuscript presents a formal approach, using the
Sherman–Morrison–Woodbury identity, to construct a more
efficient alternative form of update equations of GLS and
Levenberg–Marquardt ?LM? minimization schemes, which
generates a Hessian matrix to invert. The dimension of this
Hessian is dictated by the number of measurements, rather
than the number of parameter estimates which can be con-
siderably lower for highly under-determined problems, and
therefore, much more efficient computationally. This equiva-
lent form is also shown for other common minimization
methods, namely Tikhonov.
The later part of this article describes a way to character-
ize the systematic noise using a simple analytical formula,
when the photomultiplier tube ?PMT? is used as a detector.
Characterizing noise behavior of the experimental data leads
to the use of the GLS technique in the experimental data case
and it is also shown that usage of noise characteristics will
lead to better quality and quantification of target in a experi-
mental test case.
II. DIFFUSE OPTICAL TOMOGRAPHY: FORWARD
PROBLEM
Near-infrared ?NIR? light propagation in a biological tis-
sue like breast can be modeled using the diffusion equation
?DE?6,20which in the frequency domain becomes
− ? · D?r? ? ??r,?? + ??a?r? + i?/c???r,?? = Qo?r,??,
?1?
where the optical diffusion and absorption coefficients are
given by D?r? and ?a?r?, respectively. The light source, rep-
resented by Qo?r,??, is modeled as isotropic. ??r,?? is the
photon fluence rate at a given position r. The light modula-
tion frequency is denoted by ?, where ?=2?f, ?here f
=100 MHz?. The velocity of light in tissue is represented by
c, which is assumed to be constant. Note that
D?r? =
1
3??a?r? + ?s??r??,
?2?
where ?s??r? is the reduced scattering coefficient which is
defined as ?s?=?s?1−g?; ?sis the scattering coefficient and g
is the anisotropy factor. The finite element method ?FEM? is
used to solve Eq. ?1? to generate modeled data ?G???? for a
given distribution of optical properties ???,11,13,21where ?
=??s??r?;?a?r??. A Type-III boundary condition is employed
to account for the refractive-index mismatch at the
boundary.22Under the Rytov approximation, the data ?y? be-
comes the natural logarithm of the amplitude ?A? and phase
??? of the frequency-domain signal; y=?lnA;??.
III. DIFFUSE OPTICAL TOMOGRAPHY: INVERSE
PROBLEM
III.A. Levenberg–Marquardt „LM… minimization
The most-common approach for solving the inverse prob-
lem in DOT is LM minimization.1,6,11,13,17,20,23,24A detailed
discussion of this method is available in Ref. 17 and it is
only briefly reviewed here.
The objective function25,26for this approach is defined as
? =?y − G????2,
?3?
where y is the experimental data and G??? is the modeled
response. Minimization of this objective function with re-
spect to ? is achieved by setting the first-order derivative
equal to zero
??
??= JT? = 0,
?4?
where ? is the data-model misfit, ?=y−G???, and J repre-
sents theJacobian
?J=?G???/???.
conditioned nature of the problem, the update equation for
the optical properties at iteration “i” is regularized to be
Duetotheill-
??i= ?JTJ + ?I?−1JT?i−1
?5?
or equivalently ?See Appendix 2?
??i= JT?JJT+ ?I?−1?i−1,
?6?
where ??irepresents the update of the optical property pa-
rameters at the ith step; ? is the regularization parameter,
which monotonically decreases with increasing iteration
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Medical Physics, Vol. 35, No. 5, May 2008
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?always?0?.26In this approach ?Eq. ?5??, the Jacobian is
normalized by its optical properties. Moreover, ? is chosen
empirically ?it typically starts at 10 and reduced by a factor
of 100.25at every following iteration after being multiplied
by the maximum of the diagonal values of JJT.17,27?. The
iterative procedure is stopped when the L2 norm of the data-
model misfit ??? does not improve ?in our experience, by
more than 1% because beyond these values the LM proce-
dure can become unstable17?.
Even though LM minimization or its modified versions
have beenusedsuccessfully
reconstruction,1,6,11,13,17,20,23,24,27the final image depends on
the choice of ? due to the ill-conditioned nature of the prob-
lem. Moreover, the approach ignores the noise characteristics
of the data and optical properties. A more systematic and
generalized method for image reconstruction can be based on
GLS minimization. The GLS scheme is discussed exten-
sively in Ref. 17 and is only briefly reviewed here.
for DOTimage
III.B. Generalized least squares „GLS… minimization
In GLS, the objective function is given by17,28
? = ?y − G????TW??y − G????
+ ?? + ?0?TW?−?0?? − ?0?,
?7?
where W?and W?−?0are the weight matrices for the data-
model misfit ??? and optical properties ??−?0?, respectively.
Note that W?=?C??−1, where C represents the covariance
matrix and similarly W?−?0=?C?−?0?−1?see Appendix A-4 of
Ref. 28?. These weight matrices; are symmetric and positive
definite ?because they are, inverses of covariance matrices?.
No regularization parameter is involved because the weight
matrices include the noise characteristics of the experimental
data and optical properties.17Similarly to the LM approach,
minimization of ? ?Eq. ?7?? is accomplished by setting the
first derivative of ? with respect to ? equal to zero:
??
??= JTW?? − W?−?0?? − ?0? = 0.
?8?
Linearizing the problem leads to the iterative update equa-
tion ?for ith iteration?17
??i= ?JTW?J + W?−?0?−1?JTW??i−1− W?−?0??i−1− ?0??.
?9?
Explicit definitions of the weight matrices ?W?and W?−?0?
are given in Ref. 17. Although any number of forms for
W?−?0can exist, only one is considered here, specifically,
where the covariance matrix is defined as17
?C?−?0?ij= ???−?0?2?1 +rij
l?e−?rij/l?
?10?
with l being the correlation length ?here l=15 mm? and rij
being the distance between the FEM nodes i and j???−?0?2is
the expected variance of ?−?0. Strategies for calculating the
expected variances are given in Ref. 17. In this work, the
expected variance is determined from the prior knowledge
that the expected contrast between tumor and normal tissue
is about 50%–400%. To demonstrate the robustness of the
GLS reconstruction procedure, for the results discussed here,
the variance was chosen to be ?4*??2. Both weight matrices,
W?and W?−?0, are computed before the reconstruction pro-
cedure begins, whereas the Jacobian ?J?, and modeled data,
G???, are calculated at each iteration. The iterative proce-
dure is stopped when the L2 norm of the data-model misfit
??? does not improve by more than 0.001%. Beyond these
values, the round-off error dominates.
III.B.1. GLS implementation
The parameters recovered in the case of this GLS scheme
are ??s?;?a?, which is different from some previous ap-
proaches that estimate ?D;?a?. The later case has a mismatch
because the units of D are mm whereas those of ?aare
mm−1. In its implementation typically the whole equation is
normalized by the optical properties ?outlined in Ref. 17?
which becomes computationally intensive especially for
GLS in 3D because the update equation must be left and
right multiplied by the optical properties at every iteration.
Here, the GLS problem was reformulated in terms of
??s?;?a?, so that both parameters have the same units
?mm−1?. While this is a relatively minor alteration in the
form of the algorithm, it has important implications for the
computational time required for matrix preconditioning.
A simple transformation converts the diffusion part of the
Jacobian
??G???/?D?
to
??G???/??s??:
its scatteringcomponent
?G???
??s?
=?G???
?D
?D
??s?.
?11?
Using Eq. ?2?
?D
??s?=1
3?
− 1
???a+ ?s???2?= − 3D2
?12?
and substituting Eq. ?12? in Eq. ?11?, leads to
?G???
??s?
=?G???
?D
?− 3D2?.
?13?
After this transformation ?Eq. ?13??, the Jacobian ?J? has the
form
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Medical Physics, Vol. 35, No. 5, May 2008
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J =?
?lnA1
??s1?
??1
??s1?
?lnA2
??s1?
??2
??s1?
]
?lnANM
??s1?
??NM
??s1?
?lnA1
??s2?
??1
??s2?
?lnA2
??s2?
??2
??s2?
]
?lnANM
??s2?
??NM
??s2?
¯
?lnA1
??sNN
??1
??sNN
?lnA2
??sNN
??2
??sNN
]
?lnANM
??sNN
??NM
??sNN
?
?lnA1
??a1
??1
??a1
?lnA2
??a1
??2
??a1
]
?lnANM
??a1
??NM
??a1
?lnA1
??a2
??1
??a2
?lnA2
??a2
??2
??a2
]
?lnANM
??a2
??NM
??a2
¯
?lnA1
??aNN
??1
??aNN
?lnA2
??aNN
??2
??aNN
]
?lnANM
??aNN
??NM
¯
?
¯
¯
?
¯
¯
?
¯
...
¯
...
¯
?
¯
?
¯
??aNN?
,
?14?
where NM and NN represent the number of measurements
and number of property parameters associated with the FEM
mesh, respectively. The implementation of the GLS update
equation ?Eq. ?9?? requires assembly of the weight matrix
?W?−?0? for simultaneous reconstruction of ?s? and ?aand is
accomplished by writing Eq. ?9? in block matrix form
??
=?
H?s?2
H?s??a
?JT??s?W??i−1
H?s??a
H?a
2?+?
W?s?−?s0?
0
W?a−?a0??ai−1−?a0??,
0
W?a−?a0?????si?
W?s?−?s0???si−1
??ai?
?JT??aW??i−1?−?
? −?s0??
?15?
where H represents the Hessian matrix ?JTW?J?. Here, the
cross terms in the weight matrix ?W?−?0? are zero because
?s? and ?aare independent parameters in the estimation pro-
cedure. Note that the dimensions of the matrices in Eq. ?9?
are: J:2NM?2NN, W?−?0:2NN?2NN, W?:2NM?2NM,
?=2NM?1, and ??=2NN?1. Most 3D-D0T problems are
ill determined, i.e., NM?NN.
Computing an update of the optical properties ???i, from
Eq. ?9?? requires an inversion ?or its equivalent? of a large
matrix with dimensions 2NN?2NN. Inverting a matrix of
dimension N?N typically requires an order of N3operations
and N2memory.28Hence, any gain in reducing the dimen-
sionality of the matrix to be inverted will reduce the compu-
tation time cubically and the memory requirement quadrati-
cally. An alternative form of Eq. ?9?, which requires few
operations, is
??i= ?I − C?−?0JT?JC?−?0JT+ C??−1J?
??C?−?0JTW??i−1− ??i−1− ?0??.
?16?
Full derivation of this alternative form is given in the Appen-
dix 1, along with the equivalent expressions for other mini-
mization methods. Equation ?16? requires an inversion of a
matrix with dimensions 2NM?2NM ?same is true for Eq.
?6??.
Note that the covariance/weight matrices are calculated
before the start of the iterative procedure and are used
throughout the iteration. For nodes where the sensitivity ?col-
umn sum of Jacobian? fell below 1% of the maximum sen-
sitivity, the expected variance of the optical properties were
chosen to be 1% of background ? ?in Eq. ?10??.
IV. SIMULATION STUDIES: THREE-DIMENSIONAL
TEST PROBLEM
For all numerical experiments discussed here, the imaging
domain was chosen to be cylindrical ?as shown in Fig. 1?
with a diameter of 84 mm and height of 109 mm. The back-
ground optical properties were ?a=0.01 mm−1and ?s?
=1.0 mm−1. Two meshes were used: ?1? a cylinder consisting
of 21 440 nodes corresponding to 110 483 linear tetrahedral
FIG. 1. Schematic diagram of the three-dimensional cylindrical imaging
domain.
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Medical Physics, Vol. 35, No. 5, May 2008
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elements for the forward model and ?2? a cylinder having
9211 nodes corresponding to 45 980 linear tetrahedral ele-
ments for the reconstruction. The data-collection geometry
consisted of 48 fibers that were arranged in a circular,
equally spaced fashion in three layers spaced 10 mm apart
?Fig. 1?, with 16 fibers per plane. One fiber was used at a
time as the source while the fibers in the same “source fiber
plane” were used as detectors to generate 720 ?3?16?15?
measurement locations or a total of 1440 values ?720 lnA
data points and 720 ? data points?. The sources were mod-
eled as a Gaussian profile with a full width at half maximum
of 3 mm to represent the distribution used in an experimental
setup.29The source was also placed one mean transport
length inside the boundary.
Both spherical and cylindrical objects were considered as
targets. The cylindrical target had a contrast of 2:1 with the
background in both ?aand ?s? and a diameter of 15 mm. It
extended in Z direction throughout the domain ?height of
109 mm? and was placed at the center ?at ?0,0?, first row of
Fig. 5? and near the boundary ?at ?30,0??. Two-dimensional
cross sections of both reconstructed and actual 3D volumes,
displayed in increments of 5 mm spanning from z=
−25 mm to z=25 mm ?from left-hand side to right-hand
side?, are shown in Figs. 3, 5, and 6. Cross section below z
=−25 mm and beyond z=25 mm did not show any deviation
from the starting values of iterations as the sensitivity in this
region is almost negligible compared to the rest of the do-
main, so these cross sections are omitted for display pur-
poses. Measurements with a noise level of 1% were assumed
as the experimental data ?y? in most of the cases discussed
here. The noise variance was also used with GLS reconstruc-
tion algorithm.17The background optical properties were se-
lected as starting values for the iterative image reconstruction
procedures discussed in Sec. III. All computations were car-
ried out on a Linux work station with an AMD Dual Core
Opteron 280 processor ?2.2 GHz? with 8 GB of random ac-
cess memory ?RAM?.
V. PHANTOM STUDIES
V.A. Data variance estimation
Use of weight matrices in the GLS scheme ?W?in Eq.
?9?? requires an estimation of data variance, which requires
experimental characterization of the expected values, prior to
patient/phantom imaging. This was achieved by tracking the
detected voltage measured at the photomultiplier tube
?PMT?. PMTs are used as a detectors in the experimental
system at Dartmouth, details of the experimental system are
given in Ref. 29. Note that this characterization includes only
systematic errors associated with low signal levels, but errors
due to poor fiber-tissue coupling are not accounted for in this
model.
Starting from the assumption that the detected signal at
the PMT in diffuse optical imaging is shot noise limited
leads to
??N? =?N,
?17?
where ? is the standard deviation of N and N is the number
of photons reaching the PMT. The voltage ?V; representing
the detected ac intensity signal29? measured at the PMT is
directly proportional to N, which also implies that measured
amplitude ?A? of the detected frequency domain signal ?y? is
proportional to this voltage ?V?. This is written as
A ? V ⇒ A = kV
??A? = k??V?.
?18?
Here, k acts as a proportionality constant. In the reconstruc-
tion procedure, the Rytov approximation is used, leading to
data being represented as lnA rather than A. If f?x? is a func-
tion of x and is continuous and differentiate, then
??f?x?? =?f?x?
?x
??x?.
?19?
Similar to the previous equation ?Eq. ?19??, writing the stan-
dard deviation of InA leads to
??lnA? =1
A??A?.
?20?
now using Eq. ?18? leads to
??lnA? =??V?
V
.
?21?
From the above equation ?Eq. ?21??, it can be concluded that
the variance in data ??2? can be known by measuring the
deviation in the PMT voltage ?V?.
To measure the deviation in the measured signal, a series
of light signal measurements were taken through homoge-
neous intra-lipid phantom experiments which were con-
ducted with increasing levels of blood ?HbT? concentration,
varying from 7.3 to 36 ?M, leading to a decrease in the
measured PMT voltage. To achieve this, the gain of the PMT
was kept at 0.9. A concise discussion of the PMT gain setting
in the system characterization is given in Ref. 29. A single
source and the farthest detector was used for these transmis-
sion measurements. For every concentration, 200 data points
were collected to estimate the deviation in the measured volt-
age using the same gain settings. The approach for the char-
acterization is similar to the one described in Ref. 29, except
the raw detected voltage was used here for estimation of the
error ?or deviation ??. Note also that two sets of diameters,
56 and 84 mm, were used to get the voltage in the range of
0–1 V. This was repeated for all the wavelengths to ensure
uniformity of performance in the signal, and to ensure that
the observed trend was independent of wavelength and gain
setting.
Figure 9 gives a plot of error ???V?/V? as a function of
measured PMT voltage for 785 nm wavelength. A similar
trend was observed for other wavelengths. This plot also
gives a deviation in phase ?????? in degrees for the same
voltage. Each of these points represents a sample size of 200.
The lowest measured voltage of PMT was 0.001 V. The
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Medical Physics, Vol. 35, No. 5, May 2008
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measured deviations were 1% for lnA and 0.5° for ? for PMT
voltages above 0.005 V. These values are similar to the ones
reported in the literature ?Mcbride et al.29reported 0.32% for
the PMT voltage and 0.48° in phase?. A solid line in Fig. 9
shows these average deviation using 1/V2fitting ?following
the shot-noise model?. From this plot, the weight matrix for
the data-model misfit ?W?in Eq. ?9?? can be written as
?W??ij
=?
0
ifi ? j
1
??5.7/V2? + 0.6?2for ln A with V ? 0.001 if i = j
1
??1.3/V2? + 0.4?2for ? with V ? 0.001
1 for lnA with V ? 0.001
1
???2for ? with V ? 0.001
if i = j
if i = j
if i = j
.
?22?
This implies that if the PMT voltage is below 0.001 V, the
signal is considered to be in the noise floor. For the signals
above 0.001 V, the variance can be estimated using 1/V2
fitted curve ?Eq. ?22??. This characterization enables the us-
age of the GLS scheme in the case of experimental data, with
a requirement that the voltage measured at the PMT is avail-
able.
V.B. Gelatin phantom
A multilayer cylindrical gelatin phantom of diameter
86 mm, height 25 mm was fabricated using different mix-
tures of India ink for absorption and Titanium oxide ?TiO2?
for scattering. These different layers of gelatin were fabri-
cated by successively hardening heated gelatin solutions
?typically 80% deionized water and 20% gelatin ?G2625,
Sigma Inc?? along with different amounts of ink and TiO2
?Sigma Inc?. A cylindrical hole extending in Z direction ?di-
ameter of 16 mm and height of 24 mm? filled with intra-lipid
mixed with India ink acted as a target having the optical
properties ?a=0.02 mm−1and ?s?=1.2 mm−1. The outer
layer with optical properties ?a=0.0065 mm−1and ?s?
=0.65 mm−1had a thickness of 10 mm mimicking the typi-
cal fatty layer of the breast.30The middle layer with 76 mm
diameter, mimicking fibro-glandular layer, had optical prop-
erties ?a=0.01 mm−1and ?s?=1.0 mm−1. Validation of indi-
vidual layer optical properties was performed by the data
collected on large cylindrical samples of each layer using
785 nm wavelength laser diode as the source. Two-
dimensional cross sections of this gelatin phantom optical
properties are displayed in increments of 2.5 mm spanning
from z=−12.5 mm to z=12.5 mm ?from left-hand side to
right-hand side? in top rows of Figs. 10?a? and 10?b?. In this
10
−2
10
−1
10
0
10
1
10
2
10
7
10
10
10
13
10
16
NN/NM
(b)
No. of operations
(a)
GLS
GLS−AF
10
−2
10
−1
10
0
10
1
10
2
10
4
10
7
10
10
10
13
10
16
NN/NM
(c)
No. of operations
LM
LM−AF
10
−2
10
−1
10
0
10
1
10
2
10
0
10
3
10
6
10
9
NN/NM
Memory required
LM; GLS
LM−AF; GLS−AF
FIG. 2. Comparison of the number of operations required for the original
update equation and its alternative form ?a?; for GLS ?Eq. ?9?, represented
by GLS? and its alternative form ?Eq. ?16?, represented by GLS-AF? ?b?; for
LM ?Eq. ?5?, represented by LM? and its alternative form ?Eq. ?6?, repre-
sented by LM-AF? as a function of the ratio of number of estimation pa-
rameters to number of measurements ?represented by NN/NM?. Memory
required for implementing the inversion procedure is plotted in ?c?.
(b)
(a)
FIG. 3. Actual and reconstructed ?a? ?aand ?b? ?s? distributions of a spheri-
cal target having a diameter of 15 mm located at the center ?at ?0,0,0?? using
1% noisy data. Two-dimensional cross sections of the 3D volume in 5 mm
increments spanning from z=−25 mm to z=25 mm ?from left to right? are
shown. Actual distributions are given in the first row. Reconstructed distri-
bution using the LM minimization scheme and GLS minimization scheme
are presented in the middle and last rows, respectively.
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Medical Physics, Vol. 35, No. 5, May 2008
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phantom case, dater were collected using only one layer of
fibers ?at z=0 mm? leading to 240 ln A data points and 240 ?
data points. A cylindrical mesh consisting 8990 nodes corre-
sponding to 44 803 linear tetrahedral elements was used and
the experimental data were also calibrated using a reference
homogenous phantom data. The outer layer ?mimicking fatty
layer? optical properties were used as initial guess for recon-
struction procedures. A second mesh with the same geometry
containing 3718 nodes ?16 627 linear tetrahedral elements?
was used as a reconstruction mesh.
VI. RESULTS
The number of operations required to produce an optical
property update ???i? for both the original and alternative
GLS update equations ?Eqs. ?9? and ?16?, respectively? was
compared as a function of the ratio of the number of estima-
tion parameters ?NN? to the number of measurements ?NM?.
A similar comparison for LM update equations ?Eqs. ?5? and
?6?? was also performed. The results are plotted in Figs. 2?a?
and 2?b?, respectively. The expressions used for calculating
the number of operations are given in Appendix 4. Memory
required for implementing these inversions is presented in
Fig. 2?c?. To relate the analysis to existing experiments, NM
was chosen to be 720. The number of nodes ?equivalently
estimation parameters? was varied from 2 to 80 000. The
number of computations increases for both LM and GLS
cases as NN increases, but the alternative form for GLS
?GLS-AF? has a lower computational cost when NN/NM is
greater than 6. In the case of alternative form LM ?LM-AF?,
this is when NN/NM is greater than 2. In terms of memory,
as soon as NN?NM, both GLS-AF and LM-AF requirement
is less than GLS and LM counterparts.
In order to assess algorithm performance, a series of test
reconstructions were evaluated. Data with 1% noise from the
cylinder containing a 15-mm-diam spherical target ?first
rows of Figs. 3?a? and 3?b?? was reconstructed for the optical
properties using LM and GLS techniques ?middle and last
rows and Figs. 3?a? and 3?b??. In the case of the GLS
scheme, the reformulated update equation ?alternative form -
Eq. ?16?? was used ?last rows of Figs. 3?a? and 3?b??. How-
ever, it was also important to confirm that the two forms of
the update equation ?Eqs. ?9? and ?16?? produced numerically
equivalent solutions. Figure 4 shows a comparison of results
generated with the original GLS update equation ?Eq. ?9??
and its alternative form ?Eq. ?16?? in terms of data-model
misfit ??? and reconstructed optical properties. The differ-
ence plots ?Figs. 4?b? and 4?d?? demonstrate that Eqs. ?9? and
?16? are equal within the limits of the numerical precision to
be expected ??10−8of the L2-norm value, after the first few
iterations?. A similar analysis between the original LM up-
date equation ?Eq. ?5?? and its alternative form ?Eq. ?6?? was
performed and gave similar results ?not shown here?. These
results indicate that there were no unexpected numerical dif-
ficulties in proving that these alternative forms of update
equations ?GLS: Eq. ?16?; LM: Eq. ?6?? are equivalent to its
original forms ?GLS: Eq. ?9?; LM: Eq. ?5??. Reconstruction
results with different target shapes and positions are summa-
TABLE I. Mean and standard deviation of the reconstructed ?aand ?s? values in mm−1for the background and target with LM and GLS techniques using data
with 1% noise. The spherical target had a diameter of 15 mm. One set of reconstructed images for the target in the center is presented in Fig. 3. The cylindrical
target diameter was 15 mm and extended throughout the imaging domain in the z direction.
TargetTargetBackground
Methods
Actual
Shape
¯
Position
¯
?a
0.01
?s?
1.0
?a
0.02
?s?
2.0
LM
Spherical
?0,0,0?
?30,0,0?
?0,0?
?30,0?
?0,0,0?
?30,0,0?
?0,0?
?30,0?
0.0101?0.003
0.0101?0.0006
0.0102?0.0010
0.0101?0.006
0.0101?0.0001
0.0100?0.0004
0.0102?0.0008
0.0101?0.0008
1.0079?0.0322
1.0063?0.0500
1.0120?0.0874
1.0030?0.0663
1.0100?0.0212
1.0108?0.0250
1.0055?0.0500
1.0043?0.0588
0.0104?0.0002
0.0126?0.0009
0.0151?0.0012
0.0148?0.0015
0.0122?0.0004
0.0241?0.0006
0.0159?0.0009
0.0170?0.0012
1.1259?0.0160
1.4514?0.1152
1.4308?0.0854
1.8406?0.2470
1.2903?0.0326
1.4498?0.0853
1.4750?0.0941
1.6793?0.1848
Cylindrical
GLSSpherical
Cylindrical
12345678
0
2
4
6
8
Iteration number
L2−norm of δ
(a)
GLS
GLS−AF
12345678
10
−10
10
−8
10
−6
10
−4
Iteration number
(d)
Difference (in %)
(b)
12345678
10
2.1697
10
2.1706
Iteration number
L2−norm of (μactual− μ)
(c)
GLS
GLS−AF
12345678
10
−7
10
−6
10
−5
10
−4
Iteration number
Difference (in %)
FIG. 4. Comparison of results from the GLS update equation ?Eq. ?9?, rep-
resented by GLS? and its alternative form ?Eq. ?16?, represented by GLS-
AF?. ?a? L2 norm of data-model misfit ??? as a function of iteration, ?b?
difference ?in %? in the curves in ?a?, ?c? L2-norm of the solution space
??actual−?? as a function of iteration, ?d? difference ?in %? in the curves in
?c?.
1688Yalavarthy et al.: 3D GLS estimation in diffuse optical tomography1688
Medical Physics, Vol. 35, No. 5, May 2008
Page 8
rized in Table I which reports the mean and standard devia-
tion of recovered ?aand ?s? values in the background and
target areas. Note that the recovered optical properties be-
tween Z=15 mm and Z=−15 mm were used because the re-
constructed optical properties were equal to the actual back-
ground value and the standard deviation was zero ?within
round-off error limits? above/below these Z values.
To show the robustness of the GLS procedure, data with
5% noise was used in the reconstruction of a cylindrical tar-
get located in the center ?as shown in first rows of Figs. 5?a?
and 5?b??. The reconstruction results using the LM and GLS
schemes are presented in middle and last rows of Figs. 5?a?
and 5?b?, respectively. The GLS minimization technique was
able to localize the target more clearly than the LM method.
To see the effect of target size on the recovery of its
contrast using these reconstruction techniques, a series of
simulations were performed where the diameter of the
spherical target located at the center was varied ?from 10 to
35 mm?. One set of results is presented for the 15-mm-diam
target in Fig. 3. Another sample set for a target with diameter
of 10 mm is shown in Fig. 6. A comparison plot is given in
Fig. 7. The data noise level for the cases considered here is
1%. Increase in the diameter of the spherical target increases
the contrast recovery.
A performance comparison of these reconstruction tech-
niques with increases in target ?spherical object with diam-
eter of 25 mm? contrast ?from 2 to 10 with respect to back-
ground optical properties? located at the center ?0,0,0? and
?20,0,0? is presented in Fig. 8. Again, the noise level in the
data was 1%. The recovery of contrast is much lower in the
case of the centered target compared to the off-centered lo-
cation in both LM and GLS techniques. Between the LM and
GLS methods, the latter performs better in terms of recovery
of contrast.
A study was conducted to evaluate estimation parameter
independence ?cross-talk? in these reconstruction procedures.
The spherical target having a contrast of 2:1 only in ?awas
considered at the center and near the boundary of the imag-
ing domain. Synthetic data with 1% noise were used in the
reconstructions, and the results are presented in Table II in
(b)
(a)
FIG. 5. Actual and reconstructed ?a? ?aand ?b? ?s? distributions of a cylin-
drical target located at the center ?diameter — 15 mm? using 5% noisy data.
Two-dimensional cross sections of the 3D volume in 5 mm increments span-
ning from z=−25 mm to z=25 mm ?from left to right? are shown. Actual
distributions are given in the first row. Reconstructed distribution using the
LM minimization scheme and GLS minimization scheme are presented in
the middle and last rows, respectively.
(b)
(a)
FIG. 6. Actual and reconstructed ?a? ?aand ?b? ?s? distributions of a spheri-
cal target having a diameter of 10 mm located at the center ?at ?0,0,0?? using
1% noisy data. Two-dimensional cross sections of the 3D volume in 5 mm
increments spanning from z=−25 mm to z=25 mm ?from left to right? are
shown. Actual distributions are given in the first row. Reconstructed distri-
bution using the LM minimization scheme and GLS minimization scheme
are presented in the middle and last rows, respectively.
1689Yalavarthy et al.: 3D GLS estimation in diffuse optical tomography 1689
Medical Physics, Vol. 35, No. 5, May 2008
Page 9
terms of recovered mean and standard deviation of the opti-
cal properties. The recovery of contrast was higher in the
GLS case and the amount of cross-talk was less ?roughly
50% in the LM case compared to 30% in the GLS case?. A
similar study with a contrast of only in ?s? was also con-
ducted ?not shown? and it also showed a similar trend in
terms of cross-talk.
Finally, using the experimental data ?Sec. V B? collected
using a multilayered gelatin phantom, reconstructions using
both LM and GLS techniques were performed. For the GLS
technique, experimental data variance was estimated ?Sec.
V A? using an analytical equation, given in Eq. ?22?. Two-
dimensional cross sections of the actual and reconstructed
optical properties distribution are plotted in Figs. 10?a? and
10?b?. The middle and bottom rows correspond to the recon-
struction results obtained by LM and GLS techniques. The
variance in the data ?noise charters tics? is embedded in the
GLS reconstruction procedure, resulting in optimal weight-
ing ?highly noisy data points get less weightage and vice
versa for less noisy data points?, leading to better quantifica-
tion of tumor region.
VII. DISCUSSION
Appendix 1 presents a computationally efficient form for
implementing an iterative GLS reconstruction scheme which
reduces the dimensionality of the matrix to be inverted. The
alternative forms for other minimization methods are also
developed in the appendices ?Sec. II and III?. Appendix 4
presents expressions for estimating the operations count of
both forms of the GLS update equations ?Eqs. ?9? and ?16??
for a single iteration. Appendix 4 also gives operations count
for both LM and its alternative form ?Eqs. ?5? and ?6?? as
well. Figure 2 shows a log-log plot of operations count as a
function of the ratio of number of optical property param-
eters ?NN? to number of measurements ?NM? which deter-
mines the form of the GLS and LM update equation to be
preferred ?given that both produce numerically equivalent
results as reported in Fig. 4?. For example, when spatial pri-
ors are available, the number of optical unknowns can be
reduced to the number of regions that can be segmented31
which in the case of breast tissue is typically NN=3 ?assum-
ing fatty, fibro-glandular and tumor regions?.31Here, since
NM?NN, the original LM and GLS update equation ?Eqs.
?5? and ?9?, respectively? is effective. In under-determined
problems, such as the cases considered in this article, where
NM?NN ?NN/NM ratio of 12 in the test problems?, the
GLS alternative form reduces the number of operations ?by
up to two times in Fig. 2?a? when NN/NM=12?. In fact, the
alternative form of the GLS update equation ?Eq. ?16?? be-
comes effective once NN/NM?6 and the number of opera-
tions decreases by an order of magnitude when NN/NM
reaches 100. For the LM minimization scheme, alternative
form reduces the number of operations by a factor of 6 in
Fig. 2?b? when NN/NM=12. The memory required for in-
verting such matrices is plotted in Fig. 2?c? as a function of
NN/NM. It is also important to recognize that the memory
10 2030
0.01
0.012
0.014
0.016
0.018
0.02
Spherical target diameter (in mm)
Mean Value of recoveredμa(target)
μa
10 2030
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Spherical target diameter (in mm)
Mean Value of recoveredμs’ (target)
μs’
Expected
LM
GLS
FIG. 7. Comparison of recovered optical properties as a function of inclusion size ?diameter? for the spherical target located at ?0,0,0? using both LM and GLS
techniques. Standard deviations were of the same order as given in Table I.
1690Yalavarthy et al.: 3D GLS estimation in diffuse optical tomography 1690
Medical Physics, Vol. 35, No. 5, May 2008
Page 10
required to complete these operations can become critical
because the cache sizes and RAM available on different ar-
chitectures is variable but influences the efficiency of the
computational processes executing on a given platform.
Under-determinedness of the imaging problem ?i.e.,
NN/NM?1? leads to nonuniqueness in the solution space,
but regularization helps to give a unique solution in these
cases. Typically NN/NM values are between 2 and 10 for a
typical two-dimensional ?2D? problem, as the choice de-
pends on the expected resolution in the reconstructed image,
imaging domain size, shape, data-collection geometry, and
prior information available. For a 3D problem, this choice
?NN/NM values? also depends on these factors, but one ex-
pects this ratio to be higher than the 2D case as the imaging
domain size is bigger reflecting in the number of imaging
parameters ?NN? to be larger ?Typical example: NN=600 ?in
2D? and 6000 ?in 3D??. Ideally, one would like to have this
ratio ?NN/NM? constant between 2D and 3D; this implies
that the number of measurements has to increase by the same
factor ?typical case requires a factor of 10?, which might not
be feasible due to instrumentation constrains.24This leads to
choice of NN/NM greater than 6 at least, where the derived
forms are effective ?even though in the case of LM, the al-
ternative form is effective when NN/NM?2?. It is also im-
portant to note that, ideally one would like mesh the volume
where the sensitivity is greater ?for the imaging domain dis-
cussed here z between −25 and 25 mm? finer and rest of the
domain coarser, to keep NN/NM in the same range as in 2D
?typically 3–8?. This adaptive meshing gets trickier when the
patients are imaged, in our experience. We could not find
appropriate tools that could be used in real time for this
meshing problem, even though efforts to solve this patient-
specific adaptive meshing are being pursued.32,33Even when
NN/NM=3 ?lowest ratio one expects in a 3D imaging prob-
lem?, from Fig. 2?b?, the alternative form ?LM-AF, Eq. 6? in
the case of LM minimization scheme becomes effective.
Figure 4 demonstrates that the two GLS update forms
?Eqs. ?9? and ?16?? are equivalent numerically ?within the
TABLE II. Mean and standard deviation of reconstructed ?aand ?s? in mm−1values for the background and a spherical target with no scattering contrast using
the LM and GLS techniques. The diameter of the spherical inclusion was 15 mm. Data with 1% noise were used.
Target
position BackgroundTarget
Methods
Actual
?a
0.01
?s?
1.0
?a
0.02
?s?
1.0
¯
LM
?0,0,0?
?30,0,0?
?0,0,0?
?30,0,0?
0.0101?0.0003
0.0101?0.0004
0.0101?0.0001
0.0100?0.0004
1.0025?0.0211
1.0009?0.0205
1.0016?0.0269
1.0029?0.0189
0.0109?0.0001
0.0119?0.0004
0.0126?0.0002
0.0136?0.0003
1.0500?0.0071
1.0934?0.0269
1.0924?0.0253
1.1002?0.0294
GLS
0.020.04
Expected μa(target)
0.060.08 0.1
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
Mean Value of recoveredμa(target)
μa
246810
1
2
3
4
5
6
7
8
9
10
Expected μs’ (target)
Mean Value of recoveredμs’ (target)
μs’
Expected
LM: (0,0,0)
GLS: (0,0,0)
LM: (20,0,0)
GLS: (20,0,0)
FIG. 8. Comparison of recovered optical properties with respect to expected values in a spherical target located at ?0,0,0? and ?20,0,0? having a diameter of
25 mm using both LM and GLS reconstruction techniques. Observed standard deviations are of similar order to that reported in Table I.
1691 Yalavarthy et al.: 3D GLS estimation in diffuse optical tomography 1691
Medical Physics, Vol. 35, No. 5, May 2008
Page 11
numerical precision of the L2 norm value?. For the cases
considered here, the computation time for each iterative up-
date using Eq. ?16? was approximately 46 min, which was
three times faster than with Eq. ?9? ?computation time
?126 min per iteration?. In terms of operations count, a fac-
tor of 2 reduction would be estimated from Fig. 2?a? for the
NN/NM ratio involved. The deviation in run time that occurs
in practice is likely due to the cost of memory management
alluded to above in the case of Eq. ?9?. It is also important to
note that implementation of Eq. ?9? requires an inversion of
the covariance matrix ?Eq. ?10??, whereas this matrix can be
used directly in Eq. ?16?. In the case of LM update equations,
computation time for an iteration using Eq. ?6? was approxi-
mately 21 min and for Eq. ?5? was 91 min. The deviation
from factor of 6 ?from Fig. 2?b?? is mainly due to the
memory required to perform these operations; this affects run
time in turn.
The middle and last rows of Figs. 3?a? and 3?b? indicate
that with 1% noise in the data, LM has failed to recover ?ain
a spherical target with a diameter of 15 mm located at the
center, whereas GLS was able to identify the target very
well. The failure of LM minimization is indicative of a lack
of sensitivity at the center of the domain24which is improved
through the GLS approach by including the noise character-
istics and covariances associated with the problem. When the
same target is located near the boundary ?at ?30,0,0??, both
techniques were able to recover the contrast approximately
20% better relative to the center position ?Table I?.
When 1% noisy data generated from a centered cylindri-
cal target with a diameter of 15 mm ?first rows of Figs. 5?a?
and 5?b?? were used in the reconstruction, both LM and GLS
techniques were able to recover approximately 50% of the
expected contrast ?Table I?. For the same type of target lo-
cated near the boundary ?at ?30,0??, the recovery of contrast
was approximately 70% in the case of GLS. For LM, the
recovery was only 50% for ?aand 85% for ?s? under these
conditions. The reconstruction results also show that recov-
ery of the centered target is always poor relative to an object
near the boundary. This is primarily due to the hypersensi-
tivity at the boundary in these cases.24The extended cylin-
drical target is essentially equivalent to the two-dimensional
case of a circular inclusion and the trend observed in 3D of
recovering more contrast for a target near the boundary is
similar to the behavior found in 2D.24
When 5% noisy data were used, LM reconstruction
?middle rows of Figs. 5?a? and 5?b?? performs poorly in
terms of localization of the target, whereas GLS was able to
reconstruct optical images with better quality and quantita-
tion ?up to 70%?. Even though the reconstructed results using
very noisy data were presented here from only one type of
target, similar trends were also observed in other cases that
mimic the 2D reconstructions reported in Ref. 17. These re-
sults show that GLS outperforms LM even though the data
noise level is high because stability is retained by including
the noise characteristics into the weight matrices used for
normalization.
Accuracy in contrast recovery of local targets increases as
the size of the target increases, as shown in Fig. 7. For ex-
ample, the contrast recovered for a centered target below
20 mm in diameter is only about 30% of the true value and
as low as 10% for the LM algorithm ?Fig. 7?, whereas in-
creasing the size of target to 30 mm leads to quantitative
accuracy near 100%. The GLS approach provides maximal
contrast recovery and superior image quality at all sizes rela-
tive to LM ?example: Fig. 6?. Even when the target size is as
low as 5 mm ?Fig. 6?, the object was well localized in the
GLS case, but not with LM ?Fig. 6?.
The performance comparison of the algorithms in terms
of contrast recovery ?Fig. 8? confirms that the position of the
target dictates the response. When the target had 10:1 con-
trast in comparison to the background, the maximum recov-
ery of contrast was ?5:1. GLS outperformed LM in this
regard but there is a plateau in recovery of contrast at 400%
of the background value.
Table II shows that estimation parameter dependence
?cross-talk? is lower ?by 20%? for GLS compared to LM, by
reinforcing the independence of ?aand ?s? through elimina-
tion of any cross terms in the weight matrix W?−?0?Eq.
?15??. The inter-parameter dependence is complex because of
the non-unique relationship between the optical property dis-
tribution and the incomplete boundary data, indicating that
different formulations of the inversion tend to perform dif-
ferently. Nonetheless, the estimation parameter dependence
is substantially higher in 3D data-limited situations, relative
to when the ratio of data to number of estimates is less
skewed.
In the case of phantom data ?Fig. 10?, as expected, GLS
reconstructions showed more promising results in this test
case. Characterization of the data collection system, leading
to variance estimation depending on the voltage measure at
PMT ?Fig. 9? enabled the employment of the GLS technique
for experimental data reconstructions. Both techniques ?LM
and GLS? were able to give qualitative information about the
target, in terms of quantification the GLS technique over-
takes the LM technique ?Fig. 10?. It should be noted that this
−3−2.5−2−1.5 −1−0.50
2
4
6
Measured PMT voltage (V, in volts) in log scale
σ(V)/V (in %)
1/V2fitting
−3−2.5 −2−1.5 −1−0.50
0
1
2
Measured PMT voltage (V, in volts) in log scale
σ(θ) in degrees
FIG. 9. An error ?deviation, ?? plot of the measured voltage and phase ??? as
a function of mean of measured PMT voltage. The legend of the figure
represents the fitting model used. Each data point corresponds to a sample
size of 200.
1692 Yalavarthy et al.: 3D GLS estimation in diffuse optical tomography1692
Medical Physics, Vol. 35, No. 5, May 2008
Page 12
type of characterization of the experimental system does not
take into account coupling errors between the light
collection/delivery fiber and tissue surface. These kinds of
unsystematic errors are difficult to estimate as it depends on
many parameters, such as tissue surface roughness, tissue
elastic properties, design of tissue-fiber coupling interface,
repeatability, and alignment of fibers. Recent advances, such
as inclusion of coupling errors as a part of reconstruction
scheme by making them as unknowns and iteratively solving
for them along with optical parameters, are possible.34–37But
this makes the reconstruction problem more computationally
complex in nature as the number of unknowns of this proce-
dure are larger than the techniques considered here. As the
main focus of this article is about presenting a computation-
ally efficient algorithm for reconstructing optical parameters,
this issue was not considered as a part of this work. Devel-
opment of methods, which were presented as a part of this
work, that can include systematic noise characteristics in the
reconstruction procedure, will be taking a step in the right
direction. Moreover, as can be seen from Eq. ?22?, covari-
ances among the data points were ignored, making the data
weight matrix ?W?? a simple diagonal matrix. Inclusion of
covariances can offer a better weighting in case of experi-
mental data; this is under investigation as of now. Even
though this test case showed very promising results, in our
experience, in cases where coupling errors are dominant in
the data, the GLS scheme did not yield any meaningful re-
sults. In these cases, the LM technique was able to give
reasonable results.
Even though the data used here are generated by using
in-plane data, it was collected only from the source fiber
plane. Previous investigations indicate that the use of out-of-
plane data ?when the data were collected from rest of the
fibers in all three planes when one fiber was used as source?
may not give enough advantage in terms of reconstructed
image quality given an increase in the data-acquisition time
and computational cost compared to in-plane data ?in this
case the data were collected only from those fibers which lie
in the same plane as the source fiber?.24So for all the simu-
lation studies conducted here, in-plane data collection
scheme is used. The experimental phantom study uses only
one plane of data, as the phantom optical properties were not
varying in Z direction. Even though considering a experi-
mental phantom where the optical properties are varying in Z
direction might have been ideal to demonstrate the depth
resolution of reconstruction procedures, in here only a simple
case was considered as the main aim is to prove that the
techniques developed here can be used in experimental data
reconstruction. It is also important to note that the studies
conducted here are generic in nature, especially in terms of
proving the computational efficiency of alternative forms
?Fig. 2?, as NN/NM was changed over a range of 0.0028–
100 ?spanning from well-determined to highly under-
determined problems?.
Partial volume effects can be observed in the recovery of
contrast as a function of target size. The recovery of contrast
was much higher for the extended cylinder target compared
to the spherical inclusion ?Table I?. The quantitative accuracy
of reconstructed images increases with an increase in target
size ?Fig. 7?. GLS reconstruction results of the data from a
centered cylindrical object are encouraging, demonstrating
recovery of more than 30% contrast in this case ?other
Newton-type algorithms have reported a maximum of 10%
contrast recovery7,11,13,14,24?.
VIII. CONCLUSIONS
Three-dimensional diffuse optical tomography is more
computationally intensive because of the size of the param-
eter space to be reconstructed. Newton-based inversion
methods that operate on a Hessian matrix, which has dimen-
sions of the number of measurements rather than the number
of parameters, can be derived using the Sherman–Morrison–
Woodbury identity and become computationally more effi-
cient once the number of estimation parameters exceeds two
times the number of measurements. Representative examples
demonstrate that this form of update equation can be at least
six times faster in practice in the highly under-determined
(b)
(a)
FIG. 10. Actual and reconstructed ?a? ?aand ?b? ?s? distributions of a cy-
lindrical target using experimental multilayered phantom data. Two-
dimensional cross sections of the 3D volume in 2.5 mm increments span-
ning from z=−12.5 mm to z=12.5 mm ?from left to right? are shown.Actual
distributions are given in the first row. Reconstructed distribution using the
LM minimization scheme and GLS minimization scheme are presented in
the middle and last rows, respectively.
1693Yalavarthy et al.: 3D GLS estimation in diffuse optical tomography 1693
Medical Physics, Vol. 35, No. 5, May 2008
Page 13
problems which commonly occur in 3D. Three-dimensional
diffuse optical tomographic reconstruction algorithms also
suffer from partial volume effects that significantly degrade
the accuracy with which optical properties can be quantified.
The GLS approach which incorporates structured weight ma-
trices consisting of the variance and covariance of the data-
model misfit and the optical properties, improves the quanti-
fication of optical properties by at least 20% in 3D. The GLS
estimate is also robust to data noise as high as 5%—
conditions under which other algorithms fail when the prob-
lem is highly under-determined. By characterizing the detec-
tor noise for systematic errors, using a multi-layered gelatin
phantom data, the GLS technique can be easily employed for
reconstructing experimental data and can yield better quan-
tification of targets compared to conventional reconstruction
methods. Future investigations will include thorough exami-
nation of the GLS technique when applied to phantom and
clinical data and extension of the technique to direct-spectral
reconstruction. The test data used in this article, along with
computer algorithms, are available on a web page.38
ACKNOWLEDGMENTS
P.K.Y. acknowledges the DOD Breast Cancer predoctoral
fellowship ?BC050309?. D.R.L. acknowledges support of
NSF through Grant No. DMS-0417769. This work has been
sponsored by the National Cancer Institute through Grant
No. RO1CA78734 and PO1CA80139.
APPENDIX
1. Alternative form for GLS update equation
Before deriving the alternative form, it is useful to catalog
several properties of the weight matrices and their inverses28
W?= ?C??−1;W?−?0= ?C?−?0?−1
?W??T= W?;?W?−?0?T= W?−?0
?C??T= C?;?C?−?0?T= C?−?0.
?A1?
If a square matrix A has block form
A =?
JT
− C?
J
W?−?0?
?A2?
with dimensions ?2NM+2NN???2NM+2NN?, it is readily
shown to be symmetric by invoking the relationships in Eq.
?A1?:
?AT? =?
JT
=?
JT
− C?
J
W?−?0?
W?−?0?= A.
T
=?
− C?
JT
T
?JT?T
W?−?0
T ?
− C?
J
?A3?
Since inverses of both C?and W?−?0exist, then A−1also
exists and can be expressed in block form as well
A−1=?P Q
RS?
?A4?
in which case
AA−1=?
− C?
JT
J
W?−?0??P Q
RS?=?I 0
0 I?
?A5?
requires that
− C?P + JR= I
?A6?
− C?Q + JS= 0
?A7?
JTP + W?−?0R = 0
?A8?
JTQ + W?−?0S = I.
?A9?
These relationships can be manipulated through a series of
substitutions to express the blocks of A−1in terms of combi-
nations of the block components of A. Specifically, Eqs. ?A7?
and ?A8? along with the weight matrix properties in Eqs.
?A1? imply that
Q = W?JS
?A10?
and
R = − C?−?0JTP.
?A11?
Substituting Eq. ?A11? into Eq. ?A6? to form the expression
P = − ?C?+ JC?−?0JT?−1,
?A12?
which is put back into Eq. ?A11? produces
R = C?−?0JT?C?+ JC?−?0JT?−1.
?A13?
A similar series of steps starting with Eqs. ?A10? and ?A9? to
write
S = ?W?−?0+ JTW?J?−1,
?A14?
which is combined again with Eq. ?A10? yields
Q = W?J?W?−?0+ JTW?J?−1.
?A15?
Since A is symmetric and invertible, A−1is symmetric as
well.
?A−1?T=?P Q
R
QT
S?
T
=?PT
RT
ST?=?P Q
RS?= A−1?A16?
in which case
QT= R.
?A17?
Substituting the forms of Q and R ?Eqs. ?A15? and ?A13?,
respectively? into Eq. ?A17? results in
?W?J?W?−?0+ JTW?J?−1?T= C?−?0JT?C?+ JC?−?0JT?−1
?A18?
Equation ?A16? also requires ST=S, where S is given by Eq.
?A15?, which when identified in the term on the left side of
Eq. ?A18? allows it to be rewritten as
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Medical Physics, Vol. 35, No. 5, May 2008
Page 14
?W?−?0+ JTW?J?−1JTW?= C?−?0JT?C?+ JC?−?0JT?−1.
?A19?
Alternately,
A−1A =?P Q
RS?=?
− C?
JT
J
W?−?0?=?I 0
0 I?
?A20?
or
RJ+ SW?−?0= I.
?A21?
Solving for S
S = C?−?0− RJC?−?0
?A22?
and substituting Eqs. ?A13? and ?A14? for R and S in Eq.
?A22? produces
?W?−?0+ JTW?J?−1
= C?−?0− C?−?0JT?C?+ JC?−?0JT?−1JC?−?0.
Note that this derivation was adapted from Liebelt.39A vari-
ant of Eq. ?A23? exists in the literature with many names, the
most common being the Sherm–Morrison–Woodbury
identity.40–47It is also known as the matrix inversion
lemma.48,49Even though one can start from this equation
and derive the alternative forms, the complete derivation is
presented here for completeness.
Substituting Eq. ?A23? back into Eq. ?9? yields
??i= ?C?−?0− C?−?0JT?JC?−?0JT+ C??−1JC?−?0?
??JTW??i−1− W?−?0??i−1− ?0??
?A23?
?A24?
or
??i= ?I − C?−?0JT?JC?−?0JT+ C??−1J?
??C?−?0JTW??i−1− C?−?0W?−?0??i−1− ?0??
?A25?
which results in
??i= ?I − C?−?0JT?JC?−?0JT+ C??−1J?
??C?−?0JTW??i−1− ??i−1− ?0???A26?
as the alternative form for the update equation ?Eq. ?16??.
The next two subsections show the alternative forms of
other least-squares minimization techniques, namely LM and
Tikhonov minimizations.
2. Alternative form for LM update equation
The LM update equation ?Eq. ?5?? is a special case of the
GLS update equation ?Eq. ?9?? when W?−?0=?I and W?=I
?see Sec. III.V.4 in Ref. 17?. Using these forms in Eq. ?A19?
leads to an alternative form to Eq. ?5?
??i= ??I?−1JT?J??I?−1JT+ I−1?−1?i−1.
?A27?
Rearranging the terms in Eq. ?A27? leads to
??i=JT
??JJT+ ?I
??
−1
?i−1
?A28?
which can be simplified to produce
??i= JT?JJT+ ?I?−1?i−1.
Equation ?A29? is also known as under-determined form in
the literature.11,14
?A29?
3. Alternative form for Tikhonov update equation
The objective function17,50for the Tikhonov scheme is
? =?y − G????2+ ??L?? − ?0??2.
Minimization of Eq. ?A30? and linearizing the problem leads
to update equation17,31
??i= ?JTJ + ?LTL?−1?JT?i−1− ?LTL??i−1− ?0??.
Equation ?A31? is a special case of the GLS update equation
?Eq. ?9?? with weight matrices ?see Sec. III.B.4 in Ref. 17?
W?= I;W?−?0= ?LTL.
From Eq. ?A26? one can write
??i= ?I − ??LTL?−1JT?J??LTL?−1JT+ I−1?−1J?
????LTL?−1JTI?i−1− ??i−1− ?0??.
?A30?
?A31?
?A32?
?A33?
This leads to
??i= ?I − ?LTL?−1JT?J?LTL?−1JT+ ?I?−1J?
????LTL?−1JT?i−1− ??i−1− ?0??.
?A34?
Assuming ?i−1=?0, the single-step Tikhonov update equa-
tion ?or its equivalent?14,30,51becomes
??i= ?JTJ + ?LTL?−1JT?i−1.
Using Eqs. ?A19? and ?A32? leads to
??i= ??LTL?−1JT?J??LTL?−1JT+ I−1?−1?i−1,
?A35?
?A36?
which can be rearranged to
??i= ?LTL?−1JT?J?LTL?−1JT+ ?I?−1?i−1.
Equation ?A37? is also known as the under-determined
Tikhonov single-step update equation.11,14,16
?A37?
4. Calculation of number of operations for LM
and GLS update equations
The number of operations was estimated by assuming that
divisions/multiplications consume most of the processor
cycles. Note that Gaussian elimination was used in comput-
ing matrix inversion. Typically, Gaussian elimination for an
N?N matrix requires ??N3/3?+N2−?N/3?? operations.47
The memory required to invert a matrix of dimension N
?N is N2.47The number of operations only includes solution
of the update equation and does not account for the number
of operations required to form the matrices/vectors used in
these equations.
1695Yalavarthy et al.: 3D GLS estimation in diffuse optical tomography1695
Medical Physics, Vol. 35, No. 5, May 2008
Page 15
For the GLS update equation ?Eq. ?9??, the number of
operations required for iteration i is ?using the dimensions
defined after Eq. ?15??
Number of operations
=??2NM*2NM*2NN? + ?2NN*2NM*2NN?
+??2NN?
3
+ ?2NN?2−2NN
3??+ ?2NN*2NN*1?
+ ??2NN*2NM*1? + ?2NN*2NN*1?
+ ?2NN*2NN*1??.
?A38?
For the alternative form for the GLS update equation ?Eq.
?16??, the number of operations is
Number of operations
=??2NN*2NN*2NM?
+ ?2NM*2NN*2NM? + ?2NN*2NN*2NM?
+??2NM?3
3
+ ?2NM*2NM*2NN??+ ?2NN*2NN*1?
+ ?2NM?2−2NM
3?
+ ??2NN*2NN*2NM? + ?2NN*2NM*1?
+ ?2NM*2NM*1??.
?A39?
Similarly, for the LM update equation ?Eq. ?5??, the number
of operations required for iteration i is
Number of operations
=??2NN*2NM*2NN?
+??2NN?3
3
+ ?2NN?2−2NN
3??
+ ?2NN*2NM*1?.
?A40?
The number of operations for the alternative form for the LM
update equation ?Eq. ?6?? is
Number of operations
= ?2NN*2NM*1?
+??2NM*2NN*2NM?
+??2NM?3
3
+ ?2NM?2−2NM
3??
+ ?2NM*2NM*1?.
?A41?
Note that the computation time for these update equations
is not only dependent on the number of operations needed to
be performed, but also on the memory required for imple-
menting the operations.
a?Present address: Department of Radiation Oncology, Washington Univer-
sity School of Medicine, St. Louis, Missouri 63110.
b?Author to whom correspondence should be addressed. Electronic mail:
Pogue@dartmouth.edu
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