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2388 J. Opt. Soc. Am. A/Vol. 23, No. 10 /October 2006 Su et al.

Reconstruction method with data from amultiple-site continuous-wave source

for three-dimensional optical tomography

Jianzhong Su and Hua Shan

Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019

Hanli Liu

Department of Biomedical Engineering, University of Texas at Arlington, Arlington, Texas 76019

Michael V. Klibanov

Department of Mathematics and Statistics and Center for Optoelectronics, University of North Carolinaat Charlotte, Charlotte, North Carolina 28223

Received February 2, 2006; accepted March 31, 2006; posted May 4, 2006 (Doc. ID 67660)

A method is presented for reconstruction of the optical absorption coefficient from transmission near-infrareddata with a cw source. As it is distinct from other available schemes such as optimization or Newton’s iterativemethod, this method resolves the inverse problem by solving a boundary value problem for a Volterra-typeintegral-differential equation. It is demonstrated in numerical studies that this technique has a better thanaverage stability with respect to the discrepancy between the initial guess and the actual unknown absorptioncoefficient. The method is particularly useful for reconstruction from a large data set obtained from a CCDcamera. Several numerical reconstruction examples are presented. © 2006 Optical Society of America

OCIS codes: 100.3190, 000.4430, 110.6960, 110.7050, 170.3880.

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. INTRODUCTIONor the past one to two decades, research in both labora-ory and clinical settings has been conducted using near-nfrared (NIR) spectroscopy to noninvasively image bio-ogical tissues, particularly for the human breast and therain. Although the NIR imaging techniques have limita-ions in their spatial resolution and depth of light pen-tration, they still have a realistic potential to become aseful modality for providing functional images for cer-ain biomedical applications.

The NIR studies on the brain include detections ofrain injury and/or trauma,1 determination of cerebrovas-ular hemodynamics and oxygenation,2,3 and functionalrain imaging in response to a variety of neurologicalctivations.4,5 Particularly, NIR functional brain imagingas become of great interest for studying hemodynamicesponses to brain activation in the past few years.6 Thiss mainly because the optical signals of the NIR tech-iques are sensitive to changes in the concentration ofxygenated hemoglobin (HbO) and deoxygenated hemo-lobin (Hb).

Various efforts on NIR breast and brain imaging haveeen made by several research groups7–12 in either labo-atory or clinical studies. For example, frequency-domainFD) breast imagers have been developed, and there haveeen reports of in vivo results of optical properties of ab-ormalities from female volunteers and patients.13 Onhe other hand, because of their simplicity and low cost inomparison with the FD imaging systems, cw NIR breast

1084-7529/06/102388-8/$15.00 © 2

maging systems have also been developed by Barbour etl.9 and Chance.14 Since this paper is not intended to be aeview paper, we cite only a limited number of referencesnd encourage potential readers to look in referencesited in these publications for a more advanced search.

To spatially quantify light absorption and reduced scat-ering coefficients from the NIR measurements, one needso extract these quantities from mathematical models.ince these physical properties are described by coeffi-ients in the corresponding partial differential equation,ne needs to solve an inverse problem based on that equa-ion.

In computations of forward problems, the most com-only used theoretical model is photon diffusion theory,15

hich is a partial differential equation with respect toime and spatial variables. Photon diffusion theory de-cribes the photon propagation in tissue and predicts theeasurements at the detector positions. It is well known

hat diffusion theory can simulate well the transport pro-ess of photons that migrate through tissue,16 as long ashe detectors and light sources are sufficiently far apartgreater than 5 mm). Numerical methods, such as the fi-ite difference method or finite element method (FEM),an be utilized to solve the diffusion equation numerically,epending on the exact shape and boundary conditions ofhe sample under study.

Unlike the forward problem, the inverse reconstructionf the absorption and scattering coefficient distributionas some major mathematical challenges. The inverse

006 Optical Society of America

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Su et al. Vol. 23, No. 10 /October 2006 /J. Opt. Soc. Am. A 2389

roblem in this case is nonlinear, as well as ill-posed. Aseviewed by Hielscher et al.,17 the majority of the inverseeconstruction algorithms used for NIR tomographic im-ging utilizes a perturbation approach involving the in-ersion of large Jacobian matrices. (For example, seechottland et al.,18 O’Leary et al.,19 Gryazin et al.,20 andielscher et al.17). Other commonly applied optimization

echniques include conjugated gradient descent (CGD)nd simultaneous algebraic reconstruction techniques21

SARTs) as well as their infinite-dimensionaleneralization.22 Furthermore, to deal with the ill-osedness, various regularization techniques are often ac-ompanied by the optimization algorithms since regular-zation makes the highly ill-conditioned Jacobian matrix

ore diagonally dominant.23

One common point of these mathematical schemes ishe trial and error manner in searching the true distribu-ion. Typically one needs to provide an initial guess andhen calculates the forward problem. If the residual costunctional is not zero, one proceeds to find an “improved”uess based on the previous information and then repeatshe process. Because of the highly heterogeneous naturef tissues in the breast and brain, an inverse algorithmor NIR tomographic reconstruction can be costly in com-uting and may even provide a false solution, as shown inmathematical example given in Ref. 24 that the re-

idual cost functional can possess a cluster of localinima near the true solution, which is the absoluteinimum.Furthermore, advances in CCD technology allow simul-

aneous measures at multiple sites of tissues and thus re-uire new computational capability for processing a hugeata set, such as 512�512 data points, at a single mea-urement. In a two-dimensional-reconstruction problem,he work of Gryazin et al.20 used an integral-differentialquation approach, close to our method, but was only ainear approximation by the perturbation method.

In this paper, we have developed an improved versionf the integral-differential equation method in three spa-ial dimensions. While a similar idea was proposedarlier,20 it was applied only to a linearized problem.ompared with Ref. 20, the major development in this pa-er is that we solve the full nonlinear integral-differentialquations. Unlike other previous techniques, we obtaineconstructed images using the forward problem ap-roach, not minimizing a residual cost functional. Our nu-erical experiments indicate that this method has a bet-

er than average stability property with respect to theifference between the initial guess and the actual un-nown coefficient. This method is particularly useful for

mage reconstruction from a large data set, such as im-ges obtained from a CCD camera, because of the forwardroblem nature of our method. Numerical examples of theeconstruction of the absorption coefficient will be pre-ented below.

. PHYSICAL SETTING OF THEIMULATIONSur method is for a setting of optical transmission data

econstruction. The simulated phantom is suspended in aectangular container 3.5 cm�8 cm�9 cm (in the x ,y ,z

irections, respectively) filled with a lipid solution. Theimensions of the transparent container are large enoughor a phantom study of a rat’s head. An NIR, cw lightource is simulated from the left-hand side of the con-ainer, and a CCD camera is located at the right-handide of the container.

The simulated procedure of measurement is straight-orward. We follow an idea similar to that in frequencyounding20,24 but replace the changing frequency by mov-ng the simulated light source along a straight line (seeig. 1). Theoretically, there should be as many source po-itions as possible, and they should be densely distrib-ted. However, our numerical experiments will indicatehat five positions are adequate to obtain a reasonable re-onstruction image. The relative convenience of the mea-urement procedure in an actual experiment is one of thedvantages of this method.

. MATHEMATICAL MODELor the diffusion equation, we have

· �D�x,y,z� � w�x,y,z�� − �a�x,y,z�w�x,y,z�

= − ��x,− r,y,z�, �1�

here D= 13�s�, �a and �s� are the absorption and reduced

cattering coefficients, respectively, in the tissue, and�x ,y ,z� is the photon fluence rate or photon density. In

his paper the location of light source, r, runs along the xxis for convenience, but there are other ways to move theight source. Assume now that the light source position isut of our computational domain, which is defined as theectangular container of size 3.5 cm�8 cm�9 cm. Withinhe domain, we can replace the � function in Eq. (1) withero. We make a change of variable from w�x ,y ,z� to�x ,y ,z� as u=ln w�x ,y ,z�+ 1

2 ln D�x ,y ,z�. Then Eq. (1) be-omes a nonlinear elliptic equation:

�u�x,y,z� + �u�x,y,z� · �u�x,y,z� − a�x,y,z� = 0, �2�

here a�x ,y ,z� is a new unknown coefficient, defined as

a�x,y,z� =1

2��ln D�x,y,z��

+1

4� �ln D�x,y,z�� · ��ln D�x,y,z�� +

�a�x,y,z�

D�x,y,z�.

�3�

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2390 J. Opt. Soc. Am. A/Vol. 23, No. 10 /October 2006 Su et al.

By moving the light source to different r locations, wentend to eliminate the dependency of Eq. (2) on the un-nown coefficient a�x ,y ,z�. Notice that both Eq. (2) andts solution u=u�x ,y ,z ,r� have an additional parameter r,here r is the location of the light source, but the un-nown coefficient a�x ,y ,z� is independent of r. We differ-ntiate Eq. (2) with respect to the source location r, so weransform the original inverse problem into a forwardroblem for a system of integral-differential equations. Asill be shown below, by this method we will be able to find

he distribution of a�x ,y ,z�, which is a function of distri-ution of �a�x ,y ,z� and �s��x ,y ,z� as described in Eq. (3).Let v=�u /�r, −��r��, and u−��x ,y ,z� be the limiting

olution, as the source location r=r1�0 is far away. Theimiting value u−��x ,y ,z�, even though without an asymp-ote in analytic form, has a limit numerically from obser-ation of its level curves when r=r1�0 is sufficientlyegative. Then

u�x,y,z,r� =�r1

r

v�x,y,z,��d� + u−��x,y,z�.

fter substituting v=�u /�r and the above equation intoq. (2), the function v satisfies

�v�x,y,z,r� + 2�v�x,y,z,r� · ���r1

r

v�x,y,z,��d�

+ u−��x,y,z�� = 0. �4�

Once we solve Eq. (4) for v, then we can integrate v tond u. Theoretically, with solved u, Eq. (2) will then give

nversion for the coefficient a�x ,y ,z�. Since Eq. (2) gives aather unstable numerical scheme for calculation of�x ,y ,z�, we utilize a finite-element-method (FEM) ver-ion of Eq. (1) in our inversion.

The boundary conditions (BCs) for function w are stan-ard. We assume that the Robin conditions for the lightource side of the container are valued as �f; i.e., the pho-on density at the surface of the container. The functionf=�f �x=0,y ,z ,r� varies with the location r of lightource. Also we assume homogeneous Robin conditionslsewhere in the boundaries so that

D�� �w

�n − �a�w = �f, at x = 0; D�� �w

�x − �a�w

= 0, others.

he BCs for functions u and v are formulated accordinglyrom the BCs for w. Also, see the end of Section 3 for theverimposed BC for function v.

Guided by our numerical results, we proceed to use fiveource positions in this reconstruction, which yields mea-urements of v at four midpoint locations. By having dis-rete Eq. (4) using trapezoid formula for integral with rri, i=1,2,3,4 that are equally spaced by �r, one obtainssystem of four elliptic partial differential equations:

�v�x,y,z,r � + 2�v�x,y,z,r � · ��u �x,y,z�� = 0, i = 1,

i i −�

�v�x,y,z,ri� + 2�v�x,y,z,ri�

���1

2v�x,y,z,rl��r +

j=2

i−1

v�x,y,z,rj��r

+1

2v�x,y,z,ri��r + u−��x,y,z�� = 0, i = 2,3,4. �5�

The corresponding BC will be replaced by a discreteersion of the BC for v. There is an overimposed BC fromhe measurement of photon density on the back side ofhe container (located at x=3.5 cm) as v= �� /�r��f�, wheref=ln w�x ,y ,z�+ 1

2 ln D�x ,y ,z� at x=3.5 cm. We approxi-ate our problem by replacing the Robin condition at x3.5 cm with the measurement data, rather than includ-

ng both BCs on the same side in an overimposed bound-ry value problem.

. NUMERICAL METHODe solve the system of elliptic partial differential equa-

ions in Eq. (5) iteratively. In the kth linear loop �k1,2, . . . �, we solve the linear equation:

�v�k��x,y,z,ri� + 2�v�k��x,y,z,ri� · ��u−��x,y,z�� = 0, i = 1,

�v�k��x,y,z,ri� + 2�v�k��x,y,z,ri�

���1

2v�x,y,z,rl��r +

j=2

i−1

v�x,y,z,rj��r +1

2v�k−1�

��x,y,z,ri��r + u−��x,y,z�� = 0, i = 2,3,4. �6�

quation (6) is solved by the FEM approach (weak solu-ion). For given known values of v�k−1�, one solves for v�k�

hrough

��� �v�k��x,y,z,ri� � + 2�v�k��x,y,z,ri�

� � �u−��x,y,z�� = 0, i = 1,

��� �v�k��x,y,z,ri� � + 2�v�k��x,y,z,ri�

���1

2v�x,y,z,rl��r +

j=2

i−1

v�x,y,z,rj��r +1

2v�k−1�

��x,y,z,ri��r + u−��x,y,z�� = 0, i = 2,3,4, �7�

here the test function vanishes at boundary. Both theolution v and the test function are represented by qua-ratic elements. In our numerical experiments, the itera-ions of Eq. (6) converge very quickly for a number ofhoices for v�0��x ,y ,z ,ri�. In our first two numerical ex-mples below, we uniformly set our procedure as follows:ake a�x ,y ,z�=a0�x ,y ,z�=constant, and calculate the for-ard problem to derive v�0��x ,y ,z ,ri�. In our third ex-mple, we take a�x ,y ,z�=a �x ,y ,z� that contains 10%–

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Su et al. Vol. 23, No. 10 /October 2006 /J. Opt. Soc. Am. A 2391

0% error from the actual distribution and then calculate�0��x ,y ,z ,ri� similarly.

We have observed in our numerical experiments that�k� converges to v in Eq. (6) after iterations. The solutionwill satisfy the nonlinear Eq. (5). We begin with the case

=1, which is decoupled from other equations. Once theinear loop converges and the solution v�x ,y ,z ,r1� is ob-ained, we then proceed to solve for i=2,3,4. After we de-ive the value of v, we integrate using the formulas to de-ive the values of u�x ,y ,z ,ri� and w�x ,y ,z ,ri� accordingly.

Next, we proceed to the computation of the coefficient�x ,y ,z�. Since the differentiation of an approximatelyiven function is not a stable procedure, especially com-utation of second derivatives, Eq. (2) gives a rather un-table numerical scheme for calculation of a�x ,y ,z�. In allf our problems, D�x ,y ,z�=D0 is constant, and then Eq.3) gives a�x ,y ,z�= ��a�x ,y ,z�� /D0. Hence, we utilize aEM version of Eq. (1) in our inversion as

��� � · �w�x,y,z,ri� − a�x,y,z�w�x,y,z�dV = 0,

�8�

here the test function are chosen to be quadratic ele-ents that are zero at boundaries. The coefficient

�x ,y ,z� derived in this manner is much smoother thanhe one directly from Eq. (2). Specifically, we approximaten the interior of the domain

a�x,y,z� = l=1

N

all,

nd in Eq. (8), we take the test function =l�x ,y ,z�, l1,2, . . . ,N, where the l’s are taken over all quadraticlements. Then, Eq. (8) becomes a system of linear equa-ions for al, l=1,2, . . . ,N. We reconstruct a�x ,y ,z� from al,=1,2, . . . ,N by solving the system numerically. After weeconstruct each of the functions a�x ,y ,z� based on�x ,y ,z ,ri� for i=1,2,3,4, the final reconstructed�x ,y ,z� is the average of the four.Finally, there is an important question of how to nu-erically approximate u−��x ,y ,z�, which we name the tail

unction. Initially, for convenience, we approximate theunction u−��x ,y ,z� as the solution of the forward problemith a�x ,y ,z��a0�x ,y ,z�. Then once we have calculated

he inverse problem using this initial tail function to de-ive a new a�x ,y ,z�, we proceed to calculate the forwardroblem again to get a better u−��x ,y ,z� and repeat theteration using the updated a�x ,y ,z�.

. NUMERICAL RESULTSe have conducted our inverse calculation using theethod for a rectangular box of 3.5 cm�8 cm�9 cm. The

ight source is from the left-hand side of the box, and theamera is located at the right-hand side (Fig. 1). In allhree examples, we have used an ideal light source mod-led by a formula at 0.5–1.5 cm away from the box.

We have used the range of parameters D and �a thatre typical for biological tissues. In our initial example,he coefficients are D=0.02 cm uniformly, and �a0.1 cm−1 at all grids except at one grid location, where

a=1.0 cm−1. Figure 2 shows the grid used in solving thenverse problem for diffusion tomography equation. A to-al of 7�20�20 elements are used.

Our algorithm calculates the forward problem first,iven the distribution of absorption coefficient with RobinCs. An initial guess of uniform a�x ,y ,z�=0.1 cm−1 every-here is used for the calculation of tail function. Figures(a) and 3(b) show the actual and reconstructed distribu-ions of �a coefficient within the grids, respectively. Theifferences between the true and the reconstructed coeffi-ients are within a relative maximum error of 16.8%. Theelative maximum error is obtained by comparing maxi-um values of the true and the reconstruction coeffi-

ients. Red (online color) denotes the largest value of theeconstructed coefficient.

We next used the same algorithm to test a case of aimulated phantom by placing a light-absorbing rod em-edded in a turbid medium. The purpose of this simula-ion is to test the algorithm in a problem of a realistichysical dimension. The dimension of the absorbing rodas 1 mm in diameter and 3 cm in length suspended at

he center of the turbid medium. The coefficients are D0.02 cm uniformly, and �a=0.1 cm−1 at all places exceptt the location of the rod where � =1.0 cm−1.

Fig. 2. (Color online) Finite element mesh.

ig. 3. (Color online) Comparison of the reconstructed and thectual data. (a) Actual absorption distribution in the grid; the ab-orption coefficient �a=1 inside the target region and �a=0.1therwise. The color of the absorbing object is to show the loca-ion and is not scaled to show quantitative absorption coefficient.b) Reconstructed absorption distribution (shown in cross sec-ion). The peak �a value inside target region reaches 0.832. Theolor is scaled to show quantitative absorption coefficient.

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2392 J. Opt. Soc. Am. A/Vol. 23, No. 10 /October 2006 Su et al.

Numerical reconstruction has demonstrated that theeconstruction schemes are able to effectively calculatehe location of the rod by reconstructing the absorptionoefficients (see Fig. 4). The relative maximum error be-ween the reconstruction and actual coefficient is 17.6%.ote in the first two examples, we used a higher standard

or relative error estimate, because we are interested inbserving the peak value of reconstructions. The relativems errors (a more common standard value) are muchower in these two cases.

Finally, we test our algorithm on a case with a highlyomplex distribution of absorption coefficient in 3D. Thectual absorption distribution a�x ,y ,z� is randomly gen-rated as follows. We have a background of D=0.02 cmniformly, and �a=0.1 cm−1 except for two regions (one

nside a large sphere and one inside a small sphere). Inhese two regions, we define �a to be a cosine function ofhe distance to the center of sphere plus a white noise,nd the cosine function levels off when it reaches 0.1.

hat is, we have �a�x ,y ,z�=max�cos�d�x ,y ,z�+W� ,0.1 here d�x ,y ,z� is that distance, and W is the white noise.The initial guess starts with a distribution that is a lin-

ar combination of 90% of actual distribution �a and 2%hite noise error from the actual distribution given by0�x ,y ,z�=actual a�x ,y ,z��0.9+0.02 W����. The whiteoise W��� is valued from −1 to 1 in an equal distribution.he calculation requires 15�30�30 elements (shown inig. 5), and other conditions are the same as those given

n the previous examples. The results are shown in Figs. 6nd 7. Numerical experiments were also done with initialrrors of 20% and 30% in the initial guess and a similarevel of white noise (2%). The reconstructions are illus-

Fig. 5. (Color online) Three-dimensional finite element mesh.

ig. 4. (Color online) Comparison of the reconstructed and thectual data. (a) Actual absorption distribution in the grid. Thebsorption coefficient �a=1 inside the target region and �a=0.1therwise. The color of the absorbing object is to show the loca-ion and is not scaled to show quantitative absorption coefficient.b) Reconstructed absorption distribution (shown in cross sec-ion). The peak �a value inside target region reaches 0.824. Theolor is scaled to show quantitative absorption coefficient.

ig. 6. Comparison between the reconstructed data and actual data. (a) Actual absorption distribution; the color is scaled to show quan-itative absorption coefficient. (b) Reconstructed data with 90% initial value and 2% white noise, namely, a=a0�0.9+0.02W� with a0the original data, and W=white noise between −1 and 1. The color is scaled to show quantitative absorption coefficient.

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Su et al. Vol. 23, No. 10 /October 2006 /J. Opt. Soc. Am. A 2393

rated in Figs. 8(a) and 8(b), respectively. With an in-reased order of error in the initial guess, the numericalrrors from lengthy calculation lead to a larger deviationn the reconstruction. However, even for the 30% initialuess [Fig. 8(b)], we still achieved a reconstruction with aelative rms error around 12%, well within a commonlycceptable range of 15%. Table 1 shows a comparison ofelative errors for the reconstructed data. In comparisonith other available schemes, perturbation approachessually can “tolerate” no more than 10%–15% of such de-iations in initial guesses, while our approach achievesp to 30% tolerance.To further demonstrate the robustness of our inverse

alculation, we used the boundary values with noise arti-cially added. In the boundary values, i.e., the w valuesbtained from the measurements of light density at x3.5 cm, we added 2% white noise to the data, replacing�x ,y ,z� by w�x ,y ,z��1+0.02 W����. The white noise W���

s scaled from −1 to 1 in an equal distribution. The recon-truction is presented in Fig. 8(c), and the relative rms er-

ig. 7. Comparison of the actual and the reconstructed data onorption distribution. (b) Reconstructed absorption.

ig. 8. Reconstructed data with disturbed initial value by nowhite noise between −1 and 1. The color is scaled to show quan

c) r1=0.8, r2=0.02, and 2% noise is used on the detection side.

or is 15.7% in this case. Note that the error is attributedo boundary errors near one corner. An artifact is visiblen Fig. 8(c), but the main region of interest is recon-tructed very well.

. DISCUSSIONhere are a number of advantages of our new develop-ent, in comparison with the known methods (Newton’sethod or optimization schemes). It is agreed that the

atter is an adequate inversion reconstruction tool for suc-essful reconstruction using limited source-detector pairs,uch as 8�8 channels, even for 32�32 channels. A largeret of 512�512 output data from a CCD camera poses areat challenge for optimization schemes. Our method isased on an approach to solving a forward problem (a sys-em of elliptic partial differential equations), where nu-erical methods are more mature for large-scale compu-

ations. We have demonstrated in a series of numericalxperiments [Figs. 6, 8(a), and 8(b)] that our numerical

ss section at x=0.025 through their level curves. (a) Actual ab-

the form of a=a0�r1+r2W� with a0=the original data, and We absorption coefficient. (a) r1=0.8, r2=0.02; (b) r1=0.7, r2=0.02;

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2394 J. Opt. Soc. Am. A/Vol. 23, No. 10 /October 2006 Su et al.

econstruction technique “tolerates” up to 30% differencesetween the first guess and the actual absorption coeffi-ient. Such tolerance is better than those obtained inany perturbation algorithms, which usually can tolerate

nly up to 10%–15% differences.This work is a preliminary step toward our goal of de-

eloping a method of globally convergent reconstructionGCR). A substantial additional modification of thisethod (namely, adding Carleman weight functions24)

hould lead to a GCR algorithm, i.e., a method with aathematical assurance for the convergence of the solu-

ion with respect to any initial guess for the absorptionistribution.24 In a typical optical tomography reconstruc-ion problem, the error functional may have many localinima (i.e., false solutions) along a ravine.24 Most avail-

ble algorithms have been locally convergent and needather accurate knowledge of the background medium inrder to be used as a starting point of an iterativecheme.24 On the other hand, in imaging of biological tis-ues, such as a rat’s brain where the background mediums heterogeneous and there is a wide range of variationsmong individual rats, there is no good initial guess avail-ble. Therefore, it is highly desirable to develop a possiblepproach for GCR.Indeed, the global convergence has been mathemati-

ally proven recently by Klibanov and Timonov.24 Its con-ergence is built on the fact that the error functional be-omes strictly convex after adding Carleman weightunctions, and therefore the minimum is unique. Al-hough in this paper with three-dimensional numericalxperiments we have not yet introduced Carleman weightunctions, we still have seen a broader domain of conver-ence as evidenced by our numerical examples. The keyngredient for such a broader convergence is that only theolterra integral is presented in Eq. (4), and such inte-rals are known for good convergence properties. We plano rigorously study the convergence property of theethod given in the current paper in a future paper. We

Table 1. Comparison of Relative Errors for theReconstructed Dataa

Fig. 6(b) Fig. 8(a) Fig. 8(b) Fig. 8(c)

nitial guesswith respectto the trueabsorptionvalue

90% 80% 70% 80%

nitial whitenoise

2% 2% 2% 2%

oundarynoise

none none none 2%

umber ofiterations forupdating thetail function

5–10 5–10 5–10 5–10

elative rms 0.0725 0.0996 0.121 0.157

aBackground has a0=0.1 cm−1, and the absorbing object has a maximum value of

0=1.0 cm−1 in the original data near the centers of two spheres. The rms error isomputed as �rms= �1/Npi=1

Np �a�xi ,yi ,zi�−a0�xi ,yi ,zi��2 1/2, where �xi ,yi ,zi� is thepatial coordinate of the ith grid node, Np is total number of grid nodes, and a0 is theriginal absorption distribution. The relative rms is calculated as

rms/maxi=1. . .,Np�a0�xi ,yi ,zi��.

lso intend to investigate numerically the GCR method24

nder the setting of this paper.In theory, we need to perform infinite measurements at

ifferent light source locations and then solve Eq. (4). Inractice, our numerical results suggest that five measure-ents of photon density are sufficient to obtain a good-

uality image of reconstruction. This is another advan-age of the presented method. It is conceivable that theore measurements are taken, the more stable or regu-

arized the reconstructions are for these ill-posed prob-ems. Thus, we expect that our method will work moretably if additional cameras are equipped to simulta-eously take measurements from different directions.his expectation needs to be further tested in our future

nvestigations.

CKNOWLEDGMENTS. Liu acknowledges support in part from the National

nstitutes of Health under grants 1R21CA101098-02 andR33CA101098-03. The work of M. Klibanov was sup-orted in part by the U.S. Army Research Laboratory and.S. Army Research Office under contract W911NF-05-1-378.J. Su’s e-mail address is su@uta.edu.

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