Reconstruction method with data from a multiple-site ... - OSA Publishing

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J. Opt. Soc. Am. A / Vol. 23, No. 10 / October 2006

Su et al.

Reconstruction method with data from a multiple-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 Carolina at 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-infrared data with a cw source. As it is distinct from other available schemes such as optimization or Newton’s iterative method, this method resolves the inverse problem by solving a boundary value problem for a Volterra-type integral-differential equation. It is demonstrated in numerical studies that this technique has a better than average stability with respect to the discrepancy between the initial guess and the actual unknown absorption coefficient. The method is particularly useful for reconstruction from a large data set obtained from a CCD camera. Several numerical reconstruction examples are presented. © 2006 Optical Society of America OCIS codes: 100.3190, 000.4430, 110.6960, 110.7050, 170.3880.

1. INTRODUCTION For the past one to two decades, research in both laboratory and clinical settings has been conducted using nearinfrared (NIR) spectroscopy to noninvasively image biological tissues, particularly for the human breast and the brain. Although the NIR imaging techniques have limitations in their spatial resolution and depth of light penetration, they still have a realistic potential to become a useful modality for providing functional images for certain biomedical applications. The NIR studies on the brain include detections of brain injury and/or trauma,1 determination of cerebrovascular hemodynamics and oxygenation,2,3 and functional brain imaging in response to a variety of neurological activations.4,5 Particularly, NIR functional brain imaging has become of great interest for studying hemodynamic responses to brain activation in the past few years.6 This is mainly because the optical signals of the NIR techniques are sensitive to changes in the concentration of oxygenated hemoglobin (HbO) and deoxygenated hemoglobin (Hb). Various efforts on NIR breast and brain imaging have been made by several research groups7–12 in either laboratory or clinical studies. For example, frequency-domain (FD) breast imagers have been developed, and there have been reports of in vivo results of optical properties of abnormalities from female volunteers and patients.13 On the other hand, because of their simplicity and low cost in comparison with the FD imaging systems, cw NIR breast 1084-7529/06/102388-8/$15.00

imaging systems have also been developed by Barbour et al.9 and Chance.14 Since this paper is not intended to be a review paper, we cite only a limited number of references and encourage potential readers to look in references cited in these publications for a more advanced search. To spatially quantify light absorption and reduced scattering coefficients from the NIR measurements, one needs to extract these quantities from mathematical models. Since these physical properties are described by coefficients in the corresponding partial differential equation, one needs to solve an inverse problem based on that equation. In computations of forward problems, the most commonly used theoretical model is photon diffusion theory,15 which is a partial differential equation with respect to time and spatial variables. Photon diffusion theory describes the photon propagation in tissue and predicts the measurements at the detector positions. It is well known that diffusion theory can simulate well the transport process of photons that migrate through tissue,16 as long as the detectors and light sources are sufficiently far apart (greater than 5 mm). Numerical methods, such as the finite difference method or finite element method (FEM), can be utilized to solve the diffusion equation numerically, depending on the exact shape and boundary conditions of the sample under study. Unlike the forward problem, the inverse reconstruction of the absorption and scattering coefficient distribution has some major mathematical challenges. The inverse © 2006 Optical Society of America

Su et al.

problem in this case is nonlinear, as well as ill-posed. As reviewed by Hielscher et al.,17 the majority of the inverse reconstruction algorithms used for NIR tomographic imaging utilizes a perturbation approach involving the inversion of large Jacobian matrices. (For example, see Schottland et al.,18 O’Leary et al.,19 Gryazin et al.,20 and Hielscher et al.17). Other commonly applied optimization techniques include conjugated gradient descent (CGD) and simultaneous algebraic reconstruction techniques21 (SARTs) as well as their infinite-dimensional generalization.22 Furthermore, to deal with the illposedness, various regularization techniques are often accompanied by the optimization algorithms since regularization makes the highly ill-conditioned Jacobian matrix more diagonally dominant.23 One common point of these mathematical schemes is the trial and error manner in searching the true distribution. Typically one needs to provide an initial guess and then calculates the forward problem. If the residual cost functional is not zero, one proceeds to find an “improved” guess based on the previous information and then repeats the process. Because of the highly heterogeneous nature of tissues in the breast and brain, an inverse algorithm for NIR tomographic reconstruction can be costly in computing and may even provide a false solution, as shown in a mathematical example given in Ref. 24 that the residual cost functional can possess a cluster of local minima near the true solution, which is the absolute minimum. Furthermore, advances in CCD technology allow simultaneous measures at multiple sites of tissues and thus require new computational capability for processing a huge data set, such as 512⫻ 512 data points, at a single measurement. In a two-dimensional-reconstruction problem, the work of Gryazin et al.20 used an integral-differential equation approach, close to our method, but was only a linear approximation by the perturbation method. In this paper, we have developed an improved version of the integral-differential equation method in three spatial dimensions. While a similar idea was proposed earlier,20 it was applied only to a linearized problem. Compared with Ref. 20, the major development in this paper is that we solve the full nonlinear integral-differential equations. Unlike other previous techniques, we obtain reconstructed images using the forward problem approach, not minimizing a residual cost functional. Our numerical experiments indicate that this method has a better than average stability property with respect to the difference between the initial guess and the actual unknown coefficient. This method is particularly useful for image reconstruction from a large data set, such as images obtained from a CCD camera, because of the forward problem nature of our method. Numerical examples of the reconstruction of the absorption coefficient will be presented below.

2. PHYSICAL SETTING OF THE SIMULATIONS Our method is for a setting of optical transmission data reconstruction. The simulated phantom is suspended in a rectangular container 3.5 cm⫻ 8 cm⫻ 9 cm (in the x , y , z

Vol. 23, No. 10 / October 2006 / J. Opt. Soc. Am. A

Fig. 1.

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Light source and domain.

directions, respectively) filled with a lipid solution. The dimensions of the transparent container are large enough for a phantom study of a rat’s head. An NIR, cw light source is simulated from the left-hand side of the container, and a CCD camera is located at the right-hand side of the container. The simulated procedure of measurement is straightforward. We follow an idea similar to that in frequency sounding20,24 but replace the changing frequency by moving the simulated light source along a straight line (see Fig. 1). Theoretically, there should be as many source positions as possible, and they should be densely distributed. However, our numerical experiments will indicate that five positions are adequate to obtain a reasonable reconstruction image. The relative convenience of the measurement procedure in an actual experiment is one of the advantages of this method.

3. MATHEMATICAL MODEL For the diffusion equation, we have ⵜ · 关D共x,y,z兲 ⵜ w共x,y,z兲兴 − ␮a共x,y,z兲w共x,y,z兲 = − ␦共x,− r,y,z兲,

共1兲

where D = 31 ␮⬘s , ␮a and ␮⬘s are the absorption and reduced scattering coefficients, respectively, in the tissue, and w共x , y , z兲 is the photon fluence rate or photon density. In this paper the location of light source, r, runs along the x axis for convenience, but there are other ways to move the light source. Assume now that the light source position is out of our computational domain, which is defined as the rectangular container of size 3.5 cm⫻ 8 cm⫻ 9 cm. Within the domain, we can replace the ␦ function in Eq. (1) with zero. We make a change of variable from w共x , y , z兲 to u共x , y , z兲 as u = ln w共x , y , z兲 + 21 ln D共x , y , z兲. Then Eq. (1) becomes a nonlinear elliptic equation: ⌬u共x,y,z兲 + ⵜu共x,y,z兲 · ⵜu共x,y,z兲 − a共x,y,z兲 = 0,

共2兲

where a共x , y , z兲 is a new unknown coefficient, defined as 1 a共x,y,z兲 = ⌬关ln D共x,y,z兲兴 2 1 +

4

ⵜ 关ln D共x,y,z兲兴 · ⵜ关ln D共x,y,z兲兴 +

␮a共x,y,z兲 D共x,y,z兲

.

共3兲

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By moving the light source to different r locations, we intend to eliminate the dependency of Eq. (2) on the unknown coefficient a共x , y , z兲. Notice that both Eq. (2) and its solution u = u共x , y , z , r兲 have an additional parameter r, where r is the location of the light source, but the unknown coefficient a共x , y , z兲 is independent of r. We differentiate Eq. (2) with respect to the source location r, so we transform the original inverse problem into a forward problem for a system of integral-differential equations. As will be shown below, by this method we will be able to find the distribution of a共x , y , z兲, which is a function of distribution of ␮a共x , y , z兲 and ␮⬘s 共x , y , z兲 as described in Eq. (3). Let v = ⳵u / ⳵r, −⬁ ⬍ r ⬍ ⬁, and u−⬁共x , y , z兲 be the limiting solution, as the source location r = r1 ⬍ 0 is far away. The limiting value u−⬁共x , y , z兲, even though without an asymptote in analytic form, has a limit numerically from observation of its level curves when r = r1 ⬍ 0 is sufficiently negative. Then u共x,y,z,r兲 =

冕

r

r1

After substituting v = ⳵u / ⳵r and the above equation into Eq. (2), the function v satisfies

冋冕

⳵w ⳵n

− ␮a⬘ w = ␾f,

at x = 0;

2

i−1

v共x,y,z,rl兲⌬r +

兺 v共x,y,z,r 兲⌬r j

j=2

册

i = 2,3,4. 共5兲

The corresponding BC will be replaced by a discrete version of the BC for v. There is an overimposed BC from the measurement of photon density on the back side of the container (located at x = 3.5 cm) as v = 共⳵ / ⳵r兲共␺f兲, where ␺f = ln w共x , y , z兲 + 21 ln D共x , y , z兲 at x = 3.5 cm. We approximate our problem by replacing the Robin condition at x = 3.5 cm with the measurement data, rather than including both BCs on the same side in an overimposed boundary value problem.

We solve the system of elliptic partial differential equations in Eq. (5) iteratively. In the kth linear loop 共k = 1 , 2 , . . . 兲, we solve the linear equation: ⌬v共k兲共x,y,z,ri兲 + 2⌬v共k兲共x,y,z,ri兲 · ⵜ关u−⬁共x,y,z兲兴 = 0,

i = 1,

⌬v共k兲共x,y,z,ri兲 + 2⌬v共k兲共x,y,z,ri兲

v共x,y,z, ␶兲d␶

共4兲

Once we solve Eq. (4) for v, then we can integrate v to find u. Theoretically, with solved u, Eq. (2) will then give inversion for the coefficient a共x , y , z兲. Since Eq. (2) gives a rather unstable numerical scheme for calculation of a共x , y , z兲, we utilize a finite-element-method (FEM) version of Eq. (1) in our inversion. The boundary conditions (BCs) for function w are standard. We assume that the Robin conditions for the light source side of the container are valued as ␾f; i.e., the photon density at the surface of the container. The function ␾f = ␾f 共x = 0 , y , z , r兲 varies with the location r of light source. Also we assume homogeneous Robin conditions elsewhere in the boundaries so that

冉冊

1

1 + v共x,y,z,ri兲⌬r + u−⬁共x,y,z兲 = 0, 2

r1

册

冋

r

+ u−⬁共x,y,z兲 = 0.

D⬘

⫻ⵜ

4. NUMERICAL METHOD

v共x,y,z, ␶兲d␶ + u−⬁共x,y,z兲.

⌬v共x,y,z,r兲 + 2⌬v共x,y,z,r兲 · ⵜ

⌬v共x,y,z,ri兲 + 2⌬v共x,y,z,ri兲

D⬙

冉冊 ⳵w ⳵x

− ␮a⬙ w

= 0, others. The BCs for functions u and v are formulated accordingly from the BCs for w. Also, see the end of Section 3 for the overimposed BC for function v. Guided by our numerical results, we proceed to use five source positions in this reconstruction, which yields measurements of v at four midpoint locations. By having discrete Eq. (4) using trapezoid formula for integral with r = ri, i = 1 , 2 , 3 , 4 that are equally spaced by ⌬r, one obtains a system of four elliptic partial differential equations: ⌬v共x,y,z,ri兲 + 2⌬v共x,y,z,ri兲 · ⵜ关u−⬁共x,y,z兲兴 = 0,

i = 1,

⫻ⵜ

冋

1 2

i−1


1

兺 v共x,y,z,r 兲⌬r + 2 v j

j=2

册

⫻共x,y,z,ri兲⌬r + u−⬁共x,y,z兲 = 0,

共k−1兲

i = 2,3,4.

共6兲

Equation (6) is solved by the FEM approach (weak solution). For given known values of v共k−1兲, one solves for v共k兲 through

冕冕冕

ⵜv共k兲共x,y,z,ri兲 ⵜ ␩ + 2ⵜv共k兲共x,y,z,ri兲

⫻ ⵜ 关u−⬁共x,y,z兲兴␩ = 0,

冕冕冕

i = 1,

ⵜv共k兲共x,y,z,ri兲 ⵜ ␩ + 2ⵜv共k兲共x,y,z,ri兲

⫻ⵜ

冋

1 2

i−1


1

兺 v共x,y,z,r 兲⌬r + 2 v j

j=2

册

⫻共x,y,z,ri兲⌬r + u−⬁共x,y,z兲 ␩ = 0,

共k−1兲

i = 2,3,4,

共7兲

where the test function ␩ vanishes at boundary. Both the solution v and the test function ␩ are represented by quadratic elements. In our numerical experiments, the iterations of Eq. (6) converge very quickly for a number of choices for v共0兲共x , y , z , ri兲. In our first two numerical examples below, we uniformly set our procedure as follows: take a共x , y , z兲 = a0共x , y , z兲 = constant, and calculate the forward problem to derive v共0兲共x , y , z , ri兲. In our third example, we take a共x , y , z兲 = a0共x , y , z兲 that contains 10%–

Su et al.


30% error from the actual distribution and then calculate v共0兲共x , y , z , ri兲 similarly. We have observed in our numerical experiments that v共k兲 converges to v in Eq. (6) after iterations. The solution v will satisfy the nonlinear Eq. (5). We begin with the case i = 1, which is decoupled from other equations. Once the linear loop converges and the solution v共x , y , z , r1兲 is obtained, we then proceed to solve for i = 2 , 3 , 4. After we derive the value of v, we integrate using the formulas to derive the values of u共x , y , z , ri兲 and w共x , y , z , ri兲 accordingly. Next, we proceed to the computation of the coefficient a共x , y , z兲. Since the differentiation of an approximately given function is not a stable procedure, especially computation of second derivatives, Eq. (2) gives a rather unstable numerical scheme for calculation of a共x , y , z兲. In all of our problems, D共x , y , z兲 = D0 is constant, and then Eq. (3) gives a共x , y , z兲 = 关␮a共x , y , z兲兴 / D0. Hence, we utilize a FEM version of Eq. (1) in our inversion as

冕冕冕

Fig. 2.

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(Color online) Finite element mesh.

ⵜ␩ · ⵜw共x,y,z,ri兲 − a共x,y,z兲w共x,y,z兲␩dV = 0, 共8兲

where the test function ␩ are chosen to be quadratic elements that are zero at boundaries. The coefficient a共x , y , z兲 derived in this manner is much smoother than the one directly from Eq. (2). Specifically, we approximate in the interior of the domain N

a共x,y,z兲 =

兺a␩ , l l

l=1

and in Eq. (8), we take the test function ␩ = ␩l共x , y , z兲, l = 1 , 2 , . . . , N, where the ␩l’s are taken over all quadratic elements. Then, Eq. (8) becomes a system of linear equations for al, l = 1 , 2 , . . . , N. We reconstruct a共x , y , z兲 from al, l = 1 , 2 , . . . , N by solving the system numerically. After we reconstruct each of the functions a共x , y , z兲 based on w共x , y , z , ri兲 for i = 1 , 2 , 3 , 4, the final reconstructed a共x , y , z兲 is the average of the four. Finally, there is an important question of how to numerically approximate u−⬁共x , y , z兲, which we name the tail function. Initially, for convenience, we approximate the function u−⬁共x , y , z兲 as the solution of the forward problem with a共x , y , z兲 ⬅ a0共x , y , z兲. Then once we have calculated the inverse problem using this initial tail function to derive a new a共x , y , z兲, we proceed to calculate the forward problem again to get a better u−⬁共x , y , z兲 and repeat the iteration using the updated a共x , y , z兲.

5. NUMERICAL RESULTS We have conducted our inverse calculation using the method for a rectangular box of 3.5 cm⫻ 8 cm⫻ 9 cm. The light source is from the left-hand side of the box, and the camera is located at the right-hand side (Fig. 1). In all three examples, we have used an ideal light source modeled by a formula at 0.5– 1.5 cm away from the box. We have used the range of parameters D and ␮a that are typical for biological tissues. In our initial example, the coefficients are D = 0.02 cm uniformly, and ␮a = 0.1 cm−1 at all grids except at one grid location, where

Fig. 3. (Color online) Comparison of the reconstructed and the actual data. (a) Actual absorption distribution in the grid; the absorption coefficient ␮a = 1 inside the target region and ␮a = 0.1 otherwise. The color of the absorbing object is to show the location and is not scaled to show quantitative absorption coefficient. (b) Reconstructed absorption distribution (shown in cross section). The peak ␮a value inside target region reaches 0.832. The color is scaled to show quantitative absorption coefficient.

␮a = 1.0 cm−1. Figure 2 shows the grid used in solving the inverse problem for diffusion tomography equation. A total of 7 ⫻ 20⫻ 20 elements are used. Our algorithm calculates the forward problem first, given the distribution of absorption coefficient with Robin BCs. An initial guess of uniform a共x , y , z兲 = 0.1 cm−1 everywhere is used for the calculation of tail function. Figures 3(a) and 3(b) show the actual and reconstructed distributions of ␮a coefficient within the grids, respectively. The differences between the true and the reconstructed coefficients are within a relative maximum error of 16.8%. The relative maximum error is obtained by comparing maximum values of the true and the reconstruction coefficients. Red (online color) denotes the largest value of the reconstructed coefficient. We next used the same algorithm to test a case of a simulated phantom by placing a light-absorbing rod embedded in a turbid medium. The purpose of this simulation is to test the algorithm in a problem of a realistic physical dimension. The dimension of the absorbing rod was 1 mm in diameter and 3 cm in length suspended at the center of the turbid medium. The coefficients are D = 0.02 cm uniformly, and ␮a = 0.1 cm−1 at all places except at the location of the rod where ␮a = 1.0 cm−1.

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Numerical reconstruction has demonstrated that the reconstruction schemes are able to effectively calculate the location of the rod by reconstructing the absorption coefficients (see Fig. 4). The relative maximum error between the reconstruction and actual coefficient is 17.6%. Note in the first two examples, we used a higher standard for relative error estimate, because we are interested in observing the peak value of reconstructions. The relative rms errors (a more common standard value) are much lower in these two cases. Finally, we test our algorithm on a case with a highly complex distribution of absorption coefficient in 3D. The actual absorption distribution a共x , y , z兲 is randomly generated as follows. We have a background of D = 0.02 cm uniformly, and ␮a = 0.1 cm−1 except for two regions (one inside a large sphere and one inside a small sphere). In these two regions, we define ␮a to be a cosine function of the distance to the center of sphere plus a white noise, and the cosine function levels off when it reaches 0.1.

Fig. 4. (Color online) Comparison of the reconstructed and the actual data. (a) Actual absorption distribution in the grid. The absorption coefficient ␮a = 1 inside the target region and ␮a = 0.1 otherwise. The color of the absorbing object is to show the location and is not scaled to show quantitative absorption coefficient. (b) Reconstructed absorption distribution (shown in cross section). The peak ␮a value inside target region reaches 0.824. The color is scaled to show quantitative absorption coefficient.

Su et al.

That is, we have ␮a共x , y , z兲 = max兵cos关d共x , y , z兲 + W兴 , 0.1其 where d共x , y , z兲 is that distance, and W is the white noise. The initial guess starts with a distribution that is a linear combination of 90% of actual distribution ␮a and 2% white noise error from the actual distribution given by a0共x , y , z兲 = actual a共x , y , z兲关0.9+ 0.02 W共␻兲兴. The white noise W共␻兲 is valued from −1 to 1 in an equal distribution. The calculation requires 15⫻ 30⫻ 30 elements (shown in Fig. 5), and other conditions are the same as those given in the previous examples. The results are shown in Figs. 6 and 7. Numerical experiments were also done with initial errors of 20% and 30% in the initial guess and a similar level of white noise (2%). The reconstructions are illus-

Fig. 5.

(Color online) Three-dimensional finite element mesh.

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

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Fig. 7. Comparison of the actual and the reconstructed data on a cross section at x = 0.025 through their level curves. (a) Actual absorption distribution. (b) Reconstructed absorption.

Fig. 8. Reconstructed data with disturbed initial value by noise in the form of a = a0共r1 + r2W兲 with a0 = the original data, and W = white noise between −1 and 1. The color is scaled to show quantitative absorption coefficient. (a) r1 = 0.8, r2 = 0.02; (b) r1 = 0.7, r2 = 0.02; (c) r1 = 0.8, r2 = 0.02, and 2% noise is used on the detection side.

trated in Figs. 8(a) and 8(b), respectively. With an increased order of error in the initial guess, the numerical errors from lengthy calculation lead to a larger deviation in the reconstruction. However, even for the 30% initial guess [Fig. 8(b)], we still achieved a reconstruction with a relative rms error around 12%, well within a commonly acceptable range of 15%. Table 1 shows a comparison of relative errors for the reconstructed data. In comparison with other available schemes, perturbation approaches usually can “tolerate” no more than 10%–15% of such deviations in initial guesses, while our approach achieves up to 30% tolerance. To further demonstrate the robustness of our inverse calculation, we used the boundary values with noise artificially added. In the boundary values, i.e., the w values obtained from the measurements of light density at x = 3.5 cm, we added 2% white noise to the data, replacing w共x , y , z兲 by w共x , y , z兲关1 + 0.02 W共␻兲兴. The white noise W共␻兲 is scaled from −1 to 1 in an equal distribution. The reconstruction is presented in Fig. 8(c), and the relative rms er-

ror is 15.7% in this case. Note that the error is attributed to boundary errors near one corner. An artifact is visible in Fig. 8(c), but the main region of interest is reconstructed very well.

6. DISCUSSION There are a number of advantages of our new development, in comparison with the known methods (Newton’s method or optimization schemes). It is agreed that the latter is an adequate inversion reconstruction tool for successful reconstruction using limited source-detector pairs, such as 8 ⫻ 8 channels, even for 32⫻ 32 channels. A larger set of 512⫻ 512 output data from a CCD camera poses a great challenge for optimization schemes. Our method is based on an approach to solving a forward problem (a system of elliptic partial differential equations), where numerical methods are more mature for large-scale computations. We have demonstrated in a series of numerical experiments [Figs. 6, 8(a), and 8(b)] that our numerical

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Table 1. Comparison of Relative Errors for the Reconstructed Dataa

Initial guess with respect to the true absorption value Initial white noise Boundary noise Number of iterations for updating the tail function Relative rms a

Fig. 6(b)

Fig. 8(a)

Fig. 8(b)

Fig. 8(c)

90%

80%

70%

80%

2%

2%

2%

2%

none

none

none

2%

5–10

5–10

5–10

5–10

0.0725

0.0996

0.121

0.157

−1

Background has a0 = 0.1 cm , and the absorbing object has a maximum value of a0 = 1.0 cm−1 in the original data near the centers of two spheres. The rms error is Np computed as ⑀rms = 兵1 / N p兺i=1 关a共xi , y i , zi兲 − a0共xi , y i , zi兲兴2其1/2, where 共xi , y i , zi兲 is the spatial coordinate of the ith grid node, N p is total number of grid nodes, and a0 is the original absorption distribution. The relative rms is calculated as ⑀rms / maxi=1. . .,Np关a0共xi , y i , zi兲兴.

reconstruction technique “tolerates” up to 30% differences between the first guess and the actual absorption coefficient. Such tolerance is better than those obtained in many perturbation algorithms, which usually can tolerate only up to 10%–15% differences. This work is a preliminary step toward our goal of developing a method of globally convergent reconstruction (GCR). A substantial additional modification of this method (namely, adding Carleman weight functions24) should lead to a GCR algorithm, i.e., a method with a mathematical assurance for the convergence of the solution with respect to any initial guess for the absorption distribution.24 In a typical optical tomography reconstruction problem, the error functional may have many local minima (i.e., false solutions) along a ravine.24 Most available algorithms have been locally convergent and need rather accurate knowledge of the background medium in order to be used as a starting point of an iterative scheme.24 On the other hand, in imaging of biological tissues, such as a rat’s brain where the background medium is heterogeneous and there is a wide range of variations among individual rats, there is no good initial guess available. Therefore, it is highly desirable to develop a possible approach for GCR. Indeed, the global convergence has been mathematically proven recently by Klibanov and Timonov.24 Its convergence is built on the fact that the error functional becomes strictly convex after adding Carleman weight functions, and therefore the minimum is unique. Although in this paper with three-dimensional numerical experiments we have not yet introduced Carleman weight functions, we still have seen a broader domain of convergence as evidenced by our numerical examples. The key ingredient for such a broader convergence is that only the Volterra integral is presented in Eq. (4), and such integrals are known for good convergence properties. We plan to rigorously study the convergence property of the method given in the current paper in a future paper. We

also intend to investigate numerically the GCR method24 under the setting of this paper. In theory, we need to perform infinite measurements at different light source locations and then solve Eq. (4). In practice, our numerical results suggest that five measurements of photon density are sufficient to obtain a goodquality image of reconstruction. This is another advantage of the presented method. It is conceivable that the more measurements are taken, the more stable or regularized the reconstructions are for these ill-posed problems. Thus, we expect that our method will work more stably if additional cameras are equipped to simultaneously take measurements from different directions. This expectation needs to be further tested in our future investigations.

ACKNOWLEDGMENTS H. Liu acknowledges support in part from the National Institutes of Health under grants 1R21CA101098-02 and 4R33CA101098-03. The work of M. Klibanov was supported in part by the U.S. Army Research Laboratory and U.S. Army Research Office under contract W911NF-05-10378. J. Su’s e-mail address is [email protected].

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