## Joint sparsity-driven non-iterative simultaneous reconstruction of absorption and scattering in diffuse optical tomography |

Optics Express, Vol. 21, Issue 22, pp. 26589-26604 (2013)

http://dx.doi.org/10.1364/OE.21.026589

Acrobat PDF (2469 KB)

### Abstract

Some optical properties of a highly scattering medium, such as tissue, can be reconstructed non-invasively by diffuse optical tomography (DOT). Since the inverse problem of DOT is severely ill-posed and nonlinear, iterative methods that update Green’s function have been widely used to recover accurate optical parameters. However, recent research has shown that the joint sparse recovery principle can provide an important clue in achieving reconstructions without an iterative update of Green’s function. One of the main limitations of the previous work is that it can only be applied to absorption parameter reconstruction. In this paper, we extended this theory to estimate the absorption and scattering parameters simultaneously when the background optical properties are known. The main idea for such an extension is that a joint sparse recovery step gives us unknown fluence on the estimated support set, which eliminates the nonlinearity in an integral equation for the simultaneous estimation of the optical parameters. Our numerical results show that the proposed algorithm reduces the cross-talk artifacts between the parameters and provides improved reconstruction results compared to existing methods.

© 2013 OSA

## 1. Introduction

1. A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain function,” Trends Neurosci. **20**, 435–442 (1997). [CrossRef] [PubMed]

2. D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. **18**, 57–75 (2001). [CrossRef]

3. D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. **20**, 469–477 (2000). [CrossRef] [PubMed]

5. R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med. **9**, 123–128 (2003). [CrossRef] [PubMed]

6. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**, R41–R93 (1999). [CrossRef]

7. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**, R1–R43 (2005). [CrossRef] [PubMed]

6. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**, R41–R93 (1999). [CrossRef]

8. V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. **26**, 893–895 (2001). [CrossRef]

9. V. A. Markel and J. C. Schotland, “Inverse problem in optical diffusion tomography. I. Fourier-Laplace inversion formulas,” J. Opt. Soc. Am. A **18**, 1336–1347 (2001). [CrossRef]

10. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency-domain data: Simulation and experiment,” J. Opt. Soc. Am. A **13**, 253–266 (1996). [CrossRef]

12. J. C. Ye, K. J. Webb, C. A. Bouman, and R. P. Millane, “Optical diffusion tomography using iterative coordinate descent optimization in a Bayesian framework,” J. Opt. Soc. Am. A **16**, 2400–2412 (1999). [CrossRef]

13. J. C. Ye, S. Y. Lee, and Y. Bresler, “Exact reconstruction formula for diffuse optical tomography using simultaneous sparse representation,” in *International Symposium on Biomedical Imaging: From Nano to Macro*, Paris, France (Institute of Electrical and Electronics Engineers, 2008), pp. 1621–1624.

14. O. K. Lee, J. M. Kim, Y. Bresler, and J. C. Ye, “Compressive diffuse optical tomography: Non-iterative exact reconstruction using joint sparsity,” IEEE Trans. Med. Imaging **30**, 1129–1142 (2011). [CrossRef] [PubMed]

15. J. Chen and X. Huo, “Theoretical results on sparse representations of multiple measurement vectors,” IEEE Trans. Signal Process. **54**, 4634–4643 (2006). [CrossRef]

16. J. M. Kim, O. K. Lee, and J. C. Ye, “Compressive MUSIC: Revisiting the link between compressive sensing and array signal processing,” IEEE Trans. Inf. Theory **58**, 278–301 (2012). [CrossRef]

13. J. C. Ye, S. Y. Lee, and Y. Bresler, “Exact reconstruction formula for diffuse optical tomography using simultaneous sparse representation,” in *International Symposium on Biomedical Imaging: From Nano to Macro*, Paris, France (Institute of Electrical and Electronics Engineers, 2008), pp. 1621–1624.

14. O. K. Lee, J. M. Kim, Y. Bresler, and J. C. Ye, “Compressive diffuse optical tomography: Non-iterative exact reconstruction using joint sparsity,” IEEE Trans. Med. Imaging **30**, 1129–1142 (2011). [CrossRef] [PubMed]

13. J. C. Ye, S. Y. Lee, and Y. Bresler, “Exact reconstruction formula for diffuse optical tomography using simultaneous sparse representation,” in *International Symposium on Biomedical Imaging: From Nano to Macro*, Paris, France (Institute of Electrical and Electronics Engineers, 2008), pp. 1621–1624.

14. O. K. Lee, J. M. Kim, Y. Bresler, and J. C. Ye, “Compressive diffuse optical tomography: Non-iterative exact reconstruction using joint sparsity,” IEEE Trans. Med. Imaging **30**, 1129–1142 (2011). [CrossRef] [PubMed]

*International Symposium on Biomedical Imaging: From Nano to Macro*, Paris, France (Institute of Electrical and Electronics Engineers, 2008), pp. 1621–1624.

**30**, 1129–1142 (2011). [CrossRef] [PubMed]

*International Symposium on Biomedical Imaging: From Nano to Macro*, Paris, France (Institute of Electrical and Electronics Engineers, 2008), pp. 1621–1624.

**30**, 1129–1142 (2011). [CrossRef] [PubMed]

## 2. Theory

### 2.1. Diffusion equation

*D*(

*r*) =

*D*

_{0}(

*r*) +

*δD*(

*r*),

*μ*(

_{a}*r*) =

*μ*

_{a0}(

*r*) +

*δ*

_{μa}(

*r*), and

*μ′*(

_{s}*r*) =

*μ′*

_{s0}(

*r*) +

*δμ′*(

_{s}*r*), where

*μ*

_{a0}(

*r*) and

*μ′*

_{s0}(

*r*) are the known background absorption and reduced scattering parameters, respectively.

*D*

_{0}(

*r*) = 1/3(

*μ′*

_{s0}(

*r*) +

*μ*

_{a0}(

*r*)) and

*δD*(

*r*),

*δ*

_{μa}(

*r*), and

*δμ′*(

_{s}*r*) refer to the unknown perturbations, which we need to estimate. Note that the known background optical parameter does not need to be constant, and our theory holds for any spatially varying known background. The diffusion equation can be then transformed as where the diffusion wave number is given by

*r′*denotes a position in volume Ω and the background Green’s function

*G*

_{0}(

*r*,

*r′*) satisfies and the incidence fluence

*u*

_{0}(

*r*) is given by Now, assuming that

*δD*(

*r*) is zero in the neighborhood around the boundary, we have

*u*(

*r′*) depends on the unknown optical parameter perturbation (

*δμ*,

_{a}*δD*).

### 2.2. Proposed method

*k*-th estimate of the background optical parameter, and [

*δμ*,

_{a}*δD*]

*denotes the unknown perturbation estimated from the*

^{T}*k*-th estimate of the background optical parameters. Note that the fluence

*u*(

*r′;x*) is linearized with respect to the background optical parameter

*x*

^{(}

^{k}^{)}. After the linearization, [

*δμ*,

_{a}*δD*]

*is estimated and we perform another linearization of*

^{T}*u*(

*r′;x*) at

*D*

^{(}

^{k}^{+1)}=

*D*

^{(}

^{k}^{)}+

*δD*. This procedure is repeated until convergence. It has been shown that the linearized integral equation corresponds to the Fréchet derivative and the DBIM is equivalent to the Levenberg-Marquardt algorithm [31

31. J. C. Ye, K. J. Webb, R. P. Millane, and T. J. Downar, “Modified distorted Born iterative method algorithm with an approximate Fréchet derivative for optical diffusion tomography,” J. Opt. Soc. Am. A **16**, 1814–1826 (1999). [CrossRef]

*represents areas where the absorption or scattering properties change. We assumed that the area Ω*

_{t}*is sparse compared to the total field of view (FOV) Ω. This is usually true in practice because, for example, tumors are only sparsely distributed. Furthermore, we assumed that there are*

_{t}*N*distinct illumination patterns and let

*u*(

*r;l*) be the optical fluence from the

*l*-th illumination pattern. Then, Eq. (3) can be rewritten as where

*X*(

*r;l*) = [

*δμ*(

_{a}*r*) − ∇ ·

*δD*(

*r*)∇]

*u*(

*r;l*) is the induced current from the

*l*-th illumination. In Eq. (9), the second equality comes from the following: Once the support Ω

*is estimated,*

_{t}*X*(

*r; l*) can easily be obtained by means of linear inversion from Eq. (9). Then, the unknown optical fluence

*u*(

*r; l*) can be calculated by restricting

*r*∈ Ω

*. More specifically, we have the following Foldy-Lax equation for fluence estimation, as derived from Eq. (9) by restricting*

_{t}*r*∈ Ω

*: Here,*

_{t}*X̂*(

*r;l*) denotes the estimated induced current for the

*l*-th illumination. Then, using

*û*(

*r;l*),

*δμ*and

_{a}*δD*can be reconstructed from the following linear integral equation: Note that the nonlinear coupling due to

*u*(

*r′;l*) in Eq. (7) becomes now linear because

*û*(

*r′;l*) and ∇

*û*(

*r′;l*) are calculated from Eq. (11). Note that the major difference from the conventional methods is that as the nonlinearly coupled optical fluence

*u*(

*r;l*) for

*r*∈ Ω

*is now estimated, the iterative procedure for updating Green’s function becomes unnecessary.*

_{t}*from multiple illumination patterns to obtain an accurate estimation of the unknown fluence*

_{t}*u*(

*r;l*). Another important issue is to estimate the optical parameters using Eq. (12). In our previous work [13

*International Symposium on Biomedical Imaging: From Nano to Macro*, Paris, France (Institute of Electrical and Electronics Engineers, 2008), pp. 1621–1624.

**30**, 1129–1142 (2011). [CrossRef] [PubMed]

*r*∈ Ω

*from the estimated*

_{t}*X̂*(

*r;l*) =

*δμ*(

_{a}*r*)

*û*(

*r;l*) and

*û*(

*r;l*) values for

*l*= 1, 2,··· ,

*N*. However, the presence of the perturbation in the scattering parameter inhibited the use of this step. In the following section, we describe how to address these issues.

### 2.3. Joint support recovery

*b*(

*r*,

*r*

_{(i)}) denotes the basis function. The 3-D locations

*k*-targets are selected from

*n*-possible locations

*k*≪

*n*. Then, we have the following relationship after substituting Eq. (13) into the induced current term in Eq. (9), where By collecting the measurement of Eq. (14) at detector positions

*N*illuminations, the imaging problem can be represented as the following matrix equation: In this equation,

*Y*∈ ℝ

^{m}^{×}

*is scattered fluence, and*

^{N}*A*∈ ℝ

^{m}^{×}

*and*

^{n}*X*∈ ℝ

^{n}^{×}

*are the sensing matrix and the induced current matrix, respectively. Let the component of the*

^{N}*i*-th row and

*j*-th column of matrix

*A*be defined as

*A*Then,

_{ij}*Y*=

_{ij}*u*

_{0}(

*r*

_{di};

*j*) −

*u*(

*r*

_{di};

*j*),

*A*=

_{ij}*G̃*

_{0}(

*r*

_{di},

*r*). Note that the rows of

_{j}*X*are only non-zero when

*row-diversity*measure ||

*X*||

_{0}that counts the number of rows in

*X*∈ ℝ

^{n}^{×}

*with nonzero elements. Then, the simultaneous reconstruction of the DOT problem from multiple illumination cases, where the support of the optical parameter variation is sparse, can be stated as the following joint sparse recovery problem [14*

^{N}**30**, 1129–1142 (2011). [CrossRef] [PubMed]

15. J. Chen and X. Huo, “Theoretical results on sparse representations of multiple measurement vectors,” IEEE Trans. Signal Process. **54**, 4634–4643 (2006). [CrossRef]

*X*||

*is the Frobenius norm of matrix*

_{F}*X*. Note that when

*δD*(

*r*

_{(i)}) = 0 for

*i*= 1, 2,··· ,

*k*, Eq. (17) becomes precisely the same joint sparse recovery problem in our previous work [13

*International Symposium on Biomedical Imaging: From Nano to Macro*, Paris, France (Institute of Electrical and Electronics Engineers, 2008), pp. 1621–1624.

**30**, 1129–1142 (2011). [CrossRef] [PubMed]

*δD*(

*r*

_{(i)}) ≠ 0, in terms of implementation, the support estimation step is identical to that of the previous work, since

*δD*is recovered later from the unknown

*X*.

15. J. Chen and X. Huo, “Theoretical results on sparse representations of multiple measurement vectors,” IEEE Trans. Signal Process. **54**, 4634–4643 (2006). [CrossRef]

16. J. M. Kim, O. K. Lee, and J. C. Ye, “Compressive MUSIC: Revisiting the link between compressive sensing and array signal processing,” IEEE Trans. Inf. Theory **58**, 278–301 (2012). [CrossRef]

32. M. E. Davies and Y. C. Eldar, “Rank awareness in joint sparse recovery,” IEEE Trans. Inf. Theory **58**, 1135–1146 (2012). [CrossRef]

16. J. M. Kim, O. K. Lee, and J. C. Ye, “Compressive MUSIC: Revisiting the link between compressive sensing and array signal processing,” IEEE Trans. Inf. Theory **58**, 278–301 (2012). [CrossRef]

33. S. F. Cotter, B. D. Rao, K. Engan, and K. Kreutz-Delgado, “Sparse solutions to linear inverse problems with multiple measurement vectors,” IEEE Trans. Signal Process. **53**, 2477–2488 (2005). [CrossRef]

36. J. M. Kim, O. K. Lee, and J. C. Ye, “Improving noise robustness in subspace-based joint sparse recovery,” IEEE Trans. Signal Process. **60**, 5799–5809 (2012). [CrossRef]

35. D. P. Wipf and B. D. Rao, “An empirical Bayesian strategy for solving the simultaneous sparse approximation problem,” IEEE Trans. Signal Process. **55**, 3704–3716 (2007). [CrossRef]

### 2.4. Optical parameter estimation

*û*(

*r*) at

_{i}*r*∈ Ω̂

_{i}*, the unknown optical parameters can be reconstructed by solving the following discretized linear inverse problem derived from Eq. (12): Here,*

_{t}*û*(

*r*

_{(j)};

*l*)

*δv*, and

*δv*is the volume of the discretized voxel. In addition,

*δμ*=

_{a,i}*δμ*(

_{a}*r*

_{(i)}) and

*δD*=

_{i}*δD*(

*r*

_{(i)}) denote the unknown optical parameter perturbations. Due to the restriction of the ROI to Ω̂

*, the number of unknowns in the discretized voxel domain is significantly reduced. However, the ill-posedness of the problem is still not completely overcome by the restriction since the ill-posedness of the DOT problem originates from the underlying continuous operator. In fact, it is well-known that the linear mapping from DOT is compact mapping, whose singular values are clustered around zero, which makes the inverse mapping very ill-posed. Therefore, to solve Eq. (18), we convert it into the following constrained optimization problem with*

_{t}*L*

_{1}and the total variation regularization under the assumption of the sparsity and smoothness of both absorption and scattering perturbations: In this equation,

*A*= [

*A*

_{μa},

*A*] ∈ ℝ

_{D}

^{mN}^{×2}

*, and ||*

^{k̂}*x*||

_{1}and ||

*x*||

*are*

_{TV}*L*

_{1}penalty and the total variation of

*x*[37

37. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision **20**, 89–97 (2004). [CrossRef]

*a*,

_{min}*a*] and [

_{max}*D*,

_{min}*D*] denote the ranges of the absorption and diffusion parameter variations, respectively.

_{max}## 3. Method

### 3.1. Implementation

*λ*is set to 1 for the proposed method, whereas it is reduced to 0.03 ∼ 0.05 for conventional methods since we found that the more weight in TV provides inaccurate absorption parameter reconstruction in the conventional methods. Therefore, we wanted to adjust the parameter to make the comparison fair. To solve the aforementioned constrained optimization problem, we exploited a constrained split augmented Lagrangian shrinkage algorithm (C-SALSA) [38

38. M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process. **20**, 681–695 (2011). [CrossRef]

*L*

_{1}penalty as in the C-SALSA algorithm [37

37. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision **20**, 89–97 (2004). [CrossRef]

38. M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process. **20**, 681–695 (2011). [CrossRef]

*A*= [

*A*

_{μa}

*, A*] ∈ ℝ

_{D}

^{mN}^{×2}

*, where*

^{n}*k*-th iteration. When

*k*= 1, we use the known background optical parameters as

*x*

^{(1)}= [

*μ*

_{a0},

*D*

_{0}]

*. However, note that the size of the sensing matrix for the proposed method is much smaller than that of the conventional methods due to the restriction of the problem within the estimated joint support. The pseudocode implementation of C-SALSA is described in Algorithm 1.*

^{T}_{τgi}is the Moreau proximal mapping of

*τg*[38

_{i}38. M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process. **20**, 681–695 (2011). [CrossRef]

*g*

_{1}(

*x*) = ||

*x*||

_{1},

*g*

_{2}(

*x*) = ||

*x*||

*,*

_{TV}*g*

_{3}=

*ι*

_{E}_{(}

_{ε,y}_{)}, and

*g*

_{4}=

*ι*

_{B}_{(}

_{a,D}_{)}.

*ι*

_{E}_{(}

_{ε,y}_{)}and

*ι*

_{B}_{(}

_{a,D}_{)}are the indicator functions of an

*ε*-radius Euclidean ball centered at

*y*and a Box-constraint which depends on the range of

*x*, respectively. As a stopping criterion, we used the relative change of the cost function in Eq. (19) and performed the algorithms until |(

*C*−

_{k}*C*

_{k}_{−1}) /

*C*| < 5 × 10

_{k}^{−4}is satisfied, where

*C*is the cost function at

_{k}*k*-th iteration. All simulation parameters are summarized in Table 1. In Table 1,

*ε*and

_{i}*y*are the values of

_{i}*ε*and measurements

*y*for the

*i*-th iteration of the DBIM method. The scale factor for

*ε*was forced to increase during iterations in the case of DBIM, since we found that a fixed value often makes the iteration not converge. These parameters were found manually for each method by choosing the optimal parameter among

*c*= {8, 4, 2, 1, 1/2, 1/4, 1/8} for

*c*= {0.01, 0.03, 0.05, 0.075, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6} for

*ε*

_{1}=

*c*||

*y*

_{1}||

_{2}, and we did an additional search for

*ε*

_{2}and

*ε*

_{3}from 0.7 ∼ 1.0 to make the DBIM converge. Since the sensing matrices for the proposed method and conventional methods are different, the selected optimal parameters are distinct, as shown in Table 1, for each method.

### 3.2. Simulation geometry I : simple targets

^{3}voxels and the region of interest (ROI) for reconstruction was 22 × 26 × 22mm

^{3}in size. The optical parameters for the background were set to

*μ*= 0.015mm

_{a}^{−1}, and the

*μ′*were set to be equal to 1mm

_{s}^{−1}. We conducted four simulation studies of this geometry, as described in Table 2, as CASE A to CASE D. In CASE A and CASE B, each target had optical perturbation in both absorption and scattering coefficients, at low and high contrast, respectively. On the other hand, in CASE C and CASE D, target 1 had only absorption changes, whereas target 2 had scattering changes. These cases were designed to evaluate the cross-talk artifacts [39

39. Y. Pei, H. L. Graber, and R. L. Barbour, “Normalized-constraint algorithm for minimizing inter-parameter crosstalk in DC optical tomography,” Opt. Express **9**, 97–109 (2001). [CrossRef] [PubMed]

40. Y. Xu, X. Gu, T. Khan, and H. Jiang, “Absorption and scattering images of heterogeneous scattering media can be simultaneously reconstructed by use of DC data,” Appl. Opt. **41**, 5427–5437 (2002). [CrossRef] [PubMed]

41. Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express **17**, 20178–20190 (2009). [CrossRef] [PubMed]

*dB*was added to the scattered fluence. The results were obtained by averaging 10 runs with independent noise realization for each case.

### 3.3. Simulation geometry II : mouse phantom

42. B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: A 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. **52**, 577–587 (2007). [CrossRef] [PubMed]

^{3}voxels. The ROI for reconstruction is the area between the source and detector with a size of 38 × 30 × 22mm

^{3}, where three tumors were placed, as illustrated in Fig. 3(b). Various optical parameters of the mouse organs with a wavelength of 800nm [43

43. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: A computer simulation feasibility study,” Phys. Med. Biol. **50**, 4225–4241 (2005). [CrossRef] [PubMed]

44. S. A. Prahl, “Optical Properties Spectra,” Oregon Medical Laser Clinic, 2001, http://omlc.ogi.edu/spectra/index.html.

41. Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express **17**, 20178–20190 (2009). [CrossRef] [PubMed]

*dB*was added to the scattered fluence. The results were also obtained by averaging 10 runs with independent noise realizations.

## 4. Results

*δD*, our practical interest is

*δμ′*. Therefore, we calculated

_{s}*δμ′*using the relationship of

_{s}*δμ*and

_{a}*δD*, as shown in the following: We also used the Hausdorff distance [45

45. D. P. Huttenlocher, G. A. Klanderman, and W. J. Rucklidge, “Comparing images using the Hausdorff distance,” IEEE Trans. Pattern Anal. Mach. Intell. **15**, 850–863 (1993). [CrossRef]

### 4.1. Simple target

*δμ*and

_{a}*δμ′*for each case (The horizontal and coronal sections centered on each target are indicated in the inset image in Figs. 4 and 5). Moreover, in the reconstruction images, our method reduced the cross-talk artifacts that were observed in the conventional methods, especially in the reconstructed scattering coefficients. The average reconstruction time of various methods for the simple target simulation are summarized in Table 6 (using a PC with CPU : core i7 sandy bridge and GPU : GTX 560 Geforce series). The support estimation step using the M-SBL algorithm was also added in the calculation of the run time of the proposed method. Even though the proposed method requires this additional step, the resulting computational time for the proposed method is even smaller compared to the linearized approach. In Table 6, we also describe the computation time for each part of the algorithm. Note that the total computation time includes the time required to construct the

_{s}*A*matrix, which was about 1.2 seconds.

### 4.2. Mouse phantom

45. D. P. Huttenlocher, G. A. Klanderman, and W. J. Rucklidge, “Comparing images using the Hausdorff distance,” IEEE Trans. Pattern Anal. Mach. Intell. **15**, 850–863 (1993). [CrossRef]

*δμ*and

_{a}*δμ′*(29 and 30), respectively. The conventional methods exhibited cross-talk between

_{s}*μ*and

_{a}*μ′*, whereas the proposed method did not have such artifacts.

_{s}## 5. Discussion

**30**, 1129–1142 (2011). [CrossRef] [PubMed]

## 6. Conclusion

## Acknowledgments

## References and links

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2. | D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. |

3. | D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. |

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13. | J. C. Ye, S. Y. Lee, and Y. Bresler, “Exact reconstruction formula for diffuse optical tomography using simultaneous sparse representation,” in |

14. | O. K. Lee, J. M. Kim, Y. Bresler, and J. C. Ye, “Compressive diffuse optical tomography: Non-iterative exact reconstruction using joint sparsity,” IEEE Trans. Med. Imaging |

15. | J. Chen and X. Huo, “Theoretical results on sparse representations of multiple measurement vectors,” IEEE Trans. Signal Process. |

16. | J. M. Kim, O. K. Lee, and J. C. Ye, “Compressive MUSIC: Revisiting the link between compressive sensing and array signal processing,” IEEE Trans. Inf. Theory |

17. | A. Profio and G. Navarro, “Scientific basis of breast diaphanography,” Med. Phys. |

18. | S. Fantini, S. A. Walker, M. A. Franceschini, M. Kaschke, P. M. Schlag, and K. T. Moesta, “Assessment of the size, position, and optical properties of breast tumors in vivo by noninvasive optical methods,” Appl. Opt. |

19. | A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt. |

20. | N. F. Boyd, J. W. Byng, R. A. Jong, E. K. Fishell, L. E. Little, A. B. Miller, G. A. Lockwood, D. L. Tritchler, and M. J. Yaffe, “Quantitative classification of mammographic densities and breast cancer risk: Results from the Canadian national breast screening study,” J. Natl. Cancer Inst. |

21. | J. W. Byng, M. J. Yaffe, R. A. Jong, R. S. Shumak, G. A. Lockwood, D. L. Tritchler, and N. F. Boyd, “Analysis of mammographic density and breast cancer risk from digitized mammograms,” Radiographics |

22. | B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, S. Srinivasan, X. Song, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Characterization of hemoglobin, water, and NIR scattering in breast tissue: Analysis of intersubject variability and menstrual cycle changes,” J. Biomed. Opt |

23. | M. O’Leary, D. Boas, B. Chance, and A. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett. |

24. | J. C. Hebden, F. E. W. Schmidt, M. E. Fry, M. Schweiger, E. M. C. Hillman, D. T. Delpy, and S. R. Arridge, “Simultaneous reconstruction of absorption and scattering images by multichannel measurement of purely temporal data,” Opt. Lett. |

25. | T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, “Initial studies of in vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Lett. |

26. | J. Wang, S. D. Jiang, Z. Z. Li, R. M. diFlorio Alexander, R. J. Barth, P. A. Kaufman, B. W. Pogue, and K. D. Paulsen, “In vivo quantitative imaging of normal and cancerous breast tissue using broadband diffuse optical tomography,” Med. Phys. |

27. | B. J. Hoenders, “Existence of invisible nonscattering objects and nonradiating sources,” J. Opt. Soc. Am. A |

28. | S. Arridge and W. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. |

29. | V. Markel and J. Schotland, “Inverse problem in optical diffusion tomography. II. Role of boundary conditions,” J. Opt. Soc. Am. A |

30. | L. V. Wang and H. Wu, |

31. | J. C. Ye, K. J. Webb, R. P. Millane, and T. J. Downar, “Modified distorted Born iterative method algorithm with an approximate Fréchet derivative for optical diffusion tomography,” J. Opt. Soc. Am. A |

32. | M. E. Davies and Y. C. Eldar, “Rank awareness in joint sparse recovery,” IEEE Trans. Inf. Theory |

33. | S. F. Cotter, B. D. Rao, K. Engan, and K. Kreutz-Delgado, “Sparse solutions to linear inverse problems with multiple measurement vectors,” IEEE Trans. Signal Process. |

34. | D. Malioutov, M. Cetin, and A. Willsky, “A sparse signal reconstruction perspective for source localization with sensor arrays,” IEEE Trans. Signal Process. |

35. | D. P. Wipf and B. D. Rao, “An empirical Bayesian strategy for solving the simultaneous sparse approximation problem,” IEEE Trans. Signal Process. |

36. | J. M. Kim, O. K. Lee, and J. C. Ye, “Improving noise robustness in subspace-based joint sparse recovery,” IEEE Trans. Signal Process. |

37. | A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision |

38. | M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process. |

39. | Y. Pei, H. L. Graber, and R. L. Barbour, “Normalized-constraint algorithm for minimizing inter-parameter crosstalk in DC optical tomography,” Opt. Express |

40. | Y. Xu, X. Gu, T. Khan, and H. Jiang, “Absorption and scattering images of heterogeneous scattering media can be simultaneously reconstructed by use of DC data,” Appl. Opt. |

41. | Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express |

42. | B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: A 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. |

43. | G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: A computer simulation feasibility study,” Phys. Med. Biol. |

44. | S. A. Prahl, “Optical Properties Spectra,” Oregon Medical Laser Clinic, 2001, http://omlc.ogi.edu/spectra/index.html. |

45. | D. P. Huttenlocher, G. A. Klanderman, and W. J. Rucklidge, “Comparing images using the Hausdorff distance,” IEEE Trans. Pattern Anal. Mach. Intell. |

**OCIS Codes**

(170.0170) Medical optics and biotechnology : Medical optics and biotechnology

(170.6960) Medical optics and biotechnology : Tomography

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: September 3, 2013

Revised Manuscript: October 9, 2013

Manuscript Accepted: October 17, 2013

Published: October 28, 2013

**Virtual Issues**

Vol. 9, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Okkyun Lee and Jong Chul Ye, "Joint sparsity-driven non-iterative simultaneous reconstruction of absorption and scattering in diffuse optical tomography," Opt. Express **21**, 26589-26604 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-26589

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