## Improvement of image quality of time-domain diffuse optical tomography with
_{p} |

Biomedical Optics Express, Vol. 2, Issue 12, pp. 3334-3348 (2011)

http://dx.doi.org/10.1364/BOE.2.003334

Acrobat PDF (2804 KB)

### Abstract

An *l _{p}* (0 <

*p*≤ 1) sparsity regularization is applied to time-domain diffuse optical tomography with a gradient-based nonlinear optimization scheme to improve the spatial resolution and the robustness to noise. The expression of the

*l*sparsity regularization is reformulated as a differentiable function of a parameter to avoid the difficulty in calculating its gradient in the optimization process. The regularization parameter is selected by the L-curve method. Numerical experiments show that the

_{p}*l*sparsity regularization improves the spatial resolution and recovers the difference in the absorption coefficients between two targets, although a target with a small absorption coefficient may disappear due to the strong effect of the

_{p}*l*sparsity regularization when the value of

_{p}*p*is too small. The

*l*sparsity regularization with small

_{p}*p*values strongly localizes the target, and the reconstructed region of the target becomes smaller as the value of

*p*decreases. A phantom experiment validates the numerical simulations.

© 2011 OSA

## 1. Introduction

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

2. 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]

3. D. Grosenick, H. Wabnitz, H. H. Rinneberg, T. Moesta, and P. M. Schlag, “Development of a time-domain optical mammography and first *in vivo* applications,” Appl. Opt. **38**(13), 2927–2943 (1999). [CrossRef]

7. T. Yates, C. Hebdan, A. Gibson, N. Everdell, S. R. Arridge, and M. Douek, “Optical tomography of the breast using a multi-channel time-resolved imager,” Phys. Med. Biol. **50**, 2503–2517 (2005). [CrossRef] [PubMed]

8. A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S.R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three-dimensional whole-head optical tomography for passive motor evoked responses in the neonate,” NueroImage **30**, 521–528 (2006). [CrossRef]

9. J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. **47**, 4155–4166 (2002) [CrossRef] [PubMed]

10. B. W. Pogue, T. O. McBride, J. Prewitt, U. Lösterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. **38**(13), 2950–2961, (1999). [CrossRef]

11. G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas, “Quantitative spectroscopic optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. **50**, 3941–3956 (2005). [CrossRef] [PubMed]

12. P. K. Yalavarthy, B. W. Pogue, H. Dehghani, C. M. Carpenter, S. Jiang, and K. D. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express **15**(13), 8043–8058, (2007). [CrossRef] [PubMed]

13. A. Douiri, M. Schweiger, J. Riley, and S. R. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol. **18**, 87–95 (2007). [CrossRef]

14. P. Hiltunen, D. Calvetti, and E. Somersalo, “An adaptive smoothness regularization algorithm for optical tomography,” Opt, Express **16**(24), 19957–19977, (2008). [CrossRef]

15. C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. A **26**(5), 1277–1290 (2009). [CrossRef]

*L*

_{1}norm minimization by use of an expectation maximization algorithm for a linearized DOT inverse problem, and show that the reconstructed region with abnormal optical properties are localized more than the other methods they use [16

16. N. Cao, A. Nehorai, and M. Jacob, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express , **15**(21), 13695–13708 (2007). [CrossRef] [PubMed]

*L*

_{1}norm and other methods to obtain sparse solution are used to several applications of inverse problems. Image restoration with sparsity constrained regularization is proposed by Shankar et al [17

17. P. M. Shankar and M. A. Neifeld, “Sparsity constrained regularization for multiframe image restoration,” J. Opt. Soc. Am. A **25**(5), 1199–1214 (2008). [CrossRef]

*L*

_{1}sparsity constraint is applied to fluorescence/bioluminescence diffuse optical tomography (F/BDOT) [18

18. P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt. **46**(10), 1679–1685 (2007). [CrossRef] [PubMed]

19. Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express **17**(10), 8062–8088 (2009). [CrossRef] [PubMed]

20. S. Okawa and Y. Yamada, “Reconstruction of fluorescence/bioluminescence sources in biological medium with spatial filter,” Opt. Express **18**(12), 13151–13172 (2010). [CrossRef] [PubMed]

21. S. Baillet, J. C. Mosher, and R. M. Leahy, “Electromagnetic brain mapping,” IEEE Signal Process. Mag. **18**, 14–30 (2001). [CrossRef]

22. P. Xu, Y. Tian, H. Chen, and D. Yao, “L* _{p}* Norm iterative sparse solution for EEG source localization,” IEEE Trans. Biomed. Eng.

**54**(3), 400–409 (2007). [CrossRef] [PubMed]

*l*(0 <

_{p}*p*≤ 1) sparsity regularization by partial use of the focal underdetermined system solver (FOCUSS) algorithm, which was introduced by He et al. [23

23. Z. He, A. Cichocki, R. Zdunek, and S. Xie, “Improved FOCCUS Method With conjugate gradient iterations,” IEEE Tras. Signal Process. **57** (1), 399–404 (2009). [CrossRef]

*l*minimization. And the DOT images are reconstructed by minimizing the residual error between the measurement and predicted data sets and the

_{p}*l*norm of the changes in the absorption coefficients simultaneously. Numerical simulations and a phantom experiment demonstrate that the regularization improves the localization of the changes in the distribution of the absorption coefficient.

_{p}## 2. Methods

### 2.1. Forward problem in DOT

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

*r*and time

*t*, Φ(

*r*,

*t*), is obtained, where

*D*= 1/(3

*μ*′

*) represents the diffusion coefficient with the reduced scattering coefficient*

_{s}*μ*′

*,*

_{s}*μ*the absorption coefficient,

_{a}*c*the speed of light,

*q*

_{0}the light source. The boundary condition is given as −

*n*·

*D*∇Φ = 1/(2

*A*)Φ where

*n*is the vector normal to the surface of the medium, and

*A*is the parameter depending on the internal reflection ratio. The forward solution is obtained by solving Eq.(1) by use of the finite element method (FEM).

*r*,

*t*) = −

*n*·

*D*∇Φ, are the quantities measured by the detectors located at various positions. The mean time of flight (MTF) is calculated from Γ as

*M*, is used as the input data for reconstruction of the distribution of

*μ*. An efficient method of solving the forward problem introduced by Schweiger et al [24

_{a}24. M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse model in optical tomography,” J. Math. Imaging Vis. **3**, 263–283 (1993). [CrossRef]

### 2.2. Inverse problem with l_{p} sparsity regularization

*μ*is carried out by minimizing the residual error between the measured MTF data and the MTF data calculated by solving the forward problem. Reconstruction to estimate the

_{a}*μ*distribution is conventionally achieved by solving the following optimization, where

_{a}*M*and

*M̂*are the sets of the measured and the calculated MTFs, respectively.

*λ*is a regularization parameter, and

*f*is a regularization function depending on the regularization method such as Tikhonov regularization method, the total variation method, etc [25

25. C. R. Vogel, *Computational Methods for Inverse Problems*, Frontiers in Applied Mathematics (SIAM, Philadelphia, 2002). [CrossRef]

*∂m̂*/

_{j}*∂μ*

_{ai}, is calculated by using FEM, where

*m̂*is the

_{j}*j*-th calculated datum and

*μ*

_{ai}is the absorption coefficient at the

*i*-th FEM node.

*μ*localized in the small regions, we apply an

_{a}*l*(0 <

_{p}*p*≤ 1) sparsity regularization which minimizes

*μ*

_{ai}is a change in

*μ*from the baseline at the

_{a}*i*-th FEM node, and

*I*is the number of the FEM nodes. However, the expression of the

*l*(0 <

_{p}*p*< 1) sparsity regularization has a difficulty in calculating the gradient in the optimization by use of the gradient based method, because |Δ

*μ*

_{ai}|

^{p}^{−1}tends to infinity when Δ

*μ*

_{ai}is minimized. To avoid this difficulty, Δ

*μ*

_{ai}is reformulated with a parameter

*z*as follows [23

_{i}23. Z. He, A. Cichocki, R. Zdunek, and S. Xie, “Improved FOCCUS Method With conjugate gradient iterations,” IEEE Tras. Signal Process. **57** (1), 399–404 (2009). [CrossRef]

*μ*

_{ai}is described as where

*μ̄*is the constant baseline. Then the DOT reconstruction with the

_{a}*l*sparsity regularization is represented as follows, where

_{p}*z*represents a vector consisting of

*z*. By solving this optimization problem, a solution which selects the changes in

_{i}*μ*in small localized region is obtained. As

_{a}*p*becomes smaller, the localization is expected to be improved.

26. S. R. Arridge, “A gradient-based optimization scheme for optical tomography,” Opt. Express **12**(6), 213–226 (1998). [CrossRef]

*J*(

_{ji}*z*) =

_{i}*∂m̂*/

_{j}*∂z*, is calculated with the perturbation of

_{i}*z*which is equivalent to 0.01 percent of the baseline

_{i}*μ*̄

*in this study. The regularization parameter*

_{a}*λ*is selected with the L-curve method [27

27. P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Compt. **14**(6), 1487–1503 (1993). [CrossRef]

## 3. Numerical experiments

### 3.1. Conditions

*l*sparsity regularization with

_{p}*p*= 1, 1/2, 1/4 on the image reconstruction of time-domain diffuse optical tomography and the results are compared with Tikhonov regularization which is identical to the

*l*sparsity regularization with

_{p}*p*= 2.

*μ*distribution in a 2D circular medium with a radius of 40 mm is reconstructed with the regularizations. The medium has strongly absorbing targets. The sizes and the number of the targets are varied in the simulation.

_{a}*μ*of the targets are changed, depending on the purpose of the experiments. Except the targets, the medium has

_{a}*μ*= 0.007 mm

_{a}^{−1}and

*μ*′

*= 0.8 mm*

_{s}^{−1}homogeneously. These optical properties are given as those of breast in the NIR wavelength range by referring a literature [6

6. J. C. Hebden, H. Veenstra, H. Dehghani, E. M. C. Hillman, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Three-dimensional time-resolved optical tomography of a conical breast phantom,” Appl. Opt. **40** (19), 3278–32887 (2001). [CrossRef]

*A*in the boundary condition for PDE is equal to unity because no reflection is assumed at the boundary.

### 3.2. Results and discussions

#### 3.2.1. Effect on the spatial resolution and localization of the reconstructed image

*x*,

*y*) = (20 mm,10 mm) and (20 mm,−10 mm) with the origin of the coordinate located at the center of the medium. The radius and

*μ*of both of the targets are 5 mm and 0.014 mm

_{a}^{−1}(Δ

*μ*= 0.007 mm

_{a}^{−1}), respectively.

*μ*′

*of the targets is identical to that of the background.*

_{s}*p*= 2, 1, 1/2 and 1/4. Reconstructions are carried out with

*λ*= 10

^{−10}, 10

^{−9}, 10

^{−8},···,10

^{−2}for each

*p*, and the L-curves are plotted with

*R*= log

_{10}||

*M*–

*M̂*||

^{2}for the abscissa and

*F*= log

_{10}

*f*(

*z*) for the ordinate. The corner of the L-curve, in which both terms in Eq. (5) are minimized with a good balance, is visually judged from the plots. The optimum

*λ*determined at the corner decreases with the decrease in

*p*.

*μ*distributions are shown in Fig. 2 with the same color scale for all images using (a) Tikhonov regularization with

_{a}*p*= 2 and (b) to (d) the

*l*sparsity regularization with

_{p}*p*= 1, 1/2 and 1/4, respectively. The maximum values of the reconstructed Δ

*μ*are 0.0015 mm

_{a}^{−1}, 0.0020 mm

^{−1}, 0.0036 mm

^{−1}and 0.0050 mm

^{−1}for

*p*= 2,1,1/2 and 1/4, respectively. Figure 3(a) shows the profiles of Δ

*μ*at the FEM nodes along the lines in the

_{a}*y*-direction passing through the two peaks of reconstructed

*μ*. The two targets are observable and almost equally reconstructed with their centers at the correct positions. Especially with

_{a}*p*= 1/2 and 1/4, the two targets are clearly separated.

*ζ*= Δ

*μ*

_{amin}/Δ

*μ*

_{amax}as a function of

*p*, where Δ

*μ*

_{amin}is the minimum of Δ

*μ*reconstructed between the peaks, and Δ

_{a}*μ*

_{amax}is the average of the two peaks of Δ

*μ*.

_{a}*ζ*is an index for evaluation of spatial resolution.

*ζ*= 0 indicates that the two reconstructed targets are perfectly separated while

*ζ*= 1 indicates that the two targets are not separated but combined into one large target. Figure 3 (b) shows that the degree of the separation is improved as

*p*decreases.

*μ*, Δ

_{a}*μ*

_{amax}, is plotted as a function of

*p*in Fig. 3(c). The absolute value of Δ

*μ*of the reconstructed target with small

_{a}*p*becomes closer to the true Δ

*μ*of 0.007 mm

_{a}^{−1}in the targets than that with large

*p*.

*S*, is defined as the sum of the area of the FEM elements having Δ

*μ*≥ Δ

_{a}*μ*

_{amax}/2, where Δ

*μ*

_{amax}is the maximum of Δ

*μ*, and is plotted as a function of

_{a}*p*in Fig. 3 (d).

*S*is an index for evaluation of localization by comparing with the true value of

*S. S*decreases with the decrease in

*p*, and localization of the reconstructed targets is improved.

*μ*distribution reconstructed using Tikhonov regularization (Fig. 2(a)) has small undulations around the targets, which are caused by the random noise, and the difference in the FEM meshing between that for generating the measurement data and that for solving the inverse problem.

_{a}*μ*distributions reconstructed with the

_{a}*l*sparsity regularization. Therefore, the

_{p}*l*sparsity regularization reduces the influences of noise and of the difference in the FEM meshing. From these results, it can be said that the

_{p}*l*sparsity regularization achieves a high spatial resolution. When the true changes in

_{p}*μ*are localized in small regions, the

_{a}*l*sparsity regularization provides preferable reconstruction with the robustness to noise.

_{p}#### 3.2.2. Effect on the sensitivity to small changes in the absorption coefficient

*μ*of a medium having two targets with different

_{a}*μ*. The positions and the sizes of the targets are the same as those in the previous subsection. One of the targets with its center at (

_{a}*x*,

*y*) = (20 mm, 10 mm) has

*μ*= 0.0105 mm

_{a}^{−1}, and the other target with its center at (

*x*,

*y*) = (20mm,−10mm) has

*μ*= 0.0140 mm

_{a}^{−1}. The difference in

*μ*from the background

_{a}*μ*= 0.0070 mm

_{a}^{−1}of the former target (Δ

*μ*

_{a}_{1}= 0.0035 mm

^{−1}) is half of that of the latter one (Δ

*μ*

_{a}_{2}= 0.0070 mm

^{−1}). Therefore, the ratio of the smaller changes in

*μ*from the background to larger one, defined as

_{a}*γ*= Δ

*μ*

_{a}_{1}/Δ

*μ*

_{a}_{2}, is 0.5.

*γ*is an index for evaluating sensitivity to small changes in

*μ*by comparing with the true value of

_{a}*γ*.

*λ*is selected by the L-curve method.

*μ*image reconstructed using Tikhonov regularization shown in Fig. 4(a) reveals weak two peaks with Δ

_{a}*μ*

_{amax}= 0.0009 mm

^{−1}and 0.0015 mm

^{−1}at (

*x*,

*y*) = (23.0 mm, +12.1 mm) and (25.2 mm, −12.1 mm), respectively. The positions of the targets are well reconstructed. However,

*γ*calculated from the reconstructed peaks of Δ

*μ*is 0.60, and the reconstructed

_{a}*μ*are underestimated in this case. As well as in the case of the previous subsection, the spatial resolutions of the reconstructed images are low, and the small undulations due to noise are seen in Fig. 4(a).

_{a}*l*sparsity regularization with

_{p}*p*= 1 is used, undulation due to noise is suppressed, and localization of the reconstructed targets is improved. Two peaks of the reconstructed Δ

*μ*are 0.0013 mm

_{a}^{−1}and 0.0027 mm

^{−1}, respectively.

*γ*is 0.48 which is close to the true value of 0.50. This means that the

*l*

_{1}sparsity regularization recovers the difference in

*μ*between the two targets better than Tikhonov regularization.

_{a}*l*sparsity regularization is not always successful. When

_{p}*p*= 1/2, one of the target is lost in the reconstructed image in Fig. 4(c). The changes in

*μ*are so excessively localized that the weakly absorbing target disappears. Two peaks are found in the

_{a}*μ*distribution reconstructed with

_{a}*p*= 1/4 in Fig. 4(d), but

*γ*is 0.23 which is much smaller than the true value of 0.50. Figures 5(a) and (b) show the reconstructed peaks of Δ

*μ*of the two targets, Δ

_{a}*μ*

_{a1max}and Δ

*μ*

_{a2max}, and

*γ*= Δ

*μ*

_{a1max}/Δ

*μ*

_{a2max}calculated from the reconstructed peaks of Δ

*μ*as a function of

_{a}*p*, respectively. The

*l*sparsity regularization is effective to enhance the localization, to reduce the artifacts and to recover the difference in

_{p}*μ*between the large and small changes, although small changes in

_{a}*μ*may be eliminated by the

_{a}*l*sparsity regularization when

_{p}*p*is too small because it strongly localizes the changes in

*μ*.

_{a}#### 3.2.3. Effect on the reconstruction of broad target

*l*sparsity regularization is found to be effective for reconstructing localized targets as shown in the previous subsections. However, the targets are not always localized well in practical applications. We demonstrate reconstructions of broad targets in this subsection.

_{p}*x*,

*y*) = (20 mm, 0 mm) and

*μ*of 0.014 mm

_{a}^{−1}in the background medium having

*μ*= 0.0070 mm

_{a}^{−1}. Reconstructions are carried out with the manner mentioned above.

*μ*distributions using Tikhonov and the

_{a}*l*sparsity regularizations with

_{p}*p*= 1, 1/2, and 1/4. The maximum

*μ*reconstructed using Tikhonov regularization shown in Fig. 6(a) is 0.0122 mm

_{a}^{−1}. Undulations in the

*μ*distribution appear due to the noise in the input data.

_{a}*l*sparsity regularizations remove the undulations. The shape and size of the target are well reconstructed when

_{p}*p*= 1 as shown in Fig. 6(b). In Fig. 6(c), the target reconstructed with

*p*= 1/2 has a smaller area than that with

*p*= 1, and the maximum

*μ*value of 0.0130 mm

_{a}^{−1}with

*p*= 1/2 is larger than that of 0.0117 mm

^{−1}with

*p*= 1. The maximum

*μ*value further increases to 0.0184 mm

_{a}^{−1}in Fig. 6(d) as

*p*decreases to 1/4 although the reconstructed target is localized excessively and smaller in size than the true target. Figures 7(a) and (b) plot the reconstructed peak value of Δ

*μ*, Δ

_{a}*μ*

_{amax}, and the area of the peak of Δ

*μ*,

_{a}*S*, as a function of

*p*, respectively.

*S*is defined as the sum of the area of the FEM elements having Δ

*μ*≥ Δ

_{a}*μ*

_{amax}/2, and

*S*is the index for evaluating localization of the reconstructed target by comparing with the true value of

*S*as mentioned before.

*l*sparsity regularization is effective to improve the quality of the reconstructed images of broad targets by reducing the influence of noise. However, the quality of the reconstructed images highly depends on the parameter

_{p}*p*, and the size of the reconstructed target becomes smaller than that of the true one as

*p*decreases too much.

#### 3.2.4. Determination of optimum *p* value

*p*value on the quality of the reconstructed images in the cases of various sizes and

*μ*of the targets. According to the simulation results, the optimum

_{a}*p*value depends on the cases. For the case in the section 3.2.1, the size and

*μ*of the small targets with a radius of 5 mm is reconstructed clearly when

_{a}*p*is small, and

*p*= 1/4 is a good choice in this case as shown in Figs. 3(c) and (d).

*p*does not always lead to better reconstruction. For the case of multiple targets with different

*μ*, too small

_{a}*p*values underestimate

*μ*of the target with smaller

_{a}*μ*. From the results in the subsection 3.2.2,

_{a}*p*= 1/4 may be a good choice from the Δ

*μ*point of view (Fig. 5(a)), but

_{a}*p*= 1 may be a good choice from the

*γ*point of view (Fig. 5(b)). When the target has a radius of 10 mm, the size of the target is well reconstructed with

*p*= 1, although

*p*= 1/2 reconstructs the peak of Δ

*μ*of the target better than

_{a}*p*= 1 as plotted in Fig. 7. As a whole it can be said that

*p*≤ 1 is preferable to reduce the measurement noise and the error between the true and reconstructed

*μ*values.

_{a}*p*value. Prior information provided by other imaging modalities may be useful for that purpose. Prior information of the target size may help determine the optimum

*p*value, for example.

## 4. Phantom experiment

### 4.1. Conditions

*μ*distribution.

_{a}*μ*in a 2D-like tissue simulating phantom which was a cylinder made of polyacetal resin with a height of 240 mm and a radius of 40mm. The phantom had the background optical properties of

_{a}*μ*= 0.0006 mm

_{a}^{−1}and

*μ*′

*= 0.863 mm*

_{s}^{−1}. In this phantom, there existed a target of a cylindrical hole with a radius of 10 mm and the center at (

*x*,

*y*) = (20 mm, 0 mm). The cylindrical hole was filled with 1.0 percent Intralipid solution having

*μ*= 0.0026 mm

_{a}^{−1}and

*μ*′

*= 1.054 mm*

_{s}^{−1}.

*μ*distribution in the 2D plane was reconstructed with Tikhonov or the

_{a}*l*sparsity regularizations. We used the FEM meshing identical to that used for reconstructions in the previous simulations. We set

_{p}*μ̄*= 0.0006 mm

_{a}^{−1}and selected

*λ*based on the L-curve method. The refractive index of the polyacetal resin was given as 1.54 leading to the value of

*A*= 4.26 in the boundary condition of PDE.

### 4.2. Results and discussions

*λ*= 10

^{−10},10

^{−9},10

^{−8}, ⋯ ,10

^{−2}.

*λ*values at the corners are determined as 10

^{−2}and 10

^{−5}for

*p*=2 and 1, respectively. However, it was difficult to obtain typical L-curves for

*p*= 1/2 and 1/4. The evaluating points for

*p*= 1/2 and for 1/4 in Fig. 8(c) and (d) subtly form the L-curves, and

*λ*= 10

^{−7}is at the corners in both cases. More points on the L-curve for

*p*= 1/4 were evaluated to confirm the corner as shown in Fig. 8(d). In the simulation sections and this phantom experiment, the regularized solution with small

*p*tends to change drastically within a narrow range of

*λ*. So it is better to make the L-curve with fine evaluating points of

*λ*to find an appropriate corner of the L-curve.

*μ*distributions are shown in Fig. 9. As expected from the results in the previous section, the smaller the value of

_{a}*p*is, the more localized the target is. Tikhonov regularization reconstructed the target clearly as shown in Fig. 9(a), but there are undesirable changes in

*μ*in the wide area out of the true target position which were due to unknown noise factors and took the values from 38 percent to 63 percent of the reconstructed

_{a}*μ*of the target. The area with the undesirable changes in

_{a}*μ*became smaller by the

_{a}*l*sparsity regularization, although some of the undesirable large changes still remained near the position of (

_{p}*x*,

*y*) = (−20 mm, −10 mm) and (−10 mm, 25 mm) in Figs. 9 (b), (c) and (d). The

*l*sparsity regularizations with

_{p}*p*= 1/2 and 1/4 provide highly localized targets. The maximum value of

*μ*became larger as

_{a}*p*decreased.

*l*sparsity regularization localizes the change in the

_{p}*μ*distribution and that the regularization is effective to reduce the influence of noise. Figure 10 shows Δ

_{a}*μ*

_{amax}and

*S*in this phantom experiment. Small

*p*enhances Δ

*μ*

_{amax}, and the size of the reconstructed target becomes smaller than that of the true one as

*p*decreases. These results validate the simulations in the previous section.

*μ*are 0.0047 mm

_{a}^{−1}, 0.0068 mm

^{−1}and 0.0132 mm

^{−1}for

*p*= 1, 1/2 and 1/4, respectively, and all of these values are larger than the true value of 0.002 mm

^{−1}. These differences may be caused by the small changes in

*μ*′

*of the target from that of the background because only*

_{s}*μ*is reconstructed under the assumption of homogeneous

_{a}*μ*′

*. Simultaneous reconstruction of*

_{s}*μ*and

_{a}*μ*′

*is an interesting topic for future study. Unknown noise factor must prevent a good reconstruction also for the cases of multiple targets. Although the phantom experiment was conducted only for single target, the effects of the proposed method, i.e. improvement of localization and alleviation of influence of noise, are validated and similar results will be obtained for the cases of multiple targets.*

_{s}## 5. Conclusion

*l*(0 <

_{p}*p*= 1) sparsity regularization is applied to the time-domain diffuse optical tomography with the gradient-based nonlinear optimization scheme. The expression of the

*l*sparsity regularization is reformulated as a differentiable function of a parameter to avoid the difficulty in calculating its gradient in the optimization process. The regularization parameter is selected by the L-curve method.

_{p}*l*sparsity regularization improves the spatial resolution. Two localized targets with the absorption coefficients higher than that of the background are clearly separated in the images reconstructed with the

_{p}*l*sparsity regularization, while Tikhonov regularization reconstructs the two targets as a broad single target. The reconstructed values of the absorption coefficients approaches the correct value by use of the

_{p}*l*sparsity regularization.

_{p}*l*sparsity regularization. However, it is demonstrated that a target with a small differences in the absorption coefficient from the background may disappear due to the excessive effect of the

_{p}*l*sparsity regularization.

_{p}*l*sparsity regularization is also effective for reconstructing broad targets. The influence of noise such as non-targeted small undulations in the reconstructed image is reduced. The size of the reconstructed target is reconstructed well when

_{p}*p*= 1 and 1/2 in the simulation. The

*l*sparsity regularization with a small

_{p}*p*strongly localizes the target, and the size of the reconstructed target decreases with the decrease in

*p*. A phantom experiment validates the numerical simulations.

*l*sparsity regularization can be useful to reconstruct the localized changes in the absorption coefficient. The criterion to determine the optimum

_{p}*p*value can be discussed in a future work.

## References and links

1. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Prob. |

2. | A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. |

3. | D. Grosenick, H. Wabnitz, H. H. Rinneberg, T. Moesta, and P. M. Schlag, “Development of a time-domain optical mammography and first |

4. | D. Grosenick, K. T. Moesta, M. Möller, J. Mucke, H. Wabnitz, B. Gebauer, C. Stroszczynski, B. Wassermann, P. M. Schlag, and H. Rinneberg, “Time-domain scanning optical mammography: I. Recording and assessment of mammograms of 154 patients,” Phys. Med. Biol. |

5. | D. Grosenick, H. Wabnitz, K. T. Moesta, J. Mucke, P. M. Schlag, and H. Rinneberg, “Time-domain scanning optical mammography: II. Optical properties and tissue parameters of 87 carcinomas,” Phys. Med. Biol. |

6. | J. C. Hebden, H. Veenstra, H. Dehghani, E. M. C. Hillman, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Three-dimensional time-resolved optical tomography of a conical breast phantom,” Appl. Opt. |

7. | T. Yates, C. Hebdan, A. Gibson, N. Everdell, S. R. Arridge, and M. Douek, “Optical tomography of the breast using a multi-channel time-resolved imager,” Phys. Med. Biol. |

8. | A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S.R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three-dimensional whole-head optical tomography for passive motor evoked responses in the neonate,” NueroImage |

9. | J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. |

10. | B. W. Pogue, T. O. McBride, J. Prewitt, U. Lösterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. |

11. | G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas, “Quantitative spectroscopic optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. |

12. | P. K. Yalavarthy, B. W. Pogue, H. Dehghani, C. M. Carpenter, S. Jiang, and K. D. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express |

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14. | P. Hiltunen, D. Calvetti, and E. Somersalo, “An adaptive smoothness regularization algorithm for optical tomography,” Opt, Express |

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16. | N. Cao, A. Nehorai, and M. Jacob, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express , |

17. | P. M. Shankar and M. A. Neifeld, “Sparsity constrained regularization for multiframe image restoration,” J. Opt. Soc. Am. A |

18. | P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt. |

19. | Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express |

20. | S. Okawa and Y. Yamada, “Reconstruction of fluorescence/bioluminescence sources in biological medium with spatial filter,” Opt. Express |

21. | S. Baillet, J. C. Mosher, and R. M. Leahy, “Electromagnetic brain mapping,” IEEE Signal Process. Mag. |

22. | P. Xu, Y. Tian, H. Chen, and D. Yao, “L 54 (3), 400–409 (2007). [CrossRef] [PubMed] |

23. | Z. He, A. Cichocki, R. Zdunek, and S. Xie, “Improved FOCCUS Method With conjugate gradient iterations,” IEEE Tras. Signal Process. |

24. | M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse model in optical tomography,” J. Math. Imaging Vis. |

25. | C. R. Vogel, |

26. | S. R. Arridge, “A gradient-based optimization scheme for optical tomography,” Opt. Express |

27. | P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Compt. |

28. | S. Holder, |

29. | H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, Y. Tuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and N. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(170.3880) Medical optics and biotechnology : Medical and biological imaging

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: September 23, 2011

Revised Manuscript: November 14, 2011

Manuscript Accepted: November 14, 2011

Published: November 21, 2011

**Citation**

Shinpei Okawa, Yoko Hoshi, and Yukio Yamada, "Improvement of image quality of time-domain diffuse optical tomography with lp sparsity regularization," Biomed. Opt. Express **2**, 3334-3348 (2011)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-2-12-3334

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