Comparative studies of l<sub>p</sub>-regularization-based reconstruction algorithms for bioluminescence tomography

Qitan Zhang; Xueli Chen; Xiaochao Qu; Jimin Liang; Jie Tian

doi:10.1364/BOE.3.002916

Comparative studies of l_p-regularization-based reconstruction algorithms for bioluminescence tomography

Qitan Zhang, Xueli Chen, Xiaochao Qu, Jimin Liang, and Jie Tian

Biomedical Optics Express
Vol. 3,
Issue 11,
pp. 2916-2936
(2012)
•https://doi.org/10.1364/BOE.3.002916

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Qitan Zhang, Xueli Chen, Xiaochao Qu, Jimin Liang, and Jie Tian, "Comparative studies of l_p-regularization-based reconstruction algorithms for bioluminescence tomography," Biomed. Opt. Express 3, 2916-2936 (2012)

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Related Topics
Table of Contents Category
- Image Reconstruction and Inverse Problems
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Inverse problems (100.3190)
- Medical and biological imaging (170.3880)
- Tomography (170.6960)

About this Article
History
- Original Manuscript: August 15, 2012
- Revised Manuscript: October 18, 2012
- Manuscript Accepted: October 19, 2012
- Published: October 23, 2012

Accessible

Open Access

Abstract

Inverse source reconstruction is the most challenging aspect of bioluminescence tomography (BLT) because of its ill-posedness. Although many efforts have been devoted to this problem, so far, there is no generally accepted method. Due to the ill-posedness property of the BLT inverse problem, the regularization method plays an important role in the inverse reconstruction. In this paper, six reconstruction algorithms based on l_p regularization are surveyed. The effects of the permissible source region, measurement noise, optical properties, tissue specificity and source locations on the performance of the reconstruction algorithms are investigated using a series of single source experiments. In order to further inspect the performance of the reconstruction algorithms, we present the double sources and the in vivo mouse experiments to study their resolution ability and potential for a practical heterogeneous mouse experiment. It is hoped to provide useful guidance on algorithm development and application in the related fields.

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References

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X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12(17), 3996–4000 (2004).
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J. Feng, K. Jia, C. Qin, G. Yan, S. Zhu, X. Zhang, J. Liu, and J. Tian, “Three-dimensional bioluminescence tomography based on Bayesian approach,” Opt. Express 17(19), 16834–16848 (2009).
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H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity,” Opt. Express 18(3), 2894–2912 (2010).
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K. Liu, J. Tian, Y. Lu, C. Qin, X. Yang, S. Zhu, and X. Zhang, “A fast bioluminescent source localization method based on generalized graph cuts with mouse model validations,” Opt. Express 18(4), 3732–3745 (2010).
[Crossref] [PubMed]
B. Zhang, X. Yang, C. Qin, D. Liu, S. Zhu, J. Feng, L. Sun, K. Liu, D. Han, X. Ma, X. Zhang, J. Zhong, X. Li, X. Yang, and J. Tian, “A trust region method in adaptive finite element framework for bioluminescence tomography,” Opt. Express 18(7), 6477–6491 (2010).
[Crossref] [PubMed]
W. Cong and G. Wang, “Bioluminescence tomography based on the phase approximation model,” J. Opt. Soc. Am. A 27(2), 174–179 (2010).
[Crossref] [PubMed]
H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys. 229(13), 5246–5256 (2010).
[Crossref]
X. He, Y. Hou, D. Chen, Y. Jiang, M. Shen, J. Liu, Q. Zhang, and J. Tian, “Sparse regularization-based reconstruction for bioluminescence tomography using a multilevel adaptive finite element method,” Int. J. Biomed. Imaging 2011, 203537 (2011).
[Crossref] [PubMed]
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[Crossref] [PubMed]
Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[Crossref]
X. He, J. Liang, X. Qu, H. Huang, Y. Hou, and J. Tian, “Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem,” Int. J. Biomed. Imaging 2010, 291874 (2010).
[Crossref] [PubMed]
Q. Zhang, X. Qu, D. Chen, X. Chen, J. Liang, and J. Tian, “Experimental three-dimensional bioluminescence tomography reconstruction using the lp regularization,” Adv. Sci. Lett. 16(1), 125–129 (2012).
[Crossref]
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[Crossref]
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[Crossref]
X. Chen, F. Xu, and Y. Ye, “Lower bound theory of nonzero entries in solutions of l2-lp minimization,” SIAM J. Sci. Comput. 32(5), 2832–2852 (2010).
[Crossref]
M. Wei, W. Scott, J. James, H. McClellan, and G. Larson, “Estimation of the discrete spectrum of relaxations for electromagnetic induction responses using lp-regularized least squares for 0 ≤ p ≤ 1,” IEEE Geosci. Remote Sens. Lett. 8(2), 233–237 (2011).
[Crossref]
M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54(8), 2493–2509 (2009).
[Crossref] [PubMed]
M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref] [PubMed]
Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express 14(18), 8211–8223 (2006).
[Crossref] [PubMed]
A. Ribés and F. Schmitt, “Linear inverse problems in imaging,” IEEE Signal Process. Mag. 25(4), 84–99 (2008).
[Crossref]
R. Han, J. Liang, X. Qu, Y. Hou, N. Ren, J. Mao, and J. Tian, “A source reconstruction algorithm based on adaptive hp-FEM for bioluminescence tomography,” Opt. Express 17(17), 14481–14494 (2009).
[Crossref] [PubMed]

2012 (1)

Q. Zhang, X. Qu, D. Chen, X. Chen, J. Liang, and J. Tian, “Experimental three-dimensional bioluminescence tomography reconstruction using the lp regularization,” Adv. Sci. Lett. 16(1), 125–129 (2012).
[Crossref]

2011 (3)

X. He, Y. Hou, D. Chen, Y. Jiang, M. Shen, J. Liu, Q. Zhang, and J. Tian, “Sparse regularization-based reconstruction for bioluminescence tomography using a multilevel adaptive finite element method,” Int. J. Biomed. Imaging 2011, 203537 (2011).
[Crossref] [PubMed]

Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[Crossref]

M. Wei, W. Scott, J. James, H. McClellan, and G. Larson, “Estimation of the discrete spectrum of relaxations for electromagnetic induction responses using lp-regularized least squares for 0 ≤ p ≤ 1,” IEEE Geosci. Remote Sens. Lett. 8(2), 233–237 (2011).
[Crossref]

2010 (10)

Z. Xu, H. Zhang, Y. Wang, X. Chang, and L. Yong, “L1/2 regularization,” Sci. China Inform. Sci. 53(6), 1159–1169 (2010).
[Crossref]

X. Chen, F. Xu, and Y. Ye, “Lower bound theory of nonzero entries in solutions of l2-lp minimization,” SIAM J. Sci. Comput. 32(5), 2832–2852 (2010).
[Crossref]

X. He, J. Liang, X. Qu, H. Huang, Y. Hou, and J. Tian, “Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem,” Int. J. Biomed. Imaging 2010, 291874 (2010).
[Crossref] [PubMed]

H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys. 229(13), 5246–5256 (2010).
[Crossref]

H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization,” Opt. Express 18(3), 1854–1871 (2010).
[Crossref] [PubMed]

W. Cong and G. Wang, “Bioluminescence tomography based on the phase approximation model,” J. Opt. Soc. Am. A 27(2), 174–179 (2010).
[Crossref] [PubMed]

H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity,” Opt. Express 18(3), 2894–2912 (2010).
[Crossref] [PubMed]

K. Liu, J. Tian, Y. Lu, C. Qin, X. Yang, S. Zhu, and X. Zhang, “A fast bioluminescent source localization method based on generalized graph cuts with mouse model validations,” Opt. Express 18(4), 3732–3745 (2010).
[Crossref] [PubMed]

B. Zhang, X. Yang, C. Qin, D. Liu, S. Zhu, J. Feng, L. Sun, K. Liu, D. Han, X. Ma, X. Zhang, J. Zhong, X. Li, X. Yang, and J. Tian, “A trust region method in adaptive finite element framework for bioluminescence tomography,” Opt. Express 18(7), 6477–6491 (2010).
[Crossref] [PubMed]

X. He, J. Liang, X. Wang, J. Yu, X. Qu, X. Wang, Y. Hou, D. Chen, F. Liu, and J. Tian, “Sparse reconstruction for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method,” Opt. Express 18(24), 24825–24841 (2010).
[Crossref] [PubMed]

2009 (5)

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), 8062–8080 (2009).
[Crossref] [PubMed]

R. Han, J. Liang, X. Qu, Y. Hou, N. Ren, J. Mao, and J. Tian, “A source reconstruction algorithm based on adaptive hp-FEM for bioluminescence tomography,” Opt. Express 17(17), 14481–14494 (2009).
[Crossref] [PubMed]

J. Feng, K. Jia, C. Qin, G. Yan, S. Zhu, X. Zhang, J. Liu, and J. Tian, “Three-dimensional bioluminescence tomography based on Bayesian approach,” Opt. Express 17(19), 16834–16848 (2009).
[Crossref] [PubMed]

C. Qin, J. Tian, X. Yang, J. Feng, K. Liu, J. Liu, G. Yan, S. Zhu, and M. Xu, “Adaptive improved element free Galerkin method for quasi- or multi-spectral bioluminescence tomography,” Opt. Express 17(24), 21925–21934 (2009).
[Crossref] [PubMed]

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54(8), 2493–2509 (2009).
[Crossref] [PubMed]

2008 (5)

H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys. 35(11), 4863–4871 (2008).
[Crossref] [PubMed]

R. Weissleder and M. J. Pittet, “Imaging in the era of molecular oncology,” Nature 452(7187), 580–589 (2008).
[Crossref] [PubMed]

G. Wang, W. Cong, H. Shen, X. Qian, M. Henry, and Y. Wang, “Overview of bioluminescence tomography--a new molecular imaging modality,” Front. Biosci. 13(13), 1281–1293 (2008).
[Crossref] [PubMed]

A. Ribés and F. Schmitt, “Linear inverse problems in imaging,” IEEE Signal Process. Mag. 25(4), 84–99 (2008).
[Crossref]

J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express 16(20), 15640–15654 (2008).
[Crossref] [PubMed]

2007 (1)

Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element analysis: methodology and simulation,” Phys. Med. Biol. 52(15), 4497–4512 (2007).
[Crossref] [PubMed]

2006 (5)

D. Donoho, “Compresse sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett. 31(3), 365–367 (2006).
[Crossref] [PubMed]

G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express 14(17), 7801–7809 (2006).
[Crossref] [PubMed]

Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express 14(18), 8211–8223 (2006).
[Crossref] [PubMed]

D. Donoho, “For most large underdetermined systems of linear equations the minimal l1-norm near solution is also the sparest solution,” Commun. Pure Appl. Math. 59(6), 797–829 (2006).
[Crossref]

2005 (2)

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13(18), 6756–6771 (2005).
[Crossref] [PubMed]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50(23), 5421–5441 (2005).
[Crossref] [PubMed]

2004 (2)

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. 31(8), 2289–2299 (2004).
[Crossref] [PubMed]

X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12(17), 3996–4000 (2004).
[Crossref] [PubMed]

2003 (1)

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. F. Meinel, “Development of the first bioluminescence CT scanner,” Radiology 229(P), 566 (2003).

1995 (1)

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref] [PubMed]

1963 (1)

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Soviet Math. Dokl. 4, 1035–1038 (1963).

Arridge, S. R.

Bading, J. R.

Chan, T. F.

Chang, X.

Z. Xu, H. Zhang, Y. Wang, X. Chang, and L. Yong, “L1/2 regularization,” Sci. China Inform. Sci. 53(6), 1159–1169 (2010).
[Crossref]

Chatziioannou, A. F.

Chaudhari, A. J.

Chen, D.

Chen, X.

X. Chen, F. Xu, and Y. Ye, “Lower bound theory of nonzero entries in solutions of l2-lp minimization,” SIAM J. Sci. Comput. 32(5), 2832–2852 (2010).
[Crossref]

Cherry, S. R.

Chu, M.

Cong, A.

Cong, W.

W. Cong and G. Wang, “Bioluminescence tomography based on the phase approximation model,” J. Opt. Soc. Am. A 27(2), 174–179 (2010).
[Crossref] [PubMed]

Conti, P. S.

Darvas, F.

Davis, S. C.

H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys. 35(11), 4863–4871 (2008).
[Crossref] [PubMed]

Dehghani, H.

H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys. 35(11), 4863–4871 (2008).
[Crossref] [PubMed]

Delpy, D. T.

Donoho, D.

D. Donoho, “Compresse sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

Douraghy, A.

Durairaj, K.

Feng, J.

Gao, H.

H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization,” Opt. Express 18(3), 1854–1871 (2010).
[Crossref] [PubMed]

Gu, X.

X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12(17), 3996–4000 (2004).
[Crossref] [PubMed]

Han, D.

Han, R.

He, X.

Henry, M.

Hiraoka, M.

Hoffman, E.

Hoffman, E. A.

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. F. Meinel, “Development of the first bioluminescence CT scanner,” Radiology 229(P), 566 (2003).

Hou, Y.

Hu, Z.

Huang, H.

James, J.

Jia, K.

Jiang, H.

X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12(17), 3996–4000 (2004).
[Crossref] [PubMed]

Jiang, M.

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. 31(8), 2289–2299 (2004).
[Crossref] [PubMed]

Jiang, S.

Jiang, Y.

Klose, A. D.

Kumar, D.

Larcom, L.

X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12(17), 3996–4000 (2004).
[Crossref] [PubMed]

Larson, G.

Leahy, R. M.

Li, H.

Li, W.

Li, X.

Li, Y.

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. 31(8), 2289–2299 (2004).
[Crossref] [PubMed]

Liang, J.

Liu, D.

Liu, F.

Liu, J.

Liu, K.

Liu, Y.

Lu, Y.

Luo, J.

Lv, Y.

Ma, X.

Mao, J.

McClellan, H.

McCray, P. B.

McLennan, G.

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. F. Meinel, “Development of the first bioluminescence CT scanner,” Radiology 229(P), 566 (2003).

Meinel, J. F.

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. F. Meinel, “Development of the first bioluminescence CT scanner,” Radiology 229(P), 566 (2003).

Moats, R. A.

Patterson, M. S.

Paulsen, K. D.

Pittet, M. J.

R. Weissleder and M. J. Pittet, “Imaging in the era of molecular oncology,” Nature 452(7187), 580–589 (2008).
[Crossref] [PubMed]

Pogue, B. W.

H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys. 35(11), 4863–4871 (2008).
[Crossref] [PubMed]

Qian, X.

Qin, C.

Qu, X.

Ren, N.

Ribés, A.

A. Ribés and F. Schmitt, “Linear inverse problems in imaging,” IEEE Signal Process. Mag. 25(4), 84–99 (2008).
[Crossref]

Schmitt, F.

A. Ribés and F. Schmitt, “Linear inverse problems in imaging,” IEEE Signal Process. Mag. 25(4), 84–99 (2008).
[Crossref]

Schweiger, M.

Scott, W.

Shen, H.

Shen, M.

Sinn, P.

Smith, D. J.

Stout, D.

Sun, L.

Suter, M.

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. F. Meinel, “Development of the first bioluminescence CT scanner,” Radiology 229(P), 566 (2003).

Tian, J.

Tikhonov, A. N.

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Soviet Math. Dokl. 4, 1035–1038 (1963).

Vishwanath, K.

Wang, G.

W. Cong and G. Wang, “Bioluminescence tomography based on the phase approximation model,” J. Opt. Soc. Am. A 27(2), 174–179 (2010).
[Crossref] [PubMed]

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Figures (16)

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Fig. 1 Heterogeneous phantom with a single light source composed of muscle, lungs, heart, bone, liver and the source in the right lung.

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Fig. 2 The surface flux distribution of the heterogeneous phantom from different views. (a)-(b) The coronal and sagittal views, respectively. (c) The translucency view of (a).

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Fig. 3 Axial views of the BLT reconstruction results of l₂ regularization methods using different PSRs at z = 0mm. (a)-(c) Results of TTLS; (d)-(g) Results of Tikhonov.

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Fig. 4 Axial views of the BLT reconstruction results of l₁ regularization methods using different PSRs at z = 0mm. (a)-(d) Results of TNIPM; (e)-(h) Results of IVTCG; (i)-(l) Results of PDIP.

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Fig. 5 Axial views of the BLT reconstruction results of WISTA using different PSRs at z = 0mm.

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Fig. 6 Performance metrics for the six algorithms using different PSRs. (a) The distance errors of the BLT reconstruction results; (b) The reconstruction time of various methods.

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Fig. 7 Performance metrics of various algorithms at different noise levels. (a) The distance errors of the BLT reconstruction results; (b) The reconstruction time of various methods.

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Fig. 8 Axial views of the BLT reconstruction results of six regularization methods at the 50% measurement noise level at z = 0mm. (a)-(f) are the results of TTLS, Tikhonov, TNIPM, IVTCG, PDIP and WISTA respectively.

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Fig. 9 Performance metrics for the six algorithms using different optical parameters. (a) The distance errors of the BLT reconstruction results; (b) The reconstruction time of various methods.

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Fig. 10 Tissue specificity models (various colors are for various segmented tissues).

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Fig. 11 Performance metrics for various algorithms of different tissue specificity. (a) The location errors of BLT reconstruction; (b) The reconstruction time of various methods.

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Fig. 12 Error bar chart of the Loc_Err at different source positions.

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Fig. 13 BLT reconstruction results for l₂ regularization methods in a double source case in a 3D view. (a)-(c) Results of TTLS; (d)-(f) Results of Tikhonov.

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Fig. 14 BLT reconstruction results of l₁ regularization methods in a double source case in a 3D view. (a)-(c) Results of TNIPM; (d)-(f) Results of IVTCG; (g)-(i) Results of PDIP.

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Fig. 15 BLT reconstruction results of l_p (0 < p < 1) regularization methods in a double source case in a 3D view. (a)-(c) Results of WISTA.

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Fig. 16 BLT reconstruction results of the in vivo mouse experiment. (a) the 3-D view of the segmented micro-CT slices of the imaged mouse with a luminescent source; (b)-(f) the reconstruction results of Tikhonov, TNIPM, IVTCG, PDIP and WISTA.

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Tables (7)

Table 1 Optical parameters of the heterogeneous phantom [39]

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Table 2 Reconstruction results at different measurement noise levels

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Table 3 Reconstruction results using different optical parameters

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Table 4 Reconstruction results for tissue specificity

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Table 5 Reconstruction results at different source positions

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Table 6 Reconstruction results in double sources

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Table 7 Reconstruction results of the in vivo mouse experiment

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Equations (23)

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- \nabla \cdot (D (r) \nabla Φ (r)) + μ_{a} (r) Φ (r) = S (r) (r \in Ω)

Φ (r) + 2 A_{n} (r) D (r) (v (r) \cdot \nabla Φ (r)) = 0 (r \in \partial Ω)

Q (r) = - D (r) (v (r) \cdot \nabla Φ (r)) = - \frac{1}{2 A_{n} (r)} Φ (r) (r \in \partial Ω)

Α S = b

\min {‖ A x - b ‖}_{2}^{2} + λ {‖ x ‖}_{p}

\min {‖ Α x - b ‖}_{2}^{2} + λ {‖ x ‖}_{2}^{2}

x = {(A^{T} A + λ I)}^{- 1} A^{T} b

x = \sum_{i = 1}^{n} \frac{σ_{i}^{2}}{σ_{i}^{2} + λ} \frac{u_{i}^{T} b}{σ_{i}} v_{i}

\min \frac{1}{2} {‖ Α x - b ‖}_{2}^{2} + λ {‖ x ‖}_{1}

\begin{array}{l} \min_{z} c^{T} z + \frac{1}{2} z^{T} B z \equiv F (z) \\ s . t . z \geq 0 \end{array}

Γ_{k} = {i | i \in {1, \dots, 2 N}, [{(z^{k})}_{i} > 0, {(\nabla F (z^{k}))}_{i} = 0] o r [{(z^{k})}_{i} = 0, {(\nabla F (z^{k}))}_{i} < 0]}

I_{k} = {i_{l} \in {\hat{I}}_{k} | l \leq \min {| {\hat{I}}_{k} |, N_{s}}} a n d J_{k} = {j_{l} \in {\hat{J}}_{k} | l \leq \min {| {\hat{J}}_{k} |, N_{\max} - N_{s}}}

\begin{array}{l} \min_{d_{I_{k}}^{k}} {(\nabla F (z^{k}))}_{I_{k}} d_{I_{k}}^{k} + \frac{1}{2} d_{I_{k}}^{k}^{T} B_{I_{k} I_{k}} d_{I_{k}}^{k} \equiv F_{s u b} (d_{I_{k}}^{k}) \\ s . t . z_{I_{k}}^{k} + d_{I_{k}}^{k} \geq 0 \end{array}

\begin{array}{l} \min \frac{1}{2} {‖ Α x - b ‖}_{2}^{2} + λ \sum_{i = 1}^{n} u_{i} \\ s . t . | x_{i} | \leq u_{i}, i = 1, \dots, n . \end{array}

\min \frac{1}{2} {‖ Α x - b ‖}_{2}^{2} + λ \sum_{i = 1}^{n} u_{i} - \frac{1}{t} \sum_{i = 1}^{n} [\log (u_{i} + x_{i}) + \log (u_{i} - x_{i})] \equiv F_{t} (x, u)

\nabla^{2} F_{t} (x, u) [\begin{matrix} Δ x \\ Δ u \end{matrix}] = - \nabla F_{t} (x, u)

{\begin{cases} \min {‖ x ‖}_{1} \\ s . t . Α x = b \\ x \geq 0 \end{cases}

\begin{array}{l} Primal(P) : \min c^{T} x Dual(D) : \max b^{T} y \\ s . t . Α x = b s . t . Α^{T} y + s = c \\ x \geq 0 s \geq 0 \end{array}

\begin{array}{l} P_{θ} : \min c^{T} x - θ \sum_{j = 1}^{n} In x_{j} \\ s . t . Α x = b \\ x > 0 \end{array}

{\begin{cases} Α x = b, x > 0 \\ Α^{T} y + s = c \\ \frac{1}{θ} X S e - e = 0 \end{cases}

{\begin{cases} A Δ x = b - A x^{k} \\ A^{T} Δ y + Δ s = c - A^{T} y^{k} - s^{k} \\ S^{k} Δ x + X^{k} Δ s = θ e - X^{k} S^{k} e \end{cases}

\min_{x \geq 0} ω x subject to A x = b

x^{i + 1} = soft(x^{i} {+(A}^{t} (b - A x^{i})) / α, ω λ / (2 α))

Material	Muscle	Lungs	Heart	Bone	Liver
μ_a [mm^–1]	0.010	0.350	0.200	0.002	0.035
μ_s [mm^–1]	4.000	23.000	16.000	20.000	6.000
g	0.900	0.940	0.850	0.900	0.900

Noise level	Method	TTLS	Tikhonov	TNIPM	IVTCG	PDIP	WISTA
10%	Loc_Err(mm)	1.843	2.084	1.550	2.779	1.234	2.084
10%	Time(s)	6.314	0.984	11.289	0.955	7.128	2.273
20%	Loc_Err(mm)	1.843	2.084	1.550	2.779	1.311	2.084
20%	Time(s)	4.423	0.961	13.090	0.915	8.858	3.059
30%	Loc_Err(mm)	1.843	0.406	1.550	2.779	1.311	2.084
30%	Time(s)	4.933	2.084	14.329	1.3080	5.7387	2.566
40%	Loc_Err(mm)	2.431	2.084	1.550	2.779	1.311	2.084
40%	Time(s)	6.281	2.702	14.237	1.077	6.066	3.022
50%	Loc_Err(mm)	1.258	2.084	1.550	2.779	1.536	2.084
50%	Time(s)	4.020	0.746	11.265	0.875	6.017	1.834

Case	Method	TTLS	Tikhonov	TNIPM	IVTCG	PDIP	WISTA
P1	Loc_Err(mm)	0.782	1.830	1.753	1.956	2.005	1.955
P1	Time(s)	4.741	0.804	10.003	0.408	11.223	0.031
P2	Loc_Err(mm)	2.689	1.416	1.818	0.470	2.005	1.955
P2	Time(s)	3.793	0.897	8.158	2.571	12.821	0.106
P3	Loc_Err(mm)	2.452	1.543	1.753	1.956	2.652	1.955
P3	Time(s)	3.760	0.977	9.532	0.228	6.262	0.046
P4	Loc_Err(mm)	2.345	1.543	1.995	0.470	1.485	1.955
P4	Time(s)	4.774	1.022	8.198	0.205	5.171	0.113

Case	Method	TTLS	Tikhonov	TNIPM	IVTCG	PDIP	WISTA
N1	Loc_Err(mm)	3.061	2.084	3.443	2.680	1.311	1.955
N1	Time(s)	3.687	0.945	12.664	0.978	5.063	0.154
N2	Loc_Err(mm)	3.061	2.084	3.443	2.680	1.311	1.955
N2	Time(s)	4.802	1.044	12.859	1.314	5.167	0.230
N3	Loc_Err(mm)	3.504	2.084	3.038	2.084	0.410	1.955
N3	Time(s)	5.413	0.980	21.104	0.291	6.348	0.233
N4	Loc_Err(mm)	1.602	2.084	1.550	2.779	0.410	2.084
N4	Time(s)	14.881	6.021	14.592	0.820	6.883	2.340

Case	Method	TTLS	Tikhonov	TNIPM	IVTCG	PDIP	WISTA
1	Loc_Err(mm)	2.027	1.593	1.836	1.582	2.802	1.582
1	Time(s)	4.004	0.944	10.686	0.486	10.687	1.128
2	Loc_Err(mm)	1.602	2.084	1.550	2.779	0.410	2.084
2	Time(s)	14.881	6.021	14.592	0.820	6.883	2.340
3	Loc_Err(mm)	3.293	1.089	1.089	2.941	1.448	2.272
3	Time(s)	5.443	1.926	10.980	0.326	7.362	2.020

Case	Method	Source	TTLS	Tikhonov	TNIPM	IVTCG	PDIP	WISTA
1	Loc_Err(mm)	Source1	2.125	0.603	0.603	1.631	1.687	1.851
	Loc_Err(mm)	Source2	-	1.719	0.514	1.166	1.376	1.746
	Time(s)		3.623	1.127	6.237	0.248	3.075	0.082
2	Loc_Err(mm)	Source1	1.996	1.988	0.772	0.772	1.230	1.777
	Loc_Err(mm)	Source2	2.029	2.173	0.626	1.401	1.493	1.886
	Time(s)		4.707	1.189	5.542	0.262	2.708	0.125
3	Loc_Err(mm)	Source1	-	-	1.199	1.199	0.6262	1.953
	Loc_Err(mm)	Source2	1.910	1.369	1.230	1.010	1.811	1.634
	Time(s)		6.142	0.779	13.822	0.747	2.043	0.352

Method	Tikhonov	TNIPM	IVTCG	PDIP	WISTA
Loc_Err(mm)	5.004	19.023	5.004	5.624	4.535
Time(s)	57.597	91.537	444.986	1241.296	101.235