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 lp 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.

© 2012 OSA

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2012

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

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]

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. Imaging2011, 203537 (2011).
[CrossRef] [PubMed]

2010

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]

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]

W. Cong and G. Wang, “Bioluminescence tomography based on the phase approximation model,” J. Opt. Soc. Am. A27(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. Express18(3), 2894–2912 (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. Express18(24), 24825–24841 (2010).
[CrossRef] [PubMed]

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. Imaging2010, 291874 (2010).
[CrossRef] [PubMed]

H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization,” Opt. Express18(3), 1854–1871 (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. Express18(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. Express18(7), 6477–6491 (2010).
[CrossRef] [PubMed]

2009

2008

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

R. Weissleder and M. J. Pittet, “Imaging in the era of molecular oncology,” Nature452(7187), 580–589 (2008).
[CrossRef] [PubMed]

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. Express16(20), 15640–15654 (2008).
[CrossRef] [PubMed]

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]

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]

2007

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

2005

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. Express13(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

2003

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. F. Meinel, “Development of the first bioluminescence CT scanner,” Radiology229(P), 566 (2003).

1995

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

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

Arridge, S. R.

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]

Bading, J. R.

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]

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.

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]

Chen, D.

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]

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, 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. Imaging2011, 203537 (2011).
[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. Express18(24), 24825–24841 (2010).
[CrossRef] [PubMed]

Chen, X.

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]

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. 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]

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]

Cherry, S. R.

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]

Chu, M.

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]

Cong, A.

Cong, W.

Conti, P. S.

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]

Darvas, F.

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]

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]

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]

Dehghani, H.

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]

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]

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]

Delpy, D. T.

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]

Donoho, D.

D. Donoho, “Compresse sensing,” IEEE Trans. Inf. Theory52(4), 1289–1306 (2006).
[CrossRef]

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]

Douraghy, A.

Durairaj, K.

Feng, J.

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. Express18(7), 6477–6491 (2010).
[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. Express17(24), 21925–21934 (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. Express17(19), 16834–16848 (2009).
[CrossRef] [PubMed]

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. Express16(20), 15640–15654 (2008).
[CrossRef] [PubMed]

Gao, H.

H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization,” Opt. Express18(3), 1854–1871 (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. Express18(3), 2894–2912 (2010).
[CrossRef] [PubMed]

Gu, X.

Han, D.

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. Express18(7), 6477–6491 (2010).
[CrossRef] [PubMed]

Han, R.

He, X.

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. Imaging2011, 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]

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. Imaging2010, 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]

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. Express18(24), 24825–24841 (2010).
[CrossRef] [PubMed]

Henry, M.

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]

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. Express14(17), 7801–7809 (2006).
[CrossRef] [PubMed]

Hiraoka, M.

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[CrossRef] [PubMed]

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]

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

Yan, G.

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. Express17(24), 21925–21934 (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. Express17(19), 16834–16848 (2009).
[CrossRef] [PubMed]

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. Express16(20), 15640–15654 (2008).
[CrossRef] [PubMed]

Yang, D.

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]

Yang, W.

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

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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).
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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. Express18(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. Express18(7), 6477–6491 (2010).
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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. Express17(19), 16834–16848 (2009).
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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. Express17(10), 8062–8080 (2009).
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Zhao, H.

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).
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H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization,” Opt. Express18(3), 1854–1871 (2010).
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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. Express18(3), 2894–2912 (2010).
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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. Express18(7), 6477–6491 (2010).
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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. Express18(7), 6477–6491 (2010).
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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. Express16(20), 15640–15654 (2008).
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J. Comput. Phys.

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).
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Opt. Express

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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. Express16(20), 15640–15654 (2008).
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[CrossRef] [PubMed]

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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. Express17(10), 8062–8080 (2009).
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Figures (16)

Fig. 1
Fig. 1

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

Fig. 2
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).

Fig. 3
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.

Fig. 4
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.

Fig. 5
Fig. 5

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

Fig. 6
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.

Fig. 7
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.

Fig. 8
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.

Fig. 9
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.

Fig. 10
Fig. 10

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

Fig. 11
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.

Fig. 12
Fig. 12

Error bar chart of the Loc_Err at different source positions.

Fig. 13
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.

Fig. 14
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.

Fig. 15
Fig. 15

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

Fig. 16
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.

Tables (7)

Tables Icon

Table 1 Optical parameters of the heterogeneous phantom [39]

Tables Icon

Table 2 Reconstruction results at different measurement noise levels

Tables Icon

Table 3 Reconstruction results using different optical parameters

Tables Icon

Table 4 Reconstruction results for tissue specificity

Tables Icon

Table 5 Reconstruction results at different source positions

Tables Icon

Table 6 Reconstruction results in double sources

Tables Icon

Table 7 Reconstruction results of the in vivo mouse experiment

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

Α S = b
min A x b 2 2 + λ x p
min Α x b 2 2 + λ x 2 2
min 1 2 Α x b 2 2 + λ x 1
min z c T z + 1 2 z T B z F ( z ) s . t .   z 0
Γ k = { i | i { 1 , , 2 N } , [ ( z k ) i > 0 , ( F ( z k ) ) i = 0 ] o r [ ( z k ) i = 0 , ( F ( z k ) ) i < 0 ] }
I k = { i l I ^ k | l min { | I ^ k | , N s } } a n d   J k = { j l J ^ k | l min { | J ^ k | , N max N s } }
min 1 2 Α x b 2 2 + λ i = 1 n u i s . t .     | x i | u i ,      i = 1 , , n .
min 1 2 Α x b 2 2 + λ i = 1 n u i 1 t i = 1 n [ log ( u i + x i ) + log ( u i - x i ) ] F t ( x , u )
2 F t ( x , u ) [ Δ x Δ u ] = F t ( x , u )
{ min x 1 s . t . Α x = b x 0
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 0 s 0
P θ : min c T x θ j = 1 n In x j s . t . Α x = b x > 0
{ Α x = b , x > 0 Α T y + s = c 1 θ X S e e = 0
{ 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

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