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 (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]

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]

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]

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]

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]

2009 (5)

2008 (5)

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]

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]

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)

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)

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.

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. Imaging 2011, 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. Express 18(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. Theory 52(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.

Gao, H.

Gu, X.

Han, D.

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. 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]

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]

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]

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. Express 14(17), 7801–7809 (2006).
[Crossref] [PubMed]

Hiraoka, M.

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]

Hoffman, E.

Hoffman, E. A.

Hou, Y.

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]

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]

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]

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]

Hu, Z.

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[Crossref]

Huang, H.

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).
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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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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).
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Jiang, H.

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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).
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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).
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Larcom, L.

Larson, G.

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).
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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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Liu, J.

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).
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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).
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McLennan, G.

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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).
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H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys. 35(11), 4863–4871 (2008).
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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).
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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).
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Qu, 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).
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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).
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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).
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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).
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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).
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Smith, D. J.

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Sun, L.

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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.

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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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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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).
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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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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).
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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).
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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).
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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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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).
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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).
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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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Wang, L. V.

Wang, X.

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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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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).
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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).
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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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Adv. Sci. Lett. (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).
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IEEE Geosci. Remote Sens. Lett. (1)

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).
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IEEE Signal Process. Mag. (1)

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Int. J. Biomed. Imaging (2)

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).
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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).
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J. Comput. Phys. (1)

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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J. Opt. Soc. Am. A (1)

Med. Phys. (3)

G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. 31(8), 2289–2299 (2004).
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H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys. 35(11), 4863–4871 (2008).
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Nature (1)

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Opt. Commun. (1)

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]

Opt. Express (14)

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

Opt. Lett. (1)

Phys. Med. Biol. (3)

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]

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]

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]

Radiology (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).

Sci. China Inform. Sci. (1)

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

SIAM J. Sci. Comput. (1)

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]

Soviet Math. Dokl. (1)

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

Other (4)

E. Candès, “Compressive sampling,” in Proceedings of the International Congress of Mathematicians (ICM, 2006), pp. 1433–1452.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Springer, 2000).

M. Rodriguez-Porcel, J. Wu, and S. Gambhir, “Molecular imaging of stem cells,” in StemBook [Internet] (Harvard Stem Cell Institute,Cambridge, MA, 2008), available from http://www.ncbi.nlm.nih.gov/books/NBK27079/

F. S. Azar and X. Intes, Translational Multimodality Optical Imaging (Artech House, 2008), Chap. 17.

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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)

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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

Equations (23)

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( D ( r ) Φ ( r ) ) + μ a ( r ) Φ ( r ) = S ( r ) ( r Ω )
Φ ( r ) + 2 A n ( r ) D ( r ) ( v ( r ) Φ ( r ) ) = 0 ( r Ω )
Q ( r ) = D ( r ) ( v ( r ) Φ ( r ) ) = 1 2 A n ( r ) Φ ( r ) ( r 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 = i = 1 n σ i 2 σ i 2 + λ u i T b σ i v i
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 d I k k   ( F ( z k ) ) I k d I k k + 1 2 d I k k T B I k I k d I k k F s u b ( d I k k ) s . t .   z I k k + d I k k 0
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
min x 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 ) )

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