Abstract

In this paper we study the regularization with both l1 and total-variation norm for bioluminescence tomography based on radiative transfer equation, compare l1 data fidelity with l2 data fidelity for different type of noise, and propose novel interior-point methods for solving related optimization problems. Simulations are performed to show that our approach is not only capable of preserving shapes, details and intensities of bioluminescent sources in the presence of sparse or non-sparse sources with angular-resolved or angular-averaged data, but also robust to noise, and thus is potential for efficient high-resolution imaging with only boundary data.

© 2010 OSA

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2009

H. Gao and H. K. Zhao, “A fast forward solver of radiative transfer equation,” Transp. Theory Stat. Phys. 38(3), 149–192 (2009).
[CrossRef]

J. Yang, Y. Zhang, and W. Yin, “An Efficient TVL1 Algorithm for Deblurring Multichannel Images Corrupted by Impulsive Noise,” SIAM J. Sci. Comput. 31(4), 2842–2865 (2009).
[CrossRef]

T. Goldstein and S. Osher, “The split bregman method for l1 regularized problems,” SIAM J. Imaging Sci. 2(2), 323–343 (2009).
[CrossRef]

2008

P. Stefanov and G. Uhlmann, “An inverse source problem in optical molecular imaging,” Analysis and PDE 1, 115–126 (2008).
[CrossRef]

2007

Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental study on bioluminescence tomography with multimodality fusion,” Int. J. Biomed. Imaging 2007, 1 (2007).
[CrossRef]

G. Bal and A. Tamasan, “Inverse source problems in transport equations,” SIAM J. Math. Anal. 39(1), 57–76 (2007).
[CrossRef]

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

2006

K. Ren, G. Bal, and A. H. Hielscher, “Frequency domain optical tomography based on the equation of radiative transfer,” SIAM J. Sci. Comput. 28(4), 1463–1489 (2006).
[CrossRef]

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[CrossRef] [PubMed]

G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–9 (2006).
[CrossRef]

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

E. J. Candes and T. Tao, “Near optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52(12), 5406–5425 (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]

2005

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]

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

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50(17), 4225–4241 (2005).
[CrossRef] [PubMed]

C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging 4, 370 (2005).

T. F. Chan and S. Esedoglu, “Aspects of total variation regularized L1 function approximation,” SIAM J. Appl. Math. 65(5), 1817–1837 (2005).
[CrossRef]

W. Yin, T. Chen, X. S. Zhou, and A. Chakraborty, “Background correction for cDNA microarray image using the TV + L1 model,” Bioinformatics 21(10), 2410 (2005).
[CrossRef] [PubMed]

T. Chen, T. Huang, W. Yin, and X. S. Zhou, “A new coarse-to-fine framework for 3D brain MR image registration,” Computer Vision for Biomedical Image 3765, 114–124 (2005).
[CrossRef]

2004

M. Nikolova, “A variational approach to remove outliers and impulse noise,” J. Math. Imaging Vis. 20(1/2), 99–120 (2004).
[CrossRef]

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]

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

2003

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

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19(6), S165–S187 (2003).
[CrossRef]

2002

M. Nikolova, “Minimizers of cost-functions involving non-smooth data fidelity terms. Application to the processing of outliers,” SIAM J. Numer. Anal. 40(3), 965–994 (2002).
[CrossRef]

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging 16(4), 378–387 (2002).
[CrossRef] [PubMed]

1999

A. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), 022 (1999).
[CrossRef]

1996

R. Tibshirani, “Regression shrinkage and selection via the Lasso,” J. R. Stat. Soc., B 58, 267–288 (1996).

1992

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” J. Phys. D 60(1-4), 259–268 (1992).
[CrossRef]

S. Alliney, “Digital filters as absolute norm regularizers,” IEEE Trans. Signal Process. 40(6), 1548–1562 (1992).
[CrossRef]

1963

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11(2), 431–441 (1963).
[CrossRef]

1944

K. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).

Alexandrakis, G.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50(17), 4225–4241 (2005).
[CrossRef] [PubMed]

Alliney, S.

S. Alliney, “Digital filters as absolute norm regularizers,” IEEE Trans. Signal Process. 40(6), 1548–1562 (1992).
[CrossRef]

Arridge, A. R.

A. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), 022 (1999).
[CrossRef]

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]

Bal, G.

G. Bal and A. Tamasan, “Inverse source problems in transport equations,” SIAM J. Math. Anal. 39(1), 57–76 (2007).
[CrossRef]

K. Ren, G. Bal, and A. H. Hielscher, “Frequency domain optical tomography based on the equation of radiative transfer,” SIAM J. Sci. Comput. 28(4), 1463–1489 (2006).
[CrossRef]

Boyd, S.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

Candes, E. J.

E. J. Candes and T. Tao, “Near optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[CrossRef]

Chakraborty, A.

W. Yin, T. Chen, X. S. Zhou, and A. Chakraborty, “Background correction for cDNA microarray image using the TV + L1 model,” Bioinformatics 21(10), 2410 (2005).
[CrossRef] [PubMed]

Chan, T.

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19(6), S165–S187 (2003).
[CrossRef]

Chan, T. F.

T. F. Chan and S. Esedoglu, “Aspects of total variation regularized L1 function approximation,” SIAM J. Appl. Math. 65(5), 1817–1837 (2005).
[CrossRef]

Chatziioannou, A. F.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50(17), 4225–4241 (2005).
[CrossRef] [PubMed]

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, T.

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[CrossRef] [PubMed]

W. Yin, T. Chen, X. S. Zhou, and A. Chakraborty, “Background correction for cDNA microarray image using the TV + L1 model,” Bioinformatics 21(10), 2410 (2005).
[CrossRef] [PubMed]

T. Chen, T. Huang, W. Yin, and X. S. Zhou, “A new coarse-to-fine framework for 3D brain MR image registration,” Computer Vision for Biomedical Image 3765, 114–124 (2005).
[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]

Comaniciu, D.

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[CrossRef] [PubMed]

Cong, A.

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

Cong, W.

Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental study on bioluminescence tomography with multimodality fusion,” Int. J. Biomed. Imaging 2007, 1 (2007).
[CrossRef]

G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–9 (2006).
[CrossRef]

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

Contag, C. H.

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging 16(4), 378–387 (2002).
[CrossRef] [PubMed]

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]

Coquoz, O.

C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging 4, 370 (2005).

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, 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.

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]

Donoho, D.

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

Durairaj, K.

G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–9 (2006).
[CrossRef]

Esedoglu, S.

T. F. Chan and S. Esedoglu, “Aspects of total variation regularized L1 function approximation,” SIAM J. Appl. Math. 65(5), 1817–1837 (2005).
[CrossRef]

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” J. Phys. D 60(1-4), 259–268 (1992).
[CrossRef]

Gao, H.

H. Gao and H. K. Zhao, “A fast forward solver of radiative transfer equation,” Transp. Theory Stat. Phys. 38(3), 149–192 (2009).
[CrossRef]

Goldstein, T.

T. Goldstein and S. Osher, “The split bregman method for l1 regularized problems,” SIAM J. Imaging Sci. 2(2), 323–343 (2009).
[CrossRef]

Gorinevsky, D.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

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]

Hielscher, A. H.

K. Ren, G. Bal, and A. H. Hielscher, “Frequency domain optical tomography based on the equation of radiative transfer,” SIAM J. Sci. Comput. 28(4), 1463–1489 (2006).
[CrossRef]

Hoffman, E.

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

Hoffman, E. A.

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

Huang, T.

T. Chen, T. Huang, W. Yin, and X. S. Zhou, “A new coarse-to-fine framework for 3D brain MR image registration,” Computer Vision for Biomedical Image 3765, 114–124 (2005).
[CrossRef]

Huang, T. S.

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[CrossRef] [PubMed]

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.

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

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

Jiang, S.

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]

Kim, S. J.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

Koh, K.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

Kumar, D.

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

Kuo, C.

C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging 4, 370 (2005).

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]

Leahy, R. M.

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]

Levenberg, K.

K. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).

Li, Y.

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

Liu, Y.

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

Lustig, M.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

Lv, Y.

Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental study on bioluminescence tomography with multimodality fusion,” Int. J. Biomed. Imaging 2007, 1 (2007).
[CrossRef]

Marquardt, D. W.

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11(2), 431–441 (1963).
[CrossRef]

McCray, P.

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

McLennan, G.

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

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

Meinel, J.

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

Moats, R. A.

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]

Nikolova, M.

M. Nikolova, “A variational approach to remove outliers and impulse noise,” J. Math. Imaging Vis. 20(1/2), 99–120 (2004).
[CrossRef]

M. Nikolova, “Minimizers of cost-functions involving non-smooth data fidelity terms. Application to the processing of outliers,” SIAM J. Numer. Anal. 40(3), 965–994 (2002).
[CrossRef]

Osher, S.

T. Goldstein and S. Osher, “The split bregman method for l1 regularized problems,” SIAM J. Imaging Sci. 2(2), 323–343 (2009).
[CrossRef]

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” J. Phys. D 60(1-4), 259–268 (1992).
[CrossRef]

Patterson, M. S.

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]

Paulsen, K. D.

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]

Pogue, B. W.

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]

Qian, X.

G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–9 (2006).
[CrossRef]

Rannou, F. R.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50(17), 4225–4241 (2005).
[CrossRef] [PubMed]

Ren, K.

K. Ren, G. Bal, and A. H. Hielscher, “Frequency domain optical tomography based on the equation of radiative transfer,” SIAM J. Sci. Comput. 28(4), 1463–1489 (2006).
[CrossRef]

Rice, B.

C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging 4, 370 (2005).

Ross, B. D.

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging 16(4), 378–387 (2002).
[CrossRef] [PubMed]

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” J. Phys. D 60(1-4), 259–268 (1992).
[CrossRef]

Shen, H.

G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–9 (2006).
[CrossRef]

Smith, D. 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]

Stefanov, P.

P. Stefanov and G. Uhlmann, “An inverse source problem in optical molecular imaging,” Analysis and PDE 1, 115–126 (2008).
[CrossRef]

Strong, D.

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19(6), S165–S187 (2003).
[CrossRef]

Suter, M.

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

Tamasan, A.

G. Bal and A. Tamasan, “Inverse source problems in transport equations,” SIAM J. Math. Anal. 39(1), 57–76 (2007).
[CrossRef]

Tao, T.

E. J. Candes and T. Tao, “Near optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[CrossRef]

Tian, J.

Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental study on bioluminescence tomography with multimodality fusion,” Int. J. Biomed. Imaging 2007, 1 (2007).
[CrossRef]

Tibshirani, R.

R. Tibshirani, “Regression shrinkage and selection via the Lasso,” J. R. Stat. Soc., B 58, 267–288 (1996).

Troy, T.

C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging 4, 370 (2005).

Uhlmann, G.

P. Stefanov and G. Uhlmann, “An inverse source problem in optical molecular imaging,” Analysis and PDE 1, 115–126 (2008).
[CrossRef]

Wang, G.

Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental study on bioluminescence tomography with multimodality fusion,” Int. J. Biomed. Imaging 2007, 1 (2007).
[CrossRef]

G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–9 (2006).
[CrossRef]

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

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

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

Wang, L.

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

Wang, L. V.

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

Yang, J.

J. Yang, Y. Zhang, and W. Yin, “An Efficient TVL1 Algorithm for Deblurring Multichannel Images Corrupted by Impulsive Noise,” SIAM J. Sci. Comput. 31(4), 2842–2865 (2009).
[CrossRef]

Yin, W.

J. Yang, Y. Zhang, and W. Yin, “An Efficient TVL1 Algorithm for Deblurring Multichannel Images Corrupted by Impulsive Noise,” SIAM J. Sci. Comput. 31(4), 2842–2865 (2009).
[CrossRef]

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[CrossRef] [PubMed]

W. Yin, T. Chen, X. S. Zhou, and A. Chakraborty, “Background correction for cDNA microarray image using the TV + L1 model,” Bioinformatics 21(10), 2410 (2005).
[CrossRef] [PubMed]

T. Chen, T. Huang, W. Yin, and X. S. Zhou, “A new coarse-to-fine framework for 3D brain MR image registration,” Computer Vision for Biomedical Image 3765, 114–124 (2005).
[CrossRef]

Zabner, J.

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

Zhang, Q.

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]

Zhang, Y.

J. Yang, Y. Zhang, and W. Yin, “An Efficient TVL1 Algorithm for Deblurring Multichannel Images Corrupted by Impulsive Noise,” SIAM J. Sci. Comput. 31(4), 2842–2865 (2009).
[CrossRef]

Zhao, H. K.

H. Gao and H. K. Zhao, “A fast forward solver of radiative transfer equation,” Transp. Theory Stat. Phys. 38(3), 149–192 (2009).
[CrossRef]

Zhou, X. S.

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[CrossRef] [PubMed]

W. Yin, T. Chen, X. S. Zhou, and A. Chakraborty, “Background correction for cDNA microarray image using the TV + L1 model,” Bioinformatics 21(10), 2410 (2005).
[CrossRef] [PubMed]

T. Chen, T. Huang, W. Yin, and X. S. Zhou, “A new coarse-to-fine framework for 3D brain MR image registration,” Computer Vision for Biomedical Image 3765, 114–124 (2005).
[CrossRef]

Zwarg, D.

C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging 4, 370 (2005).

Analysis and PDE

P. Stefanov and G. Uhlmann, “An inverse source problem in optical molecular imaging,” Analysis and PDE 1, 115–126 (2008).
[CrossRef]

Bioinformatics

W. Yin, T. Chen, X. S. Zhou, and A. Chakraborty, “Background correction for cDNA microarray image using the TV + L1 model,” Bioinformatics 21(10), 2410 (2005).
[CrossRef] [PubMed]

Computer Vision for Biomedical Image

T. Chen, T. Huang, W. Yin, and X. S. Zhou, “A new coarse-to-fine framework for 3D brain MR image registration,” Computer Vision for Biomedical Image 3765, 114–124 (2005).
[CrossRef]

IEEE J. Sel. Top. Signal Process.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

IEEE Trans. Inf. Theory

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

E. J. Candes and T. Tao, “Near optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

T. Chen, W. Yin, X. S. Zhou, D. Comaniciu, and T. S. Huang, “Total variation models for variable lighting face recognition,” IEEE Trans. Pattern Anal. Mach. Intell. 28(9), 1519–1524 (2006).
[CrossRef] [PubMed]

IEEE Trans. Signal Process.

S. Alliney, “Digital filters as absolute norm regularizers,” IEEE Trans. Signal Process. 40(6), 1548–1562 (1992).
[CrossRef]

Int. J. Biomed. Imaging

G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–9 (2006).
[CrossRef]

Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental study on bioluminescence tomography with multimodality fusion,” Int. J. Biomed. Imaging 2007, 1 (2007).
[CrossRef]

Inverse Probl.

A. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), 022 (1999).
[CrossRef]

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19(6), S165–S187 (2003).
[CrossRef]

J. Magn. Reson. Imaging

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging 16(4), 378–387 (2002).
[CrossRef] [PubMed]

J. Math. Imaging Vis.

M. Nikolova, “A variational approach to remove outliers and impulse noise,” J. Math. Imaging Vis. 20(1/2), 99–120 (2004).
[CrossRef]

J. Phys. D

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” J. Phys. D 60(1-4), 259–268 (1992).
[CrossRef]

J. R. Stat. Soc., B

R. Tibshirani, “Regression shrinkage and selection via the Lasso,” J. R. Stat. Soc., B 58, 267–288 (1996).

J. Soc. Ind. Appl. Math.

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11(2), 431–441 (1963).
[CrossRef]

Med. Phys.

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

Mol. Imaging

C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging 4, 370 (2005).

Opt. Express

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. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13(18), 6756–6771 (2005).
[CrossRef] [PubMed]

Opt. Lett.

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]

Phys. Med. Biol.

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]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50(17), 4225–4241 (2005).
[CrossRef] [PubMed]

Q. Appl. Math.

K. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).

Radiology

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

SIAM J. Appl. Math.

T. F. Chan and S. Esedoglu, “Aspects of total variation regularized L1 function approximation,” SIAM J. Appl. Math. 65(5), 1817–1837 (2005).
[CrossRef]

SIAM J. Imaging Sci.

T. Goldstein and S. Osher, “The split bregman method for l1 regularized problems,” SIAM J. Imaging Sci. 2(2), 323–343 (2009).
[CrossRef]

SIAM J. Math. Anal.

G. Bal and A. Tamasan, “Inverse source problems in transport equations,” SIAM J. Math. Anal. 39(1), 57–76 (2007).
[CrossRef]

SIAM J. Numer. Anal.

M. Nikolova, “Minimizers of cost-functions involving non-smooth data fidelity terms. Application to the processing of outliers,” SIAM J. Numer. Anal. 40(3), 965–994 (2002).
[CrossRef]

SIAM J. Sci. Comput.

K. Ren, G. Bal, and A. H. Hielscher, “Frequency domain optical tomography based on the equation of radiative transfer,” SIAM J. Sci. Comput. 28(4), 1463–1489 (2006).
[CrossRef]

J. Yang, Y. Zhang, and W. Yin, “An Efficient TVL1 Algorithm for Deblurring Multichannel Images Corrupted by Impulsive Noise,” SIAM J. Sci. Comput. 31(4), 2842–2865 (2009).
[CrossRef]

Transp. Theory Stat. Phys.

H. Gao and H. K. Zhao, “A fast forward solver of radiative transfer equation,” Transp. Theory Stat. Phys. 38(3), 149–192 (2009).
[CrossRef]

Other

E. Giusti, Minimal surfaces and functions of bounded variation (Birkhäuser, 1984).

S. Boyd, and L. Vandenberghe, Convex Optimization (Cambridge university press, 2004).

J. Demmel, Applied Numerical Linear Algebra (Cambidge Univ. Press, 1997).

S. J. Kim, K. Koh, M. Lustig, and S. Boyd, “An efficient method for compressed sensing,” IEEE International Conference on Image Processing 3, 117–120 (2007).

Y. Zhang, “Theory of compressive sensing via l1-minimization: a non-RIP analysis and extensions,” Rice University CAAM Technical Report TR08–11, (2008).

H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization,” In press, (2010).

S. Chandrasekhar, Radiative Transfer (Dover Publications, 1960).

K. M. Case, and P. F. P. F. Zweifel, Linear Transport Theory (Addison-Wesley Educational Publishers Inc., 1967).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, 1978).

E. E. Lewis, and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).

K. Madsen, H. B. Nielsen, and O. Tingleff, Methods for non-linear least squares problems (Technical University of Denmark, 1999).

J. Nocedal, and S. J. Wright, Numerical Optimization (Springer, 1999).

E. Esser, X. Zhang, and T. Chan, “A general framework for a class of first order primal-dual algorithms for TV minimization,” UCLA CAM Report 09–67, (2009).

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Figures (16)

Fig. 14
Fig. 14

Reconstructions with angular-resolved data for non-piecewise-constant source with g = 0.9. Figure (a) is the true source. Figure (b), (c), (d), (e) and (f) are for l2 data fidelity with l2, H1, l1, TV, l1 + TV regularization respectively. Figure (g), (h) and (i) are for l1 data fidelity with l1, TV, l1 + TV regularization respectively.

Fig. 15
Fig. 15

Reconstructions with angular-resolved data for non-piecewise-constant source with g = 0.0. Figure (a) is the true source. Figure (b), (c), (d), (e) and (f) are for l2 data fidelity with l2, H1, l1, TV, l1 + TV regularization respectively. Figure (g), (h) and (i) are for l1 data fidelity with l1, TV, l1 + TV regularization respectively.

Fig. 1
Fig. 1

l1 + TV regularized reconstructions of different square inclusions. Figures (a), (b) and (c) are true sources. Figures (d), (g) and (j) are reconstruction results of (a) with r = 0.2, 1 and 5 respectively. Figures (e), (h) and (k) are reconstruction results of (b) with r = 0.2, 1 and 5 respectively. Figures (f), (i) and (l) are reconstruction results of (c) with r = 0.2, 1 and 5 respectively.

Fig. 2
Fig. 2

Sources and meshes for non-sparse and sparse reconstructions. Source I and II are in Figure (a), (c) respectively with the corresponding meshes in Figure (b) and (d).

Fig. 13
Fig. 13

Reconstructions with angular-resolved data for Source I with different intensities with g = 0.9. Figure (a) is the true source. Figure (b) is the reconstructed image by l1 + TV regularization.

Fig. 16
Fig. 16

Comparison of l2 data fidelity with l1 data fidelity with l1 regularization. Figures (a), (c), (e) and (g) are reconstruction results with l2 data fidelity with 0% Gaussian noise, 10% Gausisan noise, 10% Gausisan noise and 20% salt-and-pepper noise, 10% Gausisan noise and 33% salt-and-pepper noise respectively. Figures (b), (d), (f) and (h) are reconstruction results with l1 data fidelity with 0% Gaussian noise, 10% Gausisan noise, 10% Gausisan noise and 20% salt-and-pepper noise, 10% Gausisan noise and 33% salt-and-pepper noise respectively.

Fig. 3
Fig. 3

Reconstructions with angular-averaged data for Source I with g = 0.9. Figures (a), (b), (c) and (d) are reconstructed images by l2, l1, TV and l1 + TV regularization respectively.

Fig. 10
Fig. 10

Reconstructions with angular-resolved data for Source II with g = 0.0. Figures (a), (b), (c) and (d) are reconstructed images by l2, l1, TV and l1 + TV regularization respectively.

Fig. 12
Fig. 12

Reconstructions with angular-resolved data for Source I with different intensities with g = 0.9. Figures (a), (b), (c) and (d) are reconstructed images by l2, l1, TV and l1 + TV regularization respectively.

Fig. 4
Fig. 4

Reconstructions with angular-resolved data for Source I with g = 0.9. Figures (a), (b), (c) and (d) are reconstructed images by l2, l1, TV and l1 + TV regularization respectively.

Fig. 5
Fig. 5

Reconstructions with angular-averaged data for Source I with g = 0.0. Figures (a), (b), (c) and (d) are reconstructed images by l2, l1, TV and l1 + TV regularization respectively.

Fig. 6
Fig. 6

Reconstructions with angular-resolved data for Source I with g = 0.0. Figures (a), (b), (c) and (d) are reconstructed images by l2, l1, TV and l1 + TV regularization respectively.

Fig. 7
Fig. 7

Reconstructions with angular-averaged data for Source II with g = 0.9. Figures (a), (b), (c) and (d) are reconstructed images by l2, l1, TV and l1 + TV regularization respectively.

Fig. 8
Fig. 8

Reconstructions with angular-resolved data for Source II with g = 0.9. Figures (a), (b), (c) and (d) are reconstructed images by l2, l1, TV and l1 + TV regularization respectively.

Fig. 9
Fig. 9

Reconstructions with angular-averaged data for Source II with g = 0.0. Figures (a), (b), (c) and (d) are reconstructed images by l2, l1, TV and l1 + TV regularization respectively.

Fig. 11
Fig. 11

Single-level reconstruction and two-level reconstruction for Source I with g = 0.0. Figures (a) is from single-level reconstruction with 360 angular-resolved measurements. Figures (b) is from two-level reconstruction with 180 angular-resolved measurements.

Equations (2)

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s ^ Ψ ( r , s ^ ) + ( μ a + μ s ) Ψ ( r , s ^ ) = μ s S ^ n 1 f ( s ^ , s ^ ) Ψ ( r , s ^ ) d s ^ + q ( r , s ^ ) Ψ ( r , s ^ ) = 0 , if s ^ n ^ < 0
min Q , u , v | | J Q X m | | 2 2 + λ 1 ( i = 1 N s A i u i + k = 1 N e L k v k ) ,

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