Abstract

Fluorescence molecular tomography (FMT) is a promising tomographic method in preclinical research, which enables noninvasive real-time three-dimensional (3-D) visualization for in vivo studies. The ill-posedness of the FMT reconstruction problem is one of the many challenges in the studies of FMT. In this paper, we propose a l2,1-norm optimization method using a priori information, mainly the structured sparsity of the fluorescent regions for FMT reconstruction. Compared to standard sparsity methods, the structured sparsity methods are often superior in reconstruction accuracy since the structured sparsity utilizes correlations or structures of the reconstructed image. To solve the problem effectively, the Nesterov’s method was used to accelerate the computation. To evaluate the performance of the proposed l2,1-norm method, numerical phantom experiments and in vivo mouse experiments are conducted. The results show that the proposed method not only achieves accurate and desirable fluorescent source reconstruction, but also demonstrates enhanced robustness to noise.

© 2016 Optical Society of America

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References

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2015 (2)

C. C. Leng and J. Tian, “Mathematical method in optical molecular imaging,” Sci. China Life Sci. 58, 1–13 (2015).
[PubMed]

Y. An, J. Liu, G. Zhang, J. Ye, Y. Du, Y. Mao, C. Chi, and J. Tian, “A Novel Region Reconstruction Method for Fluorescence Molecular Tomography,” IEEE Trans. Biomed. Eng. 62(7), 1818–1826 (2015).
[Crossref] [PubMed]

2014 (5)

J. Ye, Y. Du, Y. An, C. Chi, and J. Tian, “Reconstruction of fluorescence molecular tomography via a nonmonotone spectral projected gradient pursuit method,” J. Biomed. Opt. 19(12), 126013 (2014).
[Crossref] [PubMed]

C. Chen, F. Tian, H. Liu, and J. Huang, “Diffuse optical tomography enhanced by clustered sparsity for functional brain imaging,” IEEE Trans. Med. Imaging 33(12), 2323–2331 (2014).
[Crossref] [PubMed]

J. Prakash, C. B. Shaw, R. Manjappa, R. Kanhirodan, and P. K. Yalavarthy, “Sparse Recovery Methods Hold Promise for Diffuse Optical Tomographic Image Reconstruction,” IEEE J. Sel. Top. Quantum Electron. 20(2), 74–82 (2014).
[Crossref]

Ping Wu, Yifang Hu, Kun Wang, and Jie Tian, “Bioluminescence Tomography by an Iterative Reweighted (l)2-Norm Optimization,” IEEE Trans. Biomed. Eng. 61(1), 189–196 (2014).
[Crossref] [PubMed]

C. Chi, Y. Du, J. Ye, D. Kou, J. Qiu, J. Wang, J. Tian, and X. Chen, “Intraoperative imaging-guided cancer surgery: from current fluorescence molecular imaging methods to future multi-modality imaging technology,” Theranostics 4(11), 1072–1084 (2014).
[Crossref] [PubMed]

2011 (2)

2010 (8)

R. Baraniuk, V. Cevher, M. Duarte, and C. Hegde, “Model-based compressive sensing,” IEEE Trans. Inf. Theory 56(4), 1982–2001 (2010).
[Crossref]

J. Huang and T. Zhang, “The Benefit Of Group Sparsity,” Ann. Stat. 38(4), 1978–2004 (2010).
[Crossref] [PubMed]

D. Han, J. Tian, S. Zhu, J. Feng, C. Qin, B. Zhang, and X. Yang, “A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization,” Opt. Express 18(8), 8630–8646 (2010).
[Crossref] [PubMed]

M. Süzen, A. Giannoula, and T. Durduran, “Compressed sensing in diffuse optical tomography,” Opt. Express 18(23), 23676–23690 (2010).
[Crossref] [PubMed]

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7(8), 603–614 (2010).
[Crossref] [PubMed]

C. Qin, S. Zhu, and J. Tian, “New optical molecular imaging systems,” Curr. Pharm. Biotechnol. 11(6), 620 (2010).
[Crossref] [PubMed]

T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Ra. 43(4), 252–261 (2010).

X. Chen, X. Gao, D. Chen, X. Ma, X. Zhao, M. Shen, X. Li, X. Qu, J. Liang, J. Ripoll, and J. Tian, “3D reconstruction of light flux distribution on arbitrary surfaces from 2D multi-photographic images,” Opt. Express 18(19), 19876–19893 (2010).
[Crossref] [PubMed]

2009 (2)

2008 (4)

W. Bangerth and A. Joshi, “Adaptive finite element methods for the solution of inverse problems in optical tomography,” Inverse Probl. 24(3), 034011 (2008).
[Crossref]

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7(7), 591–607 (2008).
[Crossref] [PubMed]

W. Bangerth, “A Framework for the Adaptive Finite Element Solution of Large-Scale Inverse Problems,” SIAM J. Sci. Comput. 30(6), 2965–2989 (2008).
[Crossref]

G. Yan, J. Tian, S. Zhu, Y. Dai, and C. Qin, “Fast cone-beam CT image reconstruction using GPU hardware,” J. XRay Sci. Technol. 16(4), 225–234 (2008).

2007 (5)

2006 (3)

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

M. Yuan and Y. Lin, “Model selection and estimation in regression with grouped variables,” J. R. Stat. Soc. Ser. A Stat. Soc. 68(1), 49–67 (2006).
[Crossref]

F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “A linear, featured-data scheme for image reconstruction in time-domain fluorescence molecular tomography,” Opt. Express 14(16), 7109–7124 (2006).
[Crossref] [PubMed]

2005 (3)

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]

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (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]

2004 (2)

2003 (2)

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[Crossref] [PubMed]

D. J. Stephens and V. J. Allan, “Light microscopy techniques for live cell imaging,” Science 300(5616), 82–86 (2003).
[Crossref] [PubMed]

1997 (1)

Adibi, A.

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]

Allan, V. J.

D. J. Stephens and V. J. Allan, “Light microscopy techniques for live cell imaging,” Science 300(5616), 82–86 (2003).
[Crossref] [PubMed]

An, Y.

Y. An, J. Liu, G. Zhang, J. Ye, Y. Du, Y. Mao, C. Chi, and J. Tian, “A Novel Region Reconstruction Method for Fluorescence Molecular Tomography,” IEEE Trans. Biomed. Eng. 62(7), 1818–1826 (2015).
[Crossref] [PubMed]

J. Ye, Y. Du, Y. An, C. Chi, and J. Tian, “Reconstruction of fluorescence molecular tomography via a nonmonotone spectral projected gradient pursuit method,” J. Biomed. Opt. 19(12), 126013 (2014).
[Crossref] [PubMed]

Arridge, S. R.

T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Ra. 43(4), 252–261 (2010).

Bai, J.

Bangerth, W.

W. Bangerth and A. Joshi, “Adaptive finite element methods for the solution of inverse problems in optical tomography,” Inverse Probl. 24(3), 034011 (2008).
[Crossref]

W. Bangerth, “A Framework for the Adaptive Finite Element Solution of Large-Scale Inverse Problems,” SIAM J. Sci. Comput. 30(6), 2965–2989 (2008).
[Crossref]

A. Joshi, W. Bangerth, and E. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12(22), 5402–5417 (2004).
[Crossref] [PubMed]

Baraniuk, R.

R. Baraniuk, V. Cevher, M. Duarte, and C. Hegde, “Model-based compressive sensing,” IEEE Trans. Inf. Theory 56(4), 1982–2001 (2010).
[Crossref]

Barbour, R. L.

Boas, D. A.

Bouman, C. A.

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Cao, N.

Cevher, V.

R. Baraniuk, V. Cevher, M. Duarte, and C. Hegde, “Model-based compressive sensing,” IEEE Trans. Inf. Theory 56(4), 1982–2001 (2010).
[Crossref]

Chan, T. F.

Chang, J.

Chatziioannou, A. F.

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]

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]

Chen, C.

C. Chen, F. Tian, H. Liu, and J. Huang, “Diffuse optical tomography enhanced by clustered sparsity for functional brain imaging,” IEEE Trans. Med. Imaging 33(12), 2323–2331 (2014).
[Crossref] [PubMed]

Chen, D.

Chen, N.

Chen, X.

C. Chi, Y. Du, J. Ye, D. Kou, J. Qiu, J. Wang, J. Tian, and X. Chen, “Intraoperative imaging-guided cancer surgery: from current fluorescence molecular imaging methods to future multi-modality imaging technology,” Theranostics 4(11), 1072–1084 (2014).
[Crossref] [PubMed]

X. Chen, X. Gao, D. Chen, X. Ma, X. Zhao, M. Shen, X. Li, X. Qu, J. Liang, J. Ripoll, and J. Tian, “3D reconstruction of light flux distribution on arbitrary surfaces from 2D multi-photographic images,” Opt. Express 18(19), 19876–19893 (2010).
[Crossref] [PubMed]

Chi, C.

Y. An, J. Liu, G. Zhang, J. Ye, Y. Du, Y. Mao, C. Chi, and J. Tian, “A Novel Region Reconstruction Method for Fluorescence Molecular Tomography,” IEEE Trans. Biomed. Eng. 62(7), 1818–1826 (2015).
[Crossref] [PubMed]

J. Ye, Y. Du, Y. An, C. Chi, and J. Tian, “Reconstruction of fluorescence molecular tomography via a nonmonotone spectral projected gradient pursuit method,” J. Biomed. Opt. 19(12), 126013 (2014).
[Crossref] [PubMed]

C. Chi, Y. Du, J. Ye, D. Kou, J. Qiu, J. Wang, J. Tian, and X. Chen, “Intraoperative imaging-guided cancer surgery: from current fluorescence molecular imaging methods to future multi-modality imaging technology,” Theranostics 4(11), 1072–1084 (2014).
[Crossref] [PubMed]

Cong, A.

Cong, W.

Dai, Y.

G. Yan, J. Tian, S. Zhu, Y. Dai, and C. Qin, “Fast cone-beam CT image reconstruction using GPU hardware,” J. XRay Sci. Technol. 16(4), 225–234 (2008).

Dinkelborg, L. M.

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7(7), 591–607 (2008).
[Crossref] [PubMed]

Douraghy, A.

Du, Y.

Y. An, J. Liu, G. Zhang, J. Ye, Y. Du, Y. Mao, C. Chi, and J. Tian, “A Novel Region Reconstruction Method for Fluorescence Molecular Tomography,” IEEE Trans. Biomed. Eng. 62(7), 1818–1826 (2015).
[Crossref] [PubMed]

J. Ye, Y. Du, Y. An, C. Chi, and J. Tian, “Reconstruction of fluorescence molecular tomography via a nonmonotone spectral projected gradient pursuit method,” J. Biomed. Opt. 19(12), 126013 (2014).
[Crossref] [PubMed]

C. Chi, Y. Du, J. Ye, D. Kou, J. Qiu, J. Wang, J. Tian, and X. Chen, “Intraoperative imaging-guided cancer surgery: from current fluorescence molecular imaging methods to future multi-modality imaging technology,” Theranostics 4(11), 1072–1084 (2014).
[Crossref] [PubMed]

Duarte, M.

R. Baraniuk, V. Cevher, M. Duarte, and C. Hegde, “Model-based compressive sensing,” IEEE Trans. Inf. Theory 56(4), 1982–2001 (2010).
[Crossref]

Durduran, T.

Eftekhar, A. A.

Feng, J.

D. Han, J. Tian, S. Zhu, J. Feng, C. Qin, B. Zhang, and X. Yang, “A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization,” Opt. Express 18(8), 8630–8646 (2010).
[Crossref] [PubMed]

S. Zhu, J. Tian, G. Yan, C. Qin, and J. Feng, “Cone beam micro-CT system for small animal imaging and performance evaluation,” Int. J. Biomed. Imaging 2009, 960573 (2009).
[Crossref] [PubMed]

Gambhir, S. S.

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7(7), 591–607 (2008).
[Crossref] [PubMed]

Gao, F.

Gao, X.

Giannoula, A.

Graber, H. L.

Han, D.

Hegde, C.

R. Baraniuk, V. Cevher, M. Duarte, and C. Hegde, “Model-based compressive sensing,” IEEE Trans. Inf. Theory 56(4), 1982–2001 (2010).
[Crossref]

Hoffman, E.

Hoshi, Y.

Huang, J.

C. Chen, F. Tian, H. Liu, and J. Huang, “Diffuse optical tomography enhanced by clustered sparsity for functional brain imaging,” IEEE Trans. Med. Imaging 33(12), 2323–2331 (2014).
[Crossref] [PubMed]

J. Huang, T. Zhang, and D. Metaxas, “Learning with structured sparsity,” J. Mach. Learn. Res. 13(7), 3371–3412 (2011).

J. Huang and T. Zhang, “The Benefit Of Group Sparsity,” Ann. Stat. 38(4), 1978–2004 (2010).
[Crossref] [PubMed]

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

Jacob, L.

L. Jacob, G. Obozinski, and J. P. Vert, “Group lasso with overlap and graph lasso,” in International Conference on Machine Learning (ACM, 2009), pp. 433–440.
[Crossref]

Jacobs, M.

Ji, S.

J. Liu, S. Ji, and J. Ye, “Multi-Task Feature Learning Via Efficient L2,1-Norm Minimization,” Eprint Arxiv339–348 (2009).

Jiang, M.

Jie Tian,

Ping Wu, Yifang Hu, Kun Wang, and Jie Tian, “Bioluminescence Tomography by an Iterative Reweighted (l)2-Norm Optimization,” IEEE Trans. Biomed. Eng. 61(1), 189–196 (2014).
[Crossref] [PubMed]

Joshi, A.

Kanhirodan, R.

J. Prakash, C. B. Shaw, R. Manjappa, R. Kanhirodan, and P. K. Yalavarthy, “Sparse Recovery Methods Hold Promise for Diffuse Optical Tomographic Image Reconstruction,” IEEE J. Sel. Top. Quantum Electron. 20(2), 74–82 (2014).
[Crossref]

Kou, D.

C. Chi, Y. Du, J. Ye, D. Kou, J. Qiu, J. Wang, J. Tian, and X. Chen, “Intraoperative imaging-guided cancer surgery: from current fluorescence molecular imaging methods to future multi-modality imaging technology,” Theranostics 4(11), 1072–1084 (2014).
[Crossref] [PubMed]

Kumar, D.

Kun Wang,

Ping Wu, Yifang Hu, Kun Wang, and Jie Tian, “Bioluminescence Tomography by an Iterative Reweighted (l)2-Norm Optimization,” IEEE Trans. Biomed. Eng. 61(1), 189–196 (2014).
[Crossref] [PubMed]

Lee, J. H.

Leng, C. C.

C. C. Leng and J. Tian, “Mathematical method in optical molecular imaging,” Sci. China Life Sci. 58, 1–13 (2015).
[PubMed]

Li, X.

Liang, J.

Lin, Y.

M. Yuan and Y. Lin, “Model selection and estimation in regression with grouped variables,” J. R. Stat. Soc. Ser. A Stat. Soc. 68(1), 49–67 (2006).
[Crossref]

Liu, H.

C. Chen, F. Tian, H. Liu, and J. Huang, “Diffuse optical tomography enhanced by clustered sparsity for functional brain imaging,” IEEE Trans. Med. Imaging 33(12), 2323–2331 (2014).
[Crossref] [PubMed]

Liu, J.

Y. An, J. Liu, G. Zhang, J. Ye, Y. Du, Y. Mao, C. Chi, and J. Tian, “A Novel Region Reconstruction Method for Fluorescence Molecular Tomography,” IEEE Trans. Biomed. Eng. 62(7), 1818–1826 (2015).
[Crossref] [PubMed]

J. Liu, S. Ji, and J. Ye, “Multi-Task Feature Learning Via Efficient L2,1-Norm Minimization,” Eprint Arxiv339–348 (2009).

Liu, Y.

Lu, Y.

Ma, X.

Manjappa, R.

J. Prakash, C. B. Shaw, R. Manjappa, R. Kanhirodan, and P. K. Yalavarthy, “Sparse Recovery Methods Hold Promise for Diffuse Optical Tomographic Image Reconstruction,” IEEE J. Sel. Top. Quantum Electron. 20(2), 74–82 (2014).
[Crossref]

Mao, Y.

Y. An, J. Liu, G. Zhang, J. Ye, Y. Du, Y. Mao, C. Chi, and J. Tian, “A Novel Region Reconstruction Method for Fluorescence Molecular Tomography,” IEEE Trans. Biomed. Eng. 62(7), 1818–1826 (2015).
[Crossref] [PubMed]

McCray, P.

McLennan, G.

Metaxas, D.

J. Huang, T. Zhang, and D. Metaxas, “Learning with structured sparsity,” J. Mach. Learn. Res. 13(7), 3371–3412 (2011).

Millane, R. P.

Milstein, A. B.

Mohajerani, P.

Nehorai, A.

Ntziachristos, V.

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7(8), 603–614 (2010).
[Crossref] [PubMed]

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[Crossref] [PubMed]

Obozinski, G.

L. Jacob, G. Obozinski, and J. P. Vert, “Group lasso with overlap and graph lasso,” in International Conference on Machine Learning (ACM, 2009), pp. 433–440.
[Crossref]

Oh, S.

Okawa, S.

Ping Wu,

Ping Wu, Yifang Hu, Kun Wang, and Jie Tian, “Bioluminescence Tomography by an Iterative Reweighted (l)2-Norm Optimization,” IEEE Trans. Biomed. Eng. 61(1), 189–196 (2014).
[Crossref] [PubMed]

Prakash, J.

J. Prakash, C. B. Shaw, R. Manjappa, R. Kanhirodan, and P. K. Yalavarthy, “Sparse Recovery Methods Hold Promise for Diffuse Optical Tomographic Image Reconstruction,” IEEE J. Sel. Top. Quantum Electron. 20(2), 74–82 (2014).
[Crossref]

Qin, C.

C. Qin, S. Zhu, and J. Tian, “New optical molecular imaging systems,” Curr. Pharm. Biotechnol. 11(6), 620 (2010).
[Crossref] [PubMed]

D. Han, J. Tian, S. Zhu, J. Feng, C. Qin, B. Zhang, and X. Yang, “A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization,” Opt. Express 18(8), 8630–8646 (2010).
[Crossref] [PubMed]

S. Zhu, J. Tian, G. Yan, C. Qin, and J. Feng, “Cone beam micro-CT system for small animal imaging and performance evaluation,” Int. J. Biomed. Imaging 2009, 960573 (2009).
[Crossref] [PubMed]

G. Yan, J. Tian, S. Zhu, Y. Dai, and C. Qin, “Fast cone-beam CT image reconstruction using GPU hardware,” J. XRay Sci. Technol. 16(4), 225–234 (2008).

Qiu, J.

C. Chi, Y. Du, J. Ye, D. Kou, J. Qiu, J. Wang, J. Tian, and X. Chen, “Intraoperative imaging-guided cancer surgery: from current fluorescence molecular imaging methods to future multi-modality imaging technology,” Theranostics 4(11), 1072–1084 (2014).
[Crossref] [PubMed]

Qu, X.

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]

Ripoll, J.

Romberg, J.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Sevick-Muraca, E.

Sevick-Muraca, E. M.

Shaw, C. B.

J. Prakash, C. B. Shaw, R. Manjappa, R. Kanhirodan, and P. K. Yalavarthy, “Sparse Recovery Methods Hold Promise for Diffuse Optical Tomographic Image Reconstruction,” IEEE J. Sel. Top. Quantum Electron. 20(2), 74–82 (2014).
[Crossref]

Shen, M.

Song, X.

Stephens, D. J.

D. J. Stephens and V. J. Allan, “Light microscopy techniques for live cell imaging,” Science 300(5616), 82–86 (2003).
[Crossref] [PubMed]

Stott, J. J.

Stout, D.

Süzen, M.

Tanikawa, Y.

Tao, T.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Tarvainen, T.

T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Ra. 43(4), 252–261 (2010).

Tian, F.

C. Chen, F. Tian, H. Liu, and J. Huang, “Diffuse optical tomography enhanced by clustered sparsity for functional brain imaging,” IEEE Trans. Med. Imaging 33(12), 2323–2331 (2014).
[Crossref] [PubMed]

Tian, J.

C. C. Leng and J. Tian, “Mathematical method in optical molecular imaging,” Sci. China Life Sci. 58, 1–13 (2015).
[PubMed]

Y. An, J. Liu, G. Zhang, J. Ye, Y. Du, Y. Mao, C. Chi, and J. Tian, “A Novel Region Reconstruction Method for Fluorescence Molecular Tomography,” IEEE Trans. Biomed. Eng. 62(7), 1818–1826 (2015).
[Crossref] [PubMed]

J. Ye, Y. Du, Y. An, C. Chi, and J. Tian, “Reconstruction of fluorescence molecular tomography via a nonmonotone spectral projected gradient pursuit method,” J. Biomed. Opt. 19(12), 126013 (2014).
[Crossref] [PubMed]

C. Chi, Y. Du, J. Ye, D. Kou, J. Qiu, J. Wang, J. Tian, and X. Chen, “Intraoperative imaging-guided cancer surgery: from current fluorescence molecular imaging methods to future multi-modality imaging technology,” Theranostics 4(11), 1072–1084 (2014).
[Crossref] [PubMed]

X. Chen, X. Gao, D. Chen, X. Ma, X. Zhao, M. Shen, X. Li, X. Qu, J. Liang, J. Ripoll, and J. Tian, “3D reconstruction of light flux distribution on arbitrary surfaces from 2D multi-photographic images,” Opt. Express 18(19), 19876–19893 (2010).
[Crossref] [PubMed]

C. Qin, S. Zhu, and J. Tian, “New optical molecular imaging systems,” Curr. Pharm. Biotechnol. 11(6), 620 (2010).
[Crossref] [PubMed]

D. Han, J. Tian, S. Zhu, J. Feng, C. Qin, B. Zhang, and X. Yang, “A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization,” Opt. Express 18(8), 8630–8646 (2010).
[Crossref] [PubMed]

S. Zhu, J. Tian, G. Yan, C. Qin, and J. Feng, “Cone beam micro-CT system for small animal imaging and performance evaluation,” Int. J. Biomed. Imaging 2009, 960573 (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]

G. Yan, J. Tian, S. Zhu, Y. Dai, and C. Qin, “Fast cone-beam CT image reconstruction using GPU hardware,” J. XRay Sci. Technol. 16(4), 225–234 (2008).

van Bruggen, N.

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7(7), 591–607 (2008).
[Crossref] [PubMed]

Vauhkonen, M.

T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Ra. 43(4), 252–261 (2010).

Vert, J. P.

L. Jacob, G. Obozinski, and J. P. Vert, “Group lasso with overlap and graph lasso,” in International Conference on Machine Learning (ACM, 2009), pp. 433–440.
[Crossref]

Wang, D.

Wang, G.

Wang, H.

Wang, J.

C. Chi, Y. Du, J. Ye, D. Kou, J. Qiu, J. Wang, J. Tian, and X. Chen, “Intraoperative imaging-guided cancer surgery: from current fluorescence molecular imaging methods to future multi-modality imaging technology,” Theranostics 4(11), 1072–1084 (2014).
[Crossref] [PubMed]

Wang, L.

Wang, L. V.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[Crossref] [PubMed]

Wang, Y.

Webb, K. J.

Webb, W. W.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[Crossref] [PubMed]

Weissleder, R.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[Crossref] [PubMed]

Williams, R. M.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[Crossref] [PubMed]

Willmann, J. K.

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7(7), 591–607 (2008).
[Crossref] [PubMed]

Yalavarthy, P. K.

J. Prakash, C. B. Shaw, R. Manjappa, R. Kanhirodan, and P. K. Yalavarthy, “Sparse Recovery Methods Hold Promise for Diffuse Optical Tomographic Image Reconstruction,” IEEE J. Sel. Top. Quantum Electron. 20(2), 74–82 (2014).
[Crossref]

Yamada, Y.

Yan, G.

S. Zhu, J. Tian, G. Yan, C. Qin, and J. Feng, “Cone beam micro-CT system for small animal imaging and performance evaluation,” Int. J. Biomed. Imaging 2009, 960573 (2009).
[Crossref] [PubMed]

G. Yan, J. Tian, S. Zhu, Y. Dai, and C. Qin, “Fast cone-beam CT image reconstruction using GPU hardware,” J. XRay Sci. Technol. 16(4), 225–234 (2008).

Yang, X.

Yao, Y.

Ye, J.

Y. An, J. Liu, G. Zhang, J. Ye, Y. Du, Y. Mao, C. Chi, and J. Tian, “A Novel Region Reconstruction Method for Fluorescence Molecular Tomography,” IEEE Trans. Biomed. Eng. 62(7), 1818–1826 (2015).
[Crossref] [PubMed]

J. Ye, Y. Du, Y. An, C. Chi, and J. Tian, “Reconstruction of fluorescence molecular tomography via a nonmonotone spectral projected gradient pursuit method,” J. Biomed. Opt. 19(12), 126013 (2014).
[Crossref] [PubMed]

C. Chi, Y. Du, J. Ye, D. Kou, J. Qiu, J. Wang, J. Tian, and X. Chen, “Intraoperative imaging-guided cancer surgery: from current fluorescence molecular imaging methods to future multi-modality imaging technology,” Theranostics 4(11), 1072–1084 (2014).
[Crossref] [PubMed]

J. Liu, S. Ji, and J. Ye, “Multi-Task Feature Learning Via Efficient L2,1-Norm Minimization,” Eprint Arxiv339–348 (2009).

Yifang Hu,

Ping Wu, Yifang Hu, Kun Wang, and Jie Tian, “Bioluminescence Tomography by an Iterative Reweighted (l)2-Norm Optimization,” IEEE Trans. Biomed. Eng. 61(1), 189–196 (2014).
[Crossref] [PubMed]

Yuan, M.

M. Yuan and Y. Lin, “Model selection and estimation in regression with grouped variables,” J. R. Stat. Soc. Ser. A Stat. Soc. 68(1), 49–67 (2006).
[Crossref]

Zabner, J.

Zhang, B.

Zhang, G.

Y. An, J. Liu, G. Zhang, J. Ye, Y. Du, Y. Mao, C. Chi, and J. Tian, “A Novel Region Reconstruction Method for Fluorescence Molecular Tomography,” IEEE Trans. Biomed. Eng. 62(7), 1818–1826 (2015).
[Crossref] [PubMed]

Zhang, T.

J. Huang, T. Zhang, and D. Metaxas, “Learning with structured sparsity,” J. Mach. Learn. Res. 13(7), 3371–3412 (2011).

J. Huang and T. Zhang, “The Benefit Of Group Sparsity,” Ann. Stat. 38(4), 1978–2004 (2010).
[Crossref] [PubMed]

Zhang, X.

Zhao, H.

Zhao, X.

Zhu, S.

D. Han, J. Tian, S. Zhu, J. Feng, C. Qin, B. Zhang, and X. Yang, “A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization,” Opt. Express 18(8), 8630–8646 (2010).
[Crossref] [PubMed]

C. Qin, S. Zhu, and J. Tian, “New optical molecular imaging systems,” Curr. Pharm. Biotechnol. 11(6), 620 (2010).
[Crossref] [PubMed]

S. Zhu, J. Tian, G. Yan, C. Qin, and J. Feng, “Cone beam micro-CT system for small animal imaging and performance evaluation,” Int. J. Biomed. Imaging 2009, 960573 (2009).
[Crossref] [PubMed]

G. Yan, J. Tian, S. Zhu, Y. Dai, and C. Qin, “Fast cone-beam CT image reconstruction using GPU hardware,” J. XRay Sci. Technol. 16(4), 225–234 (2008).

Zhu, W.

Zipfel, W. R.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[Crossref] [PubMed]

Ann. Stat. (1)

J. Huang and T. Zhang, “The Benefit Of Group Sparsity,” Ann. Stat. 38(4), 1978–2004 (2010).
[Crossref] [PubMed]

Appl. Opt. (1)

Biomed. Opt. Express (1)

Curr. Pharm. Biotechnol. (1)

C. Qin, S. Zhu, and J. Tian, “New optical molecular imaging systems,” Curr. Pharm. Biotechnol. 11(6), 620 (2010).
[Crossref] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Prakash, C. B. Shaw, R. Manjappa, R. Kanhirodan, and P. K. Yalavarthy, “Sparse Recovery Methods Hold Promise for Diffuse Optical Tomographic Image Reconstruction,” IEEE J. Sel. Top. Quantum Electron. 20(2), 74–82 (2014).
[Crossref]

IEEE Trans. Biomed. Eng. (2)

Ping Wu, Yifang Hu, Kun Wang, and Jie Tian, “Bioluminescence Tomography by an Iterative Reweighted (l)2-Norm Optimization,” IEEE Trans. Biomed. Eng. 61(1), 189–196 (2014).
[Crossref] [PubMed]

Y. An, J. Liu, G. Zhang, J. Ye, Y. Du, Y. Mao, C. Chi, and J. Tian, “A Novel Region Reconstruction Method for Fluorescence Molecular Tomography,” IEEE Trans. Biomed. Eng. 62(7), 1818–1826 (2015).
[Crossref] [PubMed]

IEEE Trans. Inf. Theory (2)

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

R. Baraniuk, V. Cevher, M. Duarte, and C. Hegde, “Model-based compressive sensing,” IEEE Trans. Inf. Theory 56(4), 1982–2001 (2010).
[Crossref]

IEEE Trans. Med. Imaging (1)

C. Chen, F. Tian, H. Liu, and J. Huang, “Diffuse optical tomography enhanced by clustered sparsity for functional brain imaging,” IEEE Trans. Med. Imaging 33(12), 2323–2331 (2014).
[Crossref] [PubMed]

Int. J. Biomed. Imaging (1)

S. Zhu, J. Tian, G. Yan, C. Qin, and J. Feng, “Cone beam micro-CT system for small animal imaging and performance evaluation,” Int. J. Biomed. Imaging 2009, 960573 (2009).
[Crossref] [PubMed]

Inverse Probl. (1)

W. Bangerth and A. Joshi, “Adaptive finite element methods for the solution of inverse problems in optical tomography,” Inverse Probl. 24(3), 034011 (2008).
[Crossref]

J. Biomed. Opt. (1)

J. Ye, Y. Du, Y. An, C. Chi, and J. Tian, “Reconstruction of fluorescence molecular tomography via a nonmonotone spectral projected gradient pursuit method,” J. Biomed. Opt. 19(12), 126013 (2014).
[Crossref] [PubMed]

J. Mach. Learn. Res. (1)

J. Huang, T. Zhang, and D. Metaxas, “Learning with structured sparsity,” J. Mach. Learn. Res. 13(7), 3371–3412 (2011).

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Ra. (1)

T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Ra. 43(4), 252–261 (2010).

J. R. Stat. Soc. Ser. A Stat. Soc. (1)

M. Yuan and Y. Lin, “Model selection and estimation in regression with grouped variables,” J. R. Stat. Soc. Ser. A Stat. Soc. 68(1), 49–67 (2006).
[Crossref]

J. XRay Sci. Technol. (1)

G. Yan, J. Tian, S. Zhu, Y. Dai, and C. Qin, “Fast cone-beam CT image reconstruction using GPU hardware,” J. XRay Sci. Technol. 16(4), 225–234 (2008).

Nat. Biotechnol. (2)

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[Crossref] [PubMed]

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[Crossref] [PubMed]

Nat. Methods (1)

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7(8), 603–614 (2010).
[Crossref] [PubMed]

Nat. Rev. Drug Discov. (1)

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7(7), 591–607 (2008).
[Crossref] [PubMed]

Opt. Express (11)

A. Joshi, W. Bangerth, and E. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12(22), 5402–5417 (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]

F. Gao, H. Zhao, Y. Tanikawa, and Y. Yamada, “A linear, featured-data scheme for image reconstruction in time-domain fluorescence molecular tomography,” Opt. Express 14(16), 7109–7124 (2006).
[Crossref] [PubMed]

J. H. Lee, A. Joshi, and E. M. Sevick-Muraca, “Fully adaptive finite element based tomography using tetrahedral dual-meshing for fluorescence enhanced optical imaging in tissue,” Opt. Express 15(11), 6955–6975 (2007).
[Crossref] [PubMed]

D. Wang, X. Song, and J. Bai, “Adaptive-mesh-based algorithm for fluorescence molecular tomography using an analytical solution,” Opt. Express 15(15), 9722–9730 (2007).
[Crossref] [PubMed]

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

X. Song, D. Wang, N. Chen, J. Bai, and H. Wang, “Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm,” Opt. Express 15(26), 18300–18317 (2007).
[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]

D. Han, J. Tian, S. Zhu, J. Feng, C. Qin, B. Zhang, and X. Yang, “A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization,” Opt. Express 18(8), 8630–8646 (2010).
[Crossref] [PubMed]

X. Chen, X. Gao, D. Chen, X. Ma, X. Zhao, M. Shen, X. Li, X. Qu, J. Liang, J. Ripoll, and J. Tian, “3D reconstruction of light flux distribution on arbitrary surfaces from 2D multi-photographic images,” Opt. Express 18(19), 19876–19893 (2010).
[Crossref] [PubMed]

M. Süzen, A. Giannoula, and T. Durduran, “Compressed sensing in diffuse optical tomography,” Opt. Express 18(23), 23676–23690 (2010).
[Crossref] [PubMed]

Phys. Med. Biol. (1)

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]

Sci. China Life Sci. (1)

C. C. Leng and J. Tian, “Mathematical method in optical molecular imaging,” Sci. China Life Sci. 58, 1–13 (2015).
[PubMed]

Science (1)

D. J. Stephens and V. J. Allan, “Light microscopy techniques for live cell imaging,” Science 300(5616), 82–86 (2003).
[Crossref] [PubMed]

SIAM J. Sci. Comput. (1)

W. Bangerth, “A Framework for the Adaptive Finite Element Solution of Large-Scale Inverse Problems,” SIAM J. Sci. Comput. 30(6), 2965–2989 (2008).
[Crossref]

Theranostics (1)

C. Chi, Y. Du, J. Ye, D. Kou, J. Qiu, J. Wang, J. Tian, and X. Chen, “Intraoperative imaging-guided cancer surgery: from current fluorescence molecular imaging methods to future multi-modality imaging technology,” Theranostics 4(11), 1072–1084 (2014).
[Crossref] [PubMed]

Other (6)

J. Liu, S. Ji, and J. Ye, “Multi-Task Feature Learning Via Efficient L2,1-Norm Minimization,” Eprint Arxiv339–348 (2009).

L. Jacob, G. Obozinski, and J. P. Vert, “Group lasso with overlap and graph lasso,” in International Conference on Machine Learning (ACM, 2009), pp. 433–440.
[Crossref]

A. Nemirovski, Efficient methods in convex programming (Academic, 2005).

Y. Nesterov, Introductory lectures on convex optimization (Springer Science & Business Media, 2004).

R. Noumeir, G. E. Mailloux, H. Mallouche, and R. Lemieux, “Comparison between ML-EM and modified Newton algorithms for SPECT image reconstruction,” in OE/LASE'93: Optics, Electro-Optics, & Laser Applications in Science& Engineering, (Academic, 1993).

V. Cevher, P. Indyk, C. Hegde, and R. Baraniuk, “Recovery of clustered sparse signals from compressive measurements,” presented at the Int. Conf. Sampl. Theory Appl., Marseille, France (2009).

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Figures (13)

Fig. 1
Fig. 1 The illustration of Nesterov’s method. We set x 1 = x 0 , so s 1 = x 1 . We assume that x 6 = x 7 = x * , and x * is an optimal solution.
Fig. 2
Fig. 2 The one-source mouse-mimicking heterogeneous phantom, (a) is the 3D view of the fluorescent sources. (b) is the cross-section in the z = 0 plane. The black dots in (b) denote the excitation point sources. Fluorescence was collected from the opposite cylindrical surface within 160° FOV for each point source.
Fig. 3
Fig. 3 Reconstruction results with different regularization parameters: (a)-(j) are cross-sectional views in the z = 0 plane when iteration number is 300, and λ = 10−1, 10−2, 10−3, 10−4, 10−5, 10−6, 10−7, 10−8, 10−9 and 10−10 respectively. The red circles in the slice images represent the real locations of the fluorescent sources. (k) numerical analysis for the reconstruction errors with different iteration numbers and parameters.
Fig. 4
Fig. 4 The two-source mouse-mimicking heterogeneous phantom, (a) is the 3D view of the fluorescent sources. (b) is the cross-section in the z = 0 plane. The black dots in (b) denote the excitation point sources. Fluorescence was collected from the opposite cylindrical surface within 160° FOV for each point source.
Fig. 5
Fig. 5 Reconstruction results from the Tikhonov_L2 method (a), the IS_L1 method (b), and the proposed method (c) for two fluorescent sources and four measurement data sets. The first row illustrates the 3-D views of the reconstruction results. The second row illustrates the slice images in the z = 0 plane. The red circles in the slice images represent the real locations of the fluorescent source.
Fig. 6
Fig. 6 The two-source mouse-mimicking heterogeneous phantom, source 1 located in the right lung and source 2 located in the muscle, (a) is the 3D view of the fluorescent sources. (b) is the cross-section in the z = 0 plane. The black dots in (b) denote the excitation point sources. Fluorescence was collected from the opposite cylindrical surface within 160° FOV for each point source.
Fig. 7
Fig. 7 Reconstruction results from the Tikhonov_L2 method (a), the IS_L1 method (b), and the proposed method (c) for two fluorescent sources and four measurement data sets. The first row illustrates the 3-D views of the reconstruction results. The second row illustrates the slice images in the z = 0 plane. The red circles in the slice images represent the real locations of the fluorescent source.
Fig. 8
Fig. 8 Reconstructed results of heterogeneous numerical phantom experiments with different noise intensities. The columns denote the reconstructed results corresponding to different noise intensities (5%, 10%, 15%, 20%). The first and second rows are the reconstruction results obtained using the Tikhonov_L2 method. The third and fourth rows correspond to the IS_L1 method. The fifth and sixth rows correspond to the l2,1-norm method. _3-D denotes the 3-D views of the reconstruction results, and _CV denotes the cross-sectional views of the results. The red spheres in the 3-D views and the red circles in the cross-sectional views show the real positions of the fluorescent sources.
Fig. 9
Fig. 9 The schematic illustration of the dual-modality micro-CT and FMT imaging system.
Fig. 10
Fig. 10 Anatomical structure of the mouse. (a) 3-D visualization of the mouse. (b) Transverse view. (c) Sagittal view. (d) Coronal view. The green square marker in (b), (c), (d) illustrates the location of the fluorescent bead.
Fig. 11
Fig. 11 Heterogeneous mouse model for the in vivo experiments. (a)Heterogeneous mouse torso for imaging reconstructions, including muscle, kidneys, liver, lungs and heart. (b) Surface view of the mouse torso with the frontal view measurements mapped on it.
Fig. 12
Fig. 12 3-D views of the results utilizing the Tikhonov_L2 method (a), the IS_L1 method (b), and the proposed method (c). The blue plane in the figure is the z = 6.4 mm slice.
Fig. 13
Fig. 13 Cross-sectional views of the in vivo region reconstruction results. (a), (d), and (e) are the transverse views of the reconstruction results via the Tikhonov_L2, IS_L1 and proposed method, respectively. The results are compared with the corresponding micro-CT slices. The cross on the red line denotes the real position of the fluorescent bead.

Tables (7)

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Table 1 Nesterov’s Method For L2,1-Norm Optimization

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Table 2 The Optical Parameters Of The Heterogeneous Phantom

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Table 3 Quantitative Analysis Of The Three Methods In Numerical Phantom Experiments

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Table 4 Quantitative Analysis Of The Three Methods In Numerical Phantom Experiments with Different Source Locations

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Table 5 Quantitative analysis of the three methods in numerical phantom experiments with different noise intensities.

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Table 6 Optical Properties Of The In Vivo Mouse Model

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Table 7 Optical Properties Of The In Vivo Mouse Model

Equations (15)

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{ - [ D x ( r ) Φ x ( r ) ] + μ a x ( r ) Φ x ( r ) = Θ δ ( r r l ) ( r Ω ) - [ D m ( r ) Φ m ( r ) ] + μ a m ( r ) Φ m ( r ) = Φ x ( r ) η μ a f ( r ) ( r Ω )
Φ x , m ( r ) + 2 q D x , m ( r ) [ v ¯ ( r ) Φ x , m ( r ) ] = 0 ( r Ω )
K x Φ x = S x
K m Φ m = F X
Φ = A X
min X E ( X ) = 1 2 A X Φ 2 2 + λ X 2 , 1
min ( t , X ) D 1 2 A X Φ 2 2 + λ i = 1 n t i
s i = x i + α i ( x i x i 1 )
x i + 1 = π G ( s i 1 β i f ( s i ) )
π G ( v ) = min x G 1 2 x v 2
f β , x ( y ) = f ( x ) + f ( x ) , y x + β 2 y x 2 2
π G ( x 1 β f ( x ) ) = arg min y G f β , x ( y )
f ( π G ( ( x 1 γ 0 f ( x ) ) ) f β , x ( π G ( ( x 1 β f ( x ) ) )
P E = P r P 0 2
C N R = μ V O I μ V O B ω V O I σ V O I 2 + ω V O B σ V O B 2

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