Abstract

Determining an appropriate regularization parameter is often challenging work because it has a narrow range and varies with problems, which is likely to lead to large reconstruction errors. In this contribution, an adaptive regularized method based on homotopy is presented for sparse fluorescence tomography reconstruction. Due to the adaptive regularization strategy, the proposed method is always able to reconstruct sources accurately independent of the estimation of the regularization parameter. Moreover, the proposed method is about two orders of magnitude faster than the two contrasting methods. Numerical and in vivo mouse experiments have been employed to validate the robustness and efficiency of the proposed method.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2012 (1)

2011 (2)

F. Leblond, K. M. Tichauer, R. W. Holt, F. El-Ghussein, and B. W. Pogue, “Toward whole-body optical imaging of rats using single-photon counting fluorescence tomography,” Opt. Lett. 36, 3723–3725 (2011).
[CrossRef]

X. Ma, J. Tian, X. Yang, C. Qin, S. Zhu, and Z. Xue, “Research on liver tumor proliferation and angiogenesis based on multi-modality molecular imaging,” Acta Biophys. Sin. 27, 355–364 (2011).
[CrossRef]

2010 (5)

2009 (3)

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, 8062–8080 (2009).
[CrossRef]

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. Imag. 2009, 960573(2009).
[CrossRef]

D. Wang, X. Liu, Y. Chen, and J. Bai, “A novel finite-element-based algorithm for fluorescence molecular tomography of heterogeneous media,” IEEE Trans. Inf. Technol. Biomed. 13, 766–773 (2009).

2008 (2)

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

Y. Tan and H. Jiang, “Diffuse optical tomography guided quantitative fluorescence molecular imaging,” Appl. Opt. 47, 2011–2016 (2008).
[CrossRef]

2007 (5)

2006 (1)

2005 (4)

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 tomograph,” Opt. Express 13, 6756–6771 (2005).
[CrossRef]

A. X. Cong and G. Wang, “A finite-element-based reconstruction method for 3D fluorescence tomography,” Opt. Express 13, 9847–9857 (2005).
[CrossRef]

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, 4225–4241 (2005).
[CrossRef]

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

2004 (1)

B. Efron, T. Hastie, I. M. Johnstone, and R. Tibshirani, “Least angle regression,” Ann. Statist. 32, 407–499 (2004).
[CrossRef]

2000 (1)

M. R. Osborne, B. Presnell, and B. A. Turlach, “A new approach to variable selection in least squares problems,” IMA J. Numer. Anal. 20, 389–403 (2000).
[CrossRef]

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, 1779–1792 (1995).
[CrossRef]

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, 4225–4241 (2005).
[CrossRef]

Arridge, S. R.

T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluorescence optical tomography using data compression,” Opt. Lett. 35, 763–765 (2010).
[CrossRef]

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, 1779–1792 (1995).
[CrossRef]

Bai, J.

D. Wang, X. Liu, Y. Chen, and J. Bai, “A novel finite-element-based algorithm for fluorescence molecular tomography of heterogeneous media,” IEEE Trans. Inf. Technol. Biomed. 13, 766–773 (2009).

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, 18300–18317 (2007).
[CrossRef]

Bertsekas, D. P.

D. P. Bertsekas, Convex Analysis and Optimization (Athena Scientific, 2003).

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, 606–617(2007).
[CrossRef]

Cao, N.

Chan, T. F.

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, 8062–8080 (2009).
[CrossRef]

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, 4225–4241 (2005).
[CrossRef]

Chen, D.

Chen, N.

Chen, X.

Chen, Y.

D. Wang, X. Liu, Y. Chen, and J. Bai, “A novel finite-element-based algorithm for fluorescence molecular tomography of heterogeneous media,” IEEE Trans. Inf. Technol. Biomed. 13, 766–773 (2009).

Cong, A.

Cong, A. X.

Cong, W.

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, 1779–1792 (1995).
[CrossRef]

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, 591–607 (2008).
[CrossRef]

Douraghy, A.

Efron, B.

B. Efron, T. Hastie, I. M. Johnstone, and R. Tibshirani, “Least angle regression,” Ann. Statist. 32, 407–499 (2004).
[CrossRef]

Eftekhar, A. A.

Elad, M.

M. Elad, B. Matalon, and M. Zibulevsky, “Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization,” Appl. Comput. Harmon. Anal. 23, 346–367 (2007).
[CrossRef]

El-Ghussein, F.

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, 8630–8646 (2010).
[CrossRef]

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. Imag. 2009, 960573(2009).
[CrossRef]

Friedman, J.

J. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Software 33, 1–22 (2010).

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, 591–607 (2008).
[CrossRef]

Gao, F.

Gao, X.

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, 606–617(2007).
[CrossRef]

Hale, E. T.

E. T. Hale, W. Yin, and Y. Zhang, “A fixed-point continuation method for l1-regularized minimization with applications to compresses sensing,” Technical report 07-07 (Department of Computational and Applied Mathematics, Rice University, 2007).

Han, D.

Hastie, T.

J. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Software 33, 1–22 (2010).

B. Efron, T. Hastie, I. M. Johnstone, and R. Tibshirani, “Least angle regression,” Ann. Statist. 32, 407–499 (2004).
[CrossRef]

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, 1779–1792 (1995).
[CrossRef]

Hoffman, E. A.

Holt, R. W.

Huang, J.

Jacob, M.

Jiang, H.

Jiang, M.

Johnstone, I. M.

B. Efron, T. Hastie, I. M. Johnstone, and R. Tibshirani, “Least angle regression,” Ann. Statist. 32, 407–499 (2004).
[CrossRef]

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, 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, 606–617(2007).
[CrossRef]

Kumar, D.

Leblond, F.

Li, X.

Liang, J.

Liu, X.

D. Wang, X. Liu, Y. Chen, and J. Bai, “A novel finite-element-based algorithm for fluorescence molecular tomography of heterogeneous media,” IEEE Trans. Inf. Technol. Biomed. 13, 766–773 (2009).

Liu, Y.

Lu, Y.

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, 606–617(2007).
[CrossRef]

Ma, X.

X. Ma, J. Tian, X. Yang, C. Qin, S. Zhu, and Z. Xue, “Research on liver tumor proliferation and angiogenesis based on multi-modality molecular imaging,” Acta Biophys. Sin. 27, 355–364 (2011).
[CrossRef]

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, 19876–19893 (2010).
[CrossRef]

Matalon, B.

M. Elad, B. Matalon, and M. Zibulevsky, “Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization,” Appl. Comput. Harmon. Anal. 23, 346–367 (2007).
[CrossRef]

McCray, P. B.

McLennan, G.

Mohajerani, P.

Nehorai, A.

Ntziachristos, V.

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

Osborne, M. R.

M. R. Osborne, B. Presnell, and B. A. Turlach, “A new approach to variable selection in least squares problems,” IMA J. Numer. Anal. 20, 389–403 (2000).
[CrossRef]

Peng, K.

Pogue, B. W.

Presnell, B.

M. R. Osborne, B. Presnell, and B. A. Turlach, “A new approach to variable selection in least squares problems,” IMA J. Numer. Anal. 20, 389–403 (2000).
[CrossRef]

Qin, C.

X. Ma, J. Tian, X. Yang, C. Qin, S. Zhu, and Z. Xue, “Research on liver tumor proliferation and angiogenesis based on multi-modality molecular imaging,” Acta Biophys. Sin. 27, 355–364 (2011).
[CrossRef]

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, 8630–8646 (2010).
[CrossRef]

C. Qin, S. Zhu, and J. Tian, “New optical molecular imaging systems,” Curr. Pharm. Biotechnol. 11, 620–627 (2010).
[CrossRef]

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. Imag. 2009, 960573(2009).
[CrossRef]

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, 4225–4241 (2005).
[CrossRef]

Ripoll, J.

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, 19876–19893 (2010).
[CrossRef]

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

Rudge, T. J.

Schweiger, 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, 1779–1792 (1995).
[CrossRef]

Shen, M.

Soloviev, V. Y.

Song, X.

Stout, D.

Tan, Y.

Tanikawa, Y.

Tian, J.

Tibshirani, R.

J. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Software 33, 1–22 (2010).

B. Efron, T. Hastie, I. M. Johnstone, and R. Tibshirani, “Least angle regression,” Ann. Statist. 32, 407–499 (2004).
[CrossRef]

Tichauer, K. M.

Turlach, B. A.

M. R. Osborne, B. Presnell, and B. A. Turlach, “A new approach to variable selection in least squares problems,” IMA J. Numer. Anal. 20, 389–403 (2000).
[CrossRef]

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, 591–607 (2008).
[CrossRef]

Wang, D.

D. Wang, X. Liu, Y. Chen, and J. Bai, “A novel finite-element-based algorithm for fluorescence molecular tomography of heterogeneous media,” IEEE Trans. Inf. Technol. Biomed. 13, 766–773 (2009).

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, 18300–18317 (2007).
[CrossRef]

Wang, G.

Wang, H.

Wang, L. V.

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 tomograph,” Opt. Express 13, 6756–6771 (2005).
[CrossRef]

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

Weissleder, R.

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

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, 591–607 (2008).
[CrossRef]

Xue, Z.

X. Ma, J. Tian, X. Yang, C. Qin, S. Zhu, and Z. Xue, “Research on liver tumor proliferation and angiogenesis based on multi-modality molecular imaging,” Acta Biophys. Sin. 27, 355–364 (2011).
[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. Imag. 2009, 960573(2009).
[CrossRef]

Yang, X.

X. Ma, J. Tian, X. Yang, C. Qin, S. Zhu, and Z. Xue, “Research on liver tumor proliferation and angiogenesis based on multi-modality molecular imaging,” Acta Biophys. Sin. 27, 355–364 (2011).
[CrossRef]

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, 8630–8646 (2010).
[CrossRef]

Yi, H.

Yin, W.

E. T. Hale, W. Yin, and Y. Zhang, “A fixed-point continuation method for l1-regularized minimization with applications to compresses sensing,” Technical report 07-07 (Department of Computational and Applied Mathematics, Rice University, 2007).

Zabner, J.

Zhang, B.

Zhang, X.

Zhang, Y.

E. T. Hale, W. Yin, and Y. Zhang, “A fixed-point continuation method for l1-regularized minimization with applications to compresses sensing,” Technical report 07-07 (Department of Computational and Applied Mathematics, Rice University, 2007).

Zhao, H.

Zhao, X.

Zhou, Y.

Zhu, S.

X. Ma, J. Tian, X. Yang, C. Qin, S. Zhu, and Z. Xue, “Research on liver tumor proliferation and angiogenesis based on multi-modality molecular imaging,” Acta Biophys. Sin. 27, 355–364 (2011).
[CrossRef]

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, 8630–8646 (2010).
[CrossRef]

C. Qin, S. Zhu, and J. Tian, “New optical molecular imaging systems,” Curr. Pharm. Biotechnol. 11, 620–627 (2010).
[CrossRef]

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. Imag. 2009, 960573(2009).
[CrossRef]

Zibulevsky, M.

M. Elad, B. Matalon, and M. Zibulevsky, “Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization,” Appl. Comput. Harmon. Anal. 23, 346–367 (2007).
[CrossRef]

Acta Biophys. Sin. (1)

X. Ma, J. Tian, X. Yang, C. Qin, S. Zhu, and Z. Xue, “Research on liver tumor proliferation and angiogenesis based on multi-modality molecular imaging,” Acta Biophys. Sin. 27, 355–364 (2011).
[CrossRef]

Ann. Statist. (1)

B. Efron, T. Hastie, I. M. Johnstone, and R. Tibshirani, “Least angle regression,” Ann. Statist. 32, 407–499 (2004).
[CrossRef]

Appl. Comput. Harmon. Anal. (1)

M. Elad, B. Matalon, and M. Zibulevsky, “Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization,” Appl. Comput. Harmon. Anal. 23, 346–367 (2007).
[CrossRef]

Appl. Opt. (3)

Curr. Pharm. Biotechnol. (1)

C. Qin, S. Zhu, and J. Tian, “New optical molecular imaging systems,” Curr. Pharm. Biotechnol. 11, 620–627 (2010).
[CrossRef]

IEEE J. Sel. Top. Signal Process. (1)

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, 606–617(2007).
[CrossRef]

IEEE Trans. Inf. Technol. Biomed. (1)

D. Wang, X. Liu, Y. Chen, and J. Bai, “A novel finite-element-based algorithm for fluorescence molecular tomography of heterogeneous media,” IEEE Trans. Inf. Technol. Biomed. 13, 766–773 (2009).

IMA J. Numer. Anal. (1)

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Figures (7)

Fig. 1.
Fig. 1.

Cylindrical heterogeneous phantom with regions resembling muscle (M), lungs (L), heart (H), and bone (B), respectively. (a) 3D view of the phantom. (b) Slice image of the phantom in a z=0 plane. The black dots in (b) represent the excitation point sources. For each excitation location, fluorescence can be measured only from the opposite cylindrical side with a 160° field of view.

Fig. 2.
Fig. 2.

Reconstruction results by the proposed algorithm for a single source (top row), for double sources (middle row), and for triple sources (bottom row), respectively. The left column has the slice images in a z=0 plane. The small circles of the cross sections denote the real positions of the fluorescent sources. The right column shows 3D views of the results. The iso-surfaces of the results are for 30% of the maximum value.

Fig. 3.
Fig. 3.

Evolution curves as a function of the iteration steps with a different number of sources. (a) Evolution curve of the regularization parameter. (b) Evolution curve of the objective function.

Fig. 4.
Fig. 4.

Reconstruction results from the Newton-L2 method (top row), the IS-L1 method (middle row), and the proposed method (bottom row). These results are presented in the form of slice images in a z=0 plane (left column) and iso-surfaces for 30% of the maximum value (right column). The small circles in the slice images denote the real positions of the fluorescent sources.

Fig. 5.
Fig. 5.

Reconstruction results with different regularization parameters for the Newton-L2 method (first row), the IS-L1 method (second row), and the proposed method (third row). These results are presented in the form of slice images in the z=0 plane. The regularization parameter is set at 1e5 (left column), 1e7 (middle column), and 1e9 (right column), respectively. The small circles of the cross sections denote the real positions of the fluorescent sources.

Fig. 6.
Fig. 6.

Comparisons of the reconstruction results for in vivo mouse experiments. (b), (c), and (f) show the transverse views of the results at z=6.40mm by the Newton-L2, IS-L1, and proposed methods, respectively. (a), (d), and (e) show the corresponding CT slices. The crosshairs of the pink dashed lines denote the actual source centers.

Fig. 7.
Fig. 7.

Iso-surface 3D view of the reconstruction results by the proposed method.

Tables (8)

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Table 1. Optical Properties of the Numerical Phantoma

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Table 2. Time Comparisons of the Reconstructionsa

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Table 3. Reconstruction Comparisons for Different Parameters

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Table 4. Comparisons for Quantitative Resultsa

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Table 5. Optical Properties of the Mouse Modela

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Table 6. Reconstruction Comparisons for the Three Methods

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Table 7. Comparisons for Different Parameters

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Table 8. Comparisons for Quantitative Results for In Vivo Experimentsa

Equations (14)

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

{·(Dx(r)Φx(r))μax(r)Φx(r)=Θsδ(rrl)·(Dm(r)Φm(r))μam(r)Φm(r)=Φx(r)ημaf(r)rΩ,
AX=Φ.
minXFλ(X)=12AXΦ22+λX1,
XλFλ(Xλ)=AT(AXλΦ)+λXλXλ1,
XλXλ1={uRn|ui=sgn(Xλ,i),Xλ,i0ui[1,1],Xλ,i0}.
c(I)=λ·sgn(Xλ(I))
|c(Ic)|λ.
I={j:|ck(j)|=ck=λ}.
AITAIpk(I)=sgn(ck(I)).
γk+=miniIc{λck(i)1aiTAIpk(I)},
γk=miniI{Xk(i)/pk(i)},
γk=min{γk+,γk}.
Xk=Xk1+γkpk.
λ=λγk.

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