Abstract

Fluorescence molecular tomography (FMT) is a promising technique for in vivo small animal imaging. In this paper, the sparsity of the fluorescent sources is considered as the a priori information and is promoted by incorporating L1 regularization. Then a reconstruction algorithm based on stagewise orthogonal matching pursuit is proposed, which treats the FMT problem as the basis pursuit problem. To evaluate this method, we compare it to the iterated-shrinkage-based algorithm with L1 regularization. Numerical simulations and physical experiments show that the proposed method can obtain comparable or even slightly better results. More importantly, the proposed method was at least 2 orders of magnitude faster in these experiments, which makes it a practical reconstruction algorithm.

© 2010 Optical Society of America

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

2010 (1)

2009 (2)

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] [PubMed]

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).
[CrossRef] [PubMed]

2008 (7)

Y. Tan and H. Jiang, “DOT guided fluorescence molecular tomography of arbitrarily shaped objects,” Med. Phys. 35, 5703–5707 (2008).
[CrossRef]

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

G. Y. Panasyuk, Z. Wang, J. C. Schotland, and V. A. Markel, “Fluorescent optical tomography with large data sets,” Opt. Lett. 33, 1744–1746 (2008).
[CrossRef] [PubMed]

F. Gao, H. Zhao, L. Zhang, Y. Tanikawa, A. Marjono, and Y. Yamada, “A self-normalized, full time-resolved method for fluorescence diffuse optical tomography,” Opt. Express 16, 13104–13121 (2008).
[CrossRef] [PubMed]

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] [PubMed]

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef] [PubMed]

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

2007 (5)

2006 (2)

E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pur. Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8, 1–33 (2006).
[CrossRef] [PubMed]

2005 (2)

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

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

2001 (1)

D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inf. Theory 47, 2845–2862 (2001).
[CrossRef]

1997 (1)

I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. 45, 600–616(1997).
[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] [PubMed]

Adibi, A.

Arridge, S. R.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[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, 18300–18317 (2007).
[CrossRef] [PubMed]

Bangerth, W.

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

Bao, S.

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef] [PubMed]

Candes, E. J.

E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pur. Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Cao, N.

Chan, T. F.

Chatziioannou, A. F.

Chen, N.

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).
[CrossRef] [PubMed]

Cong, A. X.

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] [PubMed]

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] [PubMed]

Donoho, D. L.

D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inf. Theory 47, 2845–2862 (2001).
[CrossRef]

D. L. Donoho, Y. Tsaig, I. Drori, and J. L. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit,” Tech. Rep. 2006-02 (Stanford Department of Statistics, 2006).

Douraghy, A.

Drori, I.

D. L. Donoho, Y. Tsaig, I. Drori, and J. L. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit,” Tech. Rep. 2006-02 (Stanford Department of Statistics, 2006).

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]

Feng, J.

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] [PubMed]

Gao, F.

Gilbert, A. C.

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

Gorodnitsky, I. F.

I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. 45, 600–616(1997).
[CrossRef]

Han, D.

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] [PubMed]

Huang, J.

Huo, X.

D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inf. Theory 47, 2845–2862 (2001).
[CrossRef]

Jacob, M.

Jiang, H.

Y. Tan and H. Jiang, “DOT guided fluorescence molecular tomography of arbitrarily shaped objects,” Med. Phys. 35, 5703–5707 (2008).
[CrossRef]

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

Joshi, A.

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

Li, Y.

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef] [PubMed]

Liang, W.

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Lu, Y.

Marjono, A.

Markel, V. A.

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]

Mohajerani, P.

Nehorai, A.

Ntziachristos, V.

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8, 1–33 (2006).
[CrossRef] [PubMed]

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

Panasyuk, G. Y.

Qin, C.

Rao, B. D.

I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. 45, 600–616(1997).
[CrossRef]

Ripoll, J.

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

Romberg, J. K.

E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pur. Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Schotland, J. C.

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] [PubMed]

Song, X.

Starck, J. L.

D. L. Donoho, Y. Tsaig, I. Drori, and J. L. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit,” Tech. Rep. 2006-02 (Stanford Department of Statistics, 2006).

Stout, D.

Tan, Y.

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

Y. Tan and H. Jiang, “DOT guided fluorescence molecular tomography of arbitrarily shaped objects,” Med. Phys. 35, 5703–5707 (2008).
[CrossRef]

Tanikawa, Y.

Tao, T.

E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pur. Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Tian, J.

Tropp, J. A.

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

Tsaig, Y.

D. L. Donoho, Y. Tsaig, I. Drori, and J. L. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit,” Tech. Rep. 2006-02 (Stanford Department of Statistics, 2006).

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] [PubMed]

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).
[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, 18300–18317 (2007).
[CrossRef] [PubMed]

Wang, G.

Wang, H.

Wang, L. V.

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

Wang, Z.

Weissleder, R.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the revolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320(2005).
[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, 591–607 (2008).
[CrossRef] [PubMed]

Yamada, Y.

Yan, X.

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef] [PubMed]

Yang, X.

Zhang, B.

Zhang, L.

Zhang, X.

Zhao, H.

Zhu, S.

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]

Annu. Rev. Biomed. Eng. (1)

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8, 1–33 (2006).
[CrossRef] [PubMed]

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. (2)

Commun. Pur. Appl. Math. (1)

E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pur. Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

IEEE Eng. Med. Biol. Mag. (1)

J. Tian, J. Bai, X. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (2)

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inf. Theory 47, 2845–2862 (2001).
[CrossRef]

IEEE Trans. Signal Process. (1)

I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. 45, 600–616(1997).
[CrossRef]

Inverse Probl. (1)

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

Med. Phys. (2)

Y. Tan and H. Jiang, “DOT guided fluorescence molecular tomography of arbitrarily shaped objects,” Med. Phys. 35, 5703–5707 (2008).
[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] [PubMed]

Nat. Biotechnol. (1)

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the revolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313–320(2005).
[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, 591–607 (2008).
[CrossRef] [PubMed]

Opt. Express (6)

Opt. Lett. (1)

Other (1)

D. L. Donoho, Y. Tsaig, I. Drori, and J. L. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit,” Tech. Rep. 2006-02 (Stanford Department of Statistics, 2006).

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Figures (10)

Fig. 1
Fig. 1

Flow chart of the proposed algorithm.

Fig. 2
Fig. 2

Three different fluorescent source configurations. The left column is the 3D views of the configurations, and the right column is the cross sections in the z = 0 plane. The black dots in the cross sections represent the excitation point sources. For each excitation point source, fluorescence was collected from the opposite cylindrical surface within a 160 ° field of view.

Fig. 3
Fig. 3

Cross sections in z = 0 plane of the reconstruction results for the three fluorescent sources with depths of 4 (top row), 6 (middle row), and 8 mm (bottom row), respectively. The results in the left column are from the IS-L1 method, and the results in the right column are from the proposed method. The small circles in the cross sections denote the real positions of the fluorescent sources.

Fig. 4
Fig. 4

Cylindrical heterogeneous phantom with regions resembling muscle (M), lung (L), heart (H), and bone (B). (a) 3D view of the phantom. (b) Cross section of the phantom in the z = 0 plane.

Fig. 5
Fig. 5

Two different fluorescent source configurations. The left column is the 3D views of the configurations, and the right column is the cross sections in the z = 0 plane. All the fluorescent sources were spherical with a diameter of 2 mm centered in the z = 0 plane. The black dots in the cross sections represent the excitation point sources. For each excitation point source, fluorescence was collected from the opposite cylindrical surface within a 160 ° field of view.

Fig. 6
Fig. 6

Reconstruction results from the IS-L1 method (first row) and the proposed method (second row) for one spherical fluorescent source. The left column is the isosurfaces for 30% of the maximum value. The right column is the cross sections in the z = 0 plane. The small circles in the left lung regions of the cross sections denote the real positions of the fluorescent sources.

Fig. 7
Fig. 7

Reconstruction results from the IS-L1 method (first row) and the proposed method (second row) for two spherical fluorescent sources. The left column is the isosurfaces for 30% of the maximum value. The right column is the cross sections in the z = 0 plane. The small circles in the left lung regions of the cross sections denote the real positions of the fluorescent sources.

Fig. 8
Fig. 8

Sketch of the imaging system.

Fig. 9
Fig. 9

Cubic phantom with one cylindrical fluorescent source. (a) 3D view of the phantom and the source. (b) Cross section in the z = 0 plane. The black dots in (b) represent the excitation point sources.

Fig. 10
Fig. 10

Reconstruction results from the IS-L1 method (first row) and the proposed method (second row), respectively. The left column is the isosurfaces for 30% of the maximum value. The right column is the cross sections in the z = 1 mm plane. The small circles in the cross sections denote the real positions of the fluorescent sources.

Tables (4)

Tables Icon

Table 1 Optical Parameters of the Numerical Phantom (Unit: mm 1 ) [23]

Tables Icon

Table 2 Relative Intensity Errors of the Results from the StOMP-Based and IS-L1 Algorithms

Tables Icon

Table 3 Relative Intensity Errors of the Results in the Heterogeneous Simulation Experiments

Tables Icon

Table 4 Optical Parameters of the Cubic Phantom

Equations (13)

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

{ · [ D x ( r ) Φ x ( r ) ] + μ a x ( r ) Φ x ( r ) = Θ δ ( r r l ) · [ 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 1 ,
min X 1 subject to A X = Φ .
r n 1 = Φ A X n 1 .
c n = A T r n 1 .
S n = { s : s I ¯ n 1 | c n ( s ) | > t n } ,
I n = I n 1 S n .
A I n X I n = Φ .
A I n T A I n X I n = A I n T Φ .

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