Abstract

Fluorescence molecular tomography (FMT) is a promising imaging modality that enables three-dimensional visualization of fluorescent targets in vivo in small animals. L2-norm regularization methods are usually used for severely ill-posed FMT problems. However, the smoothing effects caused by these methods result in continuous distribution that lacks high-frequency edge-type features and hence limits the resolution of FMT. In this paper, the sparsity in FMT reconstruction results is exploited via compressed sensing (CS). First, in order to ensure the feasibility of CS for the FMT inverse problem, truncated singular value decomposition (TSVD) conversion is implemented for the measurement matrix of the FMT problem. Then, as one kind of greedy algorithm, an ameliorated stagewise orthogonal matching pursuit with gradually shrunk thresholds and a specific halting condition is developed for the FMT inverse problem. To evaluate the proposed algorithm, we compared it with a TSVD method based on L2-norm regularization in numerical simulation and phantom experiments. The results show that the proposed algorithm can obtain higher spatial resolution and higher signal-to-noise ratio compared with the TSVD method.

© 2013 Optical Society of America

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2012

V. C. Kavuri, Z. Lin, F. Tian, and H. Liu, “Sparsity enhanced spatial resolution and depth localization in diffuse optical tomography,” Biomed. Opt. Express 3, 943–957 (2012).
[CrossRef]

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

2011

2010

F. Liu, X. Liu, D. Wang, B. Zhang, and J. Bai, “A parallel excitation based fluorescence molecular tomography system for whole-body simultaneous imaging of small animals,” Ann. Biomed. Eng. 38, 3440–3448 (2010).
[CrossRef]

D. Han, X. Yang, K. Liu, C. Qin, B. Zhang, X. Ma, and J. Tian, “Efficient reconstruction method for L1 regularization in fluorescence molecular tomography,” Appl. Opt. 49, 6930–6937(2010).
[CrossRef]

J. Dutta, S. Ahn, A. A. Joshi, and R. M. Leahy, “Illumination pattern optimization for fluorescence tomography: theory and simulation studies,” Phys. Med. Biol. 55, 2961–2982 (2010).
[CrossRef]

J. Baritaux, K. Hassler, and M. Unser, “An efficient numerical method for general Lp regularization in fluorescence molecular tomography,” IEEE Trans. Med. Imag. 29, 1075–1087 (2010).
[CrossRef]

H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 1: l1 regularization,” Opt. Express 18, 1854–1871 (2010).
[CrossRef]

2009

D. Needell and R. Vershynin, “Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit,” Found. Comput. Math. 9, 317–334 (2009).
[CrossRef]

D. Needell and J. A. Tropp, “CoSaMP: iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal. 26, 301–321 (2009).
[CrossRef]

2008

R. Chartrand and V. Staneva, “Restricted isometry properties and nonconvex compressive sensing,” Inverse Probl. 24, 035020 (2008).
[CrossRef]

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (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, 591–607 (2008).
[CrossRef]

2007

J. Rao, A. Dragulescu-Andrasi, and H. Yao, “Fluorescence imaging in vivo: recent advances,” Curr. Opin. Biotechnol. 18, 17–25 (2007).
[CrossRef]

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, 1679–1685 (2007).
[CrossRef]

M. Lustig, D. L. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

M. A. Figueiredo, R. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Select. Topics Signal Process. 1, 586–597 (2007).
[CrossRef]

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

2006

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

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

2005

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

2004

J. Tropp, “Greed is good: algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory 50, 2231–2242 (2004).
[CrossRef]

2001

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

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
[CrossRef]

1999

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

1996

1993

S. Mallat and Z. Zhang, “Matching pursuit with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

1990

P. C. Hansen, “Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank,” SIAM J. Sci. Comput. 11, 503–518 (1990).
[CrossRef]

1987

P. C. Hansen, “The truncated SVD as a method for regularization,” BIT 27, 534–553 (1987).
[CrossRef]

Adibi, A.

Ahn, S.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

J. Dutta, S. Ahn, A. A. Joshi, and R. M. Leahy, “Illumination pattern optimization for fluorescence tomography: theory and simulation studies,” Phys. Med. Biol. 55, 2961–2982 (2010).
[CrossRef]

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

Bai, J.

Baraniuk, R.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

Baritaux, J.

J. Baritaux, K. Hassler, and M. Unser, “An efficient numerical method for general Lp regularization in fluorescence molecular tomography,” IEEE Trans. Med. Imag. 29, 1075–1087 (2010).
[CrossRef]

Boas, D. A.

Cao, X.

Chance, B.

Chartrand, R.

R. Chartrand and V. Staneva, “Restricted isometry properties and nonconvex compressive sensing,” Inverse Probl. 24, 035020 (2008).
[CrossRef]

Chen, S. S.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
[CrossRef]

Cherry, S. R.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

Davenport, M.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[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]

Do, T. T.

T. T. Do, L. Gan, N. Guyen, and T. D. Tran, “Sparsity adaptive matching pursuit algorithm for practical compressed sensing,” in Aslimore Conference on Signals, Systems and Computers (Academic, 2008), pp. 581–587.

Donoho, D.

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

Donoho, D. L.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

M. Lustig, D. L. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

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

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
[CrossRef]

D. L. Donoho, Y. Tsaig, I. Drori, and J. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit march,” IEEE Trans. Inf. Theory53, 1094–1121(2012).

Dragulescu-Andrasi, A.

J. Rao, A. Dragulescu-Andrasi, and H. Yao, “Fluorescence imaging in vivo: recent advances,” Curr. Opin. Biotechnol. 18, 17–25 (2007).
[CrossRef]

Drori, I.

D. L. Donoho, Y. Tsaig, I. Drori, and J. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit march,” IEEE Trans. Inf. Theory53, 1094–1121(2012).

Duarte, M.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

Dutta, J.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

J. Dutta, S. Ahn, A. A. Joshi, and R. M. Leahy, “Illumination pattern optimization for fluorescence tomography: theory and simulation studies,” Phys. Med. Biol. 55, 2961–2982 (2010).
[CrossRef]

Eftekhar, A. A.

Figueiredo, M. A.

M. A. Figueiredo, R. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Select. Topics Signal Process. 1, 586–597 (2007).
[CrossRef]

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]

Gan, L.

T. T. Do, L. Gan, N. Guyen, and T. D. Tran, “Sparsity adaptive matching pursuit algorithm for practical compressed sensing,” in Aslimore Conference on Signals, Systems and Computers (Academic, 2008), pp. 581–587.

Gao, H.

Gilbert, A. C.

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

Guyen, N.

T. T. Do, L. Gan, N. Guyen, and T. D. Tran, “Sparsity adaptive matching pursuit algorithm for practical compressed sensing,” in Aslimore Conference on Signals, Systems and Computers (Academic, 2008), pp. 581–587.

Han, D.

Hansen, P. C.

P. C. Hansen, “Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank,” SIAM J. Sci. Comput. 11, 503–518 (1990).
[CrossRef]

P. C. Hansen, “The truncated SVD as a method for regularization,” BIT 27, 534–553 (1987).
[CrossRef]

Hassler, K.

J. Baritaux, K. Hassler, and M. Unser, “An efficient numerical method for general Lp regularization in fluorescence molecular tomography,” IEEE Trans. Med. Imag. 29, 1075–1087 (2010).
[CrossRef]

Huang, J.

Huo, X.

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

Joshi, A. A.

J. Dutta, S. Ahn, A. A. Joshi, and R. M. Leahy, “Illumination pattern optimization for fluorescence tomography: theory and simulation studies,” Phys. Med. Biol. 55, 2961–2982 (2010).
[CrossRef]

Kavuri, V. C.

Kelly, K.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

Laska, J.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

Leahy, R. M.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

J. Dutta, S. Ahn, A. A. Joshi, and R. M. Leahy, “Illumination pattern optimization for fluorescence tomography: theory and simulation studies,” Phys. Med. Biol. 55, 2961–2982 (2010).
[CrossRef]

Li, C.

J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57, 1459–1476 (2012).
[CrossRef]

Li, X. D.

Lin, Z.

Liu, F.

Liu, H.

Liu, K.

Liu, X.

B. Zhang, X. Cao, F. Liu, X. Liu, X. Wang, and J. Bai, “Early-photon fluorescence tomography of a heterogeneous mouse model with the telegraph equation,” Appl. Opt. 50, 5397–5407 (2011).
[CrossRef]

F. Liu, X. Liu, D. Wang, B. Zhang, and J. Bai, “A parallel excitation based fluorescence molecular tomography system for whole-body simultaneous imaging of small animals,” Ann. Biomed. Eng. 38, 3440–3448 (2010).
[CrossRef]

Lustig, M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

M. Lustig, D. L. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

Ma, X.

Mallat, S.

S. Mallat and Z. Zhang, “Matching pursuit with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

Mohajerani, P.

Needell, D.

D. Needell and R. Vershynin, “Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit,” Found. Comput. Math. 9, 317–334 (2009).
[CrossRef]

D. Needell and J. A. Tropp, “CoSaMP: iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal. 26, 301–321 (2009).
[CrossRef]

Nowak, R.

M. A. Figueiredo, R. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Select. Topics Signal Process. 1, 586–597 (2007).
[CrossRef]

Ntziachristos, V.

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

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

O’Leary, M. A.

Pauly, J. M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

M. Lustig, D. L. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

Qin, C.

Rao, J.

J. Rao, A. Dragulescu-Andrasi, and H. Yao, “Fluorescence imaging in vivo: recent advances,” Curr. Opin. Biotechnol. 18, 17–25 (2007).
[CrossRef]

Ripoll, J.

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

Santos, J. M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

Saunders, M. A.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
[CrossRef]

Staneva, V.

R. Chartrand and V. Staneva, “Restricted isometry properties and nonconvex compressive sensing,” Inverse Probl. 24, 035020 (2008).
[CrossRef]

Starck, J.

D. L. Donoho, Y. Tsaig, I. Drori, and J. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit march,” IEEE Trans. Inf. Theory53, 1094–1121(2012).

Sun, T.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

Takhar, D.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

Tian, F.

Tian, J.

Tran, T. D.

T. T. Do, L. Gan, N. Guyen, and T. D. Tran, “Sparsity adaptive matching pursuit algorithm for practical compressed sensing,” in Aslimore Conference on Signals, Systems and Computers (Academic, 2008), pp. 581–587.

Tropp, J.

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

J. Tropp, “Greed is good: algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory 50, 2231–2242 (2004).
[CrossRef]

Tropp, J. A.

D. Needell and J. A. Tropp, “CoSaMP: iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal. 26, 301–321 (2009).
[CrossRef]

Tsaig, Y.

D. L. Donoho, Y. Tsaig, I. Drori, and J. Starck, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit march,” IEEE Trans. Inf. Theory53, 1094–1121(2012).

Unser, M.

J. Baritaux, K. Hassler, and M. Unser, “An efficient numerical method for general Lp regularization in fluorescence molecular tomography,” IEEE Trans. Med. Imag. 29, 1075–1087 (2010).
[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]

Vershynin, R.

D. Needell and R. Vershynin, “Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit,” Found. Comput. Math. 9, 317–334 (2009).
[CrossRef]

Wang, D.

F. Liu, X. Liu, D. Wang, B. Zhang, and J. Bai, “A parallel excitation based fluorescence molecular tomography system for whole-body simultaneous imaging of small animals,” Ann. Biomed. Eng. 38, 3440–3448 (2010).
[CrossRef]

Wang, L. V.

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Figures (10)

Fig. 1.
Fig. 1.

Intuitive reconstruction of a sparse signal contaminated by interference. (a) The reconstruction information in WTϕmMeasurement is submerged by incoherent interference. (b) Selected components calculated by using atoms that are obtained by hard threshold. (d) The weaker components of the useful information can be singularized by subtracting (a) with (c).

Fig. 2.
Fig. 2.

Flow chart of the aSTOMP algorithm.

Fig. 3.
Fig. 3.

Geometry configuration of simulation experiments with double cylinder fluorescent targets, where the edge-to-edge distance is 6 and 3 mm in (a) and (b), respectively.

Fig. 4.
Fig. 4.

Illustration of the rationality of TSVDC. The slice picture denotes true fluorescent targets (a) and the corresponding stem picture (d). The slice picture denotes matrix WTϕf (b) and the corresponding stem picture (e). The slice picture denotes matrix (Wr×nnew)Tϕr×1new (c) and the corresponding stem picture, where the red stem denotes the true targets (f).

Fig. 5.
Fig. 5.

Relationship between residual error and sparsity. Each blue circle corresponds to a certain threshold shrunk by Eq. (16).

Fig. 6.
Fig. 6.

Reconstruction results for simulation experiment. Slice (i) and stereo (ii) pictures of double fluorescent targets with an edge-to-edge distance of 6 mm. Slice (iii) and stereo (iv) pictures for double fluorescent targets with an edge-to-edge distance of 3 mm. (a) Reconstructed results by the TSVD method. (b) Reconstructed results by the proposed algorithm without TSVDC. (c) Reconstructed results by the proposed algorithm with TSVDC.

Fig. 7.
Fig. 7.

Stem pictures of the distribution of fluorescent targets. (a) Distribution of double targets with an edge-to-edge distance of 6 mm. (b) Distribution of double targets with an edge-to-edge distance of 2 mm. The red stem (with circle top) denotes the true distribution of fluorophores, and the black (with star top) stem is the reconstruction result obtained by the proposed algorithm with TSVDC.

Fig. 8.
Fig. 8.

Sketch of the free-space full-angle FMT system.

Fig. 9.
Fig. 9.

Reconstruction results for physical experiment with an edge-to-edge distance of 6 mm. Slice (a1) and stereo (a2) are pictures of the true fluorescent targets. (b1),(b2) Reconstructed results obtained by the TSVD method. (c1),(c2) Reconstructed results obtained by the proposed algorithm without TSVDC. (d1),(d2) Reconstructed results obtained by the proposed algorithm with TSVDC.

Fig. 10.
Fig. 10.

Fluorescence signal and reconstruction results with nonspecific background fluorescence. (a1), (a2) Fluorescence signal without nonspecific background fluorescence corresponding to different projection. (b1), (b2) Fluorescence signal with nonspecific background fluorescence. (c1), (c2) Reconstruction results using TSVD. (d1), (d2) Reconstruction results using the proposed greedy method without TSVDC. (e1), (e2) Reconstruction results using the proposed greedy method with TSVDC.

Equations (19)

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ϕf(rd,rs)=υS0DλfG(rd,r)x(r)G(rs,r)d3r,
Dλf=1/(3μa+3μs),
G({rl},r)=G(rl,r)drl,
ϕf=WX,
rank(Wm×n)<min(m,n).
minXX0s.t.WXϕf22<ε.
Wm×n=Um×mm×nVn×nT,
m×n=[diag{σi}000],
Um×mm×nVn×nTXn=ϕf.
Um×rr×rVr×nTXn=ϕf.
Vr×nTXn=r×r1Um×rTϕf.
Wr×nnew=Vr×nT,
ϕr×1new=r×r1Um×rTϕf,
Wr×nnewXn×1=ϕr×1new.
Sn={s:sI¯n1|cn(s)|>αmax(|cn(s)|)}α(0,1),
α{n}=Nadativen1×αNadative(0,1).
In=In1Sn.
WInnewXIn=ϕr×1new.
rn=ϕr1newWInnewXIn.

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