Abstract

Some optical properties of a highly scattering medium, such as tissue, can be reconstructed non-invasively by diffuse optical tomography (DOT). Since the inverse problem of DOT is severely ill-posed and nonlinear, iterative methods that update Green’s function have been widely used to recover accurate optical parameters. However, recent research has shown that the joint sparse recovery principle can provide an important clue in achieving reconstructions without an iterative update of Green’s function. One of the main limitations of the previous work is that it can only be applied to absorption parameter reconstruction. In this paper, we extended this theory to estimate the absorption and scattering parameters simultaneously when the background optical properties are known. The main idea for such an extension is that a joint sparse recovery step gives us unknown fluence on the estimated support set, which eliminates the nonlinearity in an integral equation for the simultaneous estimation of the optical parameters. Our numerical results show that the proposed algorithm reduces the cross-talk artifacts between the parameters and provides improved reconstruction results compared to existing methods.

© 2013 OSA

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2012 (3)

J. M. Kim, O. K. Lee, and J. C. Ye, “Compressive MUSIC: Revisiting the link between compressive sensing and array signal processing,” IEEE Trans. Inf. Theory58, 278–301 (2012).
[CrossRef]

M. E. Davies and Y. C. Eldar, “Rank awareness in joint sparse recovery,” IEEE Trans. Inf. Theory58, 1135–1146 (2012).
[CrossRef]

J. M. Kim, O. K. Lee, and J. C. Ye, “Improving noise robustness in subspace-based joint sparse recovery,” IEEE Trans. Signal Process.60, 5799–5809 (2012).
[CrossRef]

2011 (2)

M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process.20, 681–695 (2011).
[CrossRef]

O. K. Lee, J. M. Kim, Y. Bresler, and J. C. Ye, “Compressive diffuse optical tomography: Non-iterative exact reconstruction using joint sparsity,” IEEE Trans. Med. Imaging30, 1129–1142 (2011).
[CrossRef] [PubMed]

2010 (1)

J. Wang, S. D. Jiang, Z. Z. Li, R. M. diFlorio Alexander, R. J. Barth, P. A. Kaufman, B. W. Pogue, and K. D. Paulsen, “In vivo quantitative imaging of normal and cancerous breast tissue using broadband diffuse optical tomography,” Med. Phys.37, 3715–3724 (2010).
[CrossRef] [PubMed]

2009 (1)

2007 (2)

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: A 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577–587 (2007).
[CrossRef] [PubMed]

D. P. Wipf and B. D. Rao, “An empirical Bayesian strategy for solving the simultaneous sparse approximation problem,” IEEE Trans. Signal Process.55, 3704–3716 (2007).
[CrossRef]

2006 (1)

J. Chen and X. Huo, “Theoretical results on sparse representations of multiple measurement vectors,” IEEE Trans. Signal Process.54, 4634–4643 (2006).
[CrossRef]

2005 (4)

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005).
[CrossRef] [PubMed]

S. F. Cotter, B. D. Rao, K. Engan, and K. Kreutz-Delgado, “Sparse solutions to linear inverse problems with multiple measurement vectors,” IEEE Trans. Signal Process.53, 2477–2488 (2005).
[CrossRef]

D. Malioutov, M. Cetin, and A. Willsky, “A sparse signal reconstruction perspective for source localization with sensor arrays,” IEEE Trans. Signal Process.53, 3010–3022 (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] [PubMed]

2004 (2)

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision20, 89–97 (2004).
[CrossRef]

B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, S. Srinivasan, X. Song, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Characterization of hemoglobin, water, and NIR scattering in breast tissue: Analysis of intersubject variability and menstrual cycle changes,” J. Biomed. Opt9, 541–552 (2004).
[CrossRef] [PubMed]

2003 (1)

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med.9, 123–128 (2003).
[CrossRef] [PubMed]

2002 (3)

2001 (6)

2000 (1)

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab.20, 469–477 (2000).
[CrossRef] [PubMed]

1999 (4)

1998 (3)

1997 (3)

1996 (1)

1995 (2)

N. F. Boyd, J. W. Byng, R. A. Jong, E. K. Fishell, L. E. Little, A. B. Miller, G. A. Lockwood, D. L. Tritchler, and M. J. Yaffe, “Quantitative classification of mammographic densities and breast cancer risk: Results from the Canadian national breast screening study,” J. Natl. Cancer Inst.87, 670–675 (1995).
[CrossRef] [PubMed]

M. O’Leary, D. Boas, B. Chance, and A. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett.20, 426–428 (1995).
[CrossRef]

1993 (1)

D. P. Huttenlocher, G. A. Klanderman, and W. J. Rucklidge, “Comparing images using the Hausdorff distance,” IEEE Trans. Pattern Anal. Mach. Intell.15, 850–863 (1993).
[CrossRef]

1989 (1)

A. Profio and G. Navarro, “Scientific basis of breast diaphanography,” Med. Phys.16, 60–65 (1989).
[CrossRef] [PubMed]

Afonso, M.

M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process.20, 681–695 (2011).
[CrossRef]

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] [PubMed]

Arridge, S.

Arridge, S. R.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005).
[CrossRef] [PubMed]

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

J. C. Hebden, F. E. W. Schmidt, M. E. Fry, M. Schweiger, E. M. C. Hillman, D. T. Delpy, and S. R. Arridge, “Simultaneous reconstruction of absorption and scattering images by multichannel measurement of purely temporal data,” Opt. Lett.24, 534–536 (1999).
[CrossRef]

Barbour, R. L.

Barth, R. J.

J. Wang, S. D. Jiang, Z. Z. Li, R. M. diFlorio Alexander, R. J. Barth, P. A. Kaufman, B. W. Pogue, and K. D. Paulsen, “In vivo quantitative imaging of normal and cancerous breast tissue using broadband diffuse optical tomography,” Med. Phys.37, 3715–3724 (2010).
[CrossRef] [PubMed]

Benaron, D. A.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab.20, 469–477 (2000).
[CrossRef] [PubMed]

Berger, A. J.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt.7, 60–71 (2002).
[CrossRef] [PubMed]

Bevilacqua, F.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt.7, 60–71 (2002).
[CrossRef] [PubMed]

Bioucas-Dias, J.

M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process.20, 681–695 (2011).
[CrossRef]

Boas, D.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab.20, 469–477 (2000).
[CrossRef] [PubMed]

M. O’Leary, D. Boas, B. Chance, and A. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett.20, 426–428 (1995).
[CrossRef]

Boas, D. A.

Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express17, 20178–20190 (2009).
[CrossRef] [PubMed]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag.18, 57–75 (2001).
[CrossRef]

Bouman, C. A.

Boyd, N. F.

J. W. Byng, M. J. Yaffe, R. A. Jong, R. S. Shumak, G. A. Lockwood, D. L. Tritchler, and N. F. Boyd, “Analysis of mammographic density and breast cancer risk from digitized mammograms,” Radiographics18, 1587–1598 (1998).
[PubMed]

N. F. Boyd, J. W. Byng, R. A. Jong, E. K. Fishell, L. E. Little, A. B. Miller, G. A. Lockwood, D. L. Tritchler, and M. J. Yaffe, “Quantitative classification of mammographic densities and breast cancer risk: Results from the Canadian national breast screening study,” J. Natl. Cancer Inst.87, 670–675 (1995).
[CrossRef] [PubMed]

Bresler, Y.

O. K. Lee, J. M. Kim, Y. Bresler, and J. C. Ye, “Compressive diffuse optical tomography: Non-iterative exact reconstruction using joint sparsity,” IEEE Trans. Med. Imaging30, 1129–1142 (2011).
[CrossRef] [PubMed]

J. C. Ye, S. Y. Lee, and Y. Bresler, “Exact reconstruction formula for diffuse optical tomography using simultaneous sparse representation,” in International Symposium on Biomedical Imaging: From Nano to Macro, Paris, France (Institute of Electrical and Electronics Engineers, 2008), pp. 1621–1624.

Brooks, D. H.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag.18, 57–75 (2001).
[CrossRef]

Butler, J.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt.7, 60–71 (2002).
[CrossRef] [PubMed]

Byng, J. W.

J. W. Byng, M. J. Yaffe, R. A. Jong, R. S. Shumak, G. A. Lockwood, D. L. Tritchler, and N. F. Boyd, “Analysis of mammographic density and breast cancer risk from digitized mammograms,” Radiographics18, 1587–1598 (1998).
[PubMed]

N. F. Boyd, J. W. Byng, R. A. Jong, E. K. Fishell, L. E. Little, A. B. Miller, G. A. Lockwood, D. L. Tritchler, and M. J. Yaffe, “Quantitative classification of mammographic densities and breast cancer risk: Results from the Canadian national breast screening study,” J. Natl. Cancer Inst.87, 670–675 (1995).
[CrossRef] [PubMed]

Cerussi, A. E.

A. E. Cerussi, D. Jakubowski, N. Shah, F. Bevilacqua, R. Lanning, A. J. Berger, D. Hsiang, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Spectroscopy enhances the information content of optical mammography,” J. Biomed. Opt.7, 60–71 (2002).
[CrossRef] [PubMed]

Cetin, M.

D. Malioutov, M. Cetin, and A. Willsky, “A sparse signal reconstruction perspective for source localization with sensor arrays,” IEEE Trans. Signal Process.53, 3010–3022 (2005).
[CrossRef]

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision20, 89–97 (2004).
[CrossRef]

Chance, B.

V. Ntziachristos and B. Chance, “Probing physiology and molecular function using optical imaging: Applications to breast cancer,” Breast Cancer Res.3, 41–46 (2001).
[CrossRef] [PubMed]

A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain function,” Trends Neurosci.20, 435–442 (1997).
[CrossRef] [PubMed]

M. O’Leary, D. Boas, B. Chance, and A. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett.20, 426–428 (1995).
[CrossRef]

Chatziioannou, A. F.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: A 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577–587 (2007).
[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, 4225–4241 (2005).
[CrossRef] [PubMed]

Chen, J.

J. Chen and X. Huo, “Theoretical results on sparse representations of multiple measurement vectors,” IEEE Trans. Signal Process.54, 4634–4643 (2006).
[CrossRef]

Cheong, W. F.

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W. F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab.20, 469–477 (2000).
[CrossRef] [PubMed]

Cotter, S. F.

S. F. Cotter, B. D. Rao, K. Engan, and K. Kreutz-Delgado, “Sparse solutions to linear inverse problems with multiple measurement vectors,” IEEE Trans. Signal Process.53, 2477–2488 (2005).
[CrossRef]

Davies, M. E.

M. E. Davies and Y. C. Eldar, “Rank awareness in joint sparse recovery,” IEEE Trans. Inf. Theory58, 1135–1146 (2012).
[CrossRef]

Dehghani, H.

B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, S. Srinivasan, X. Song, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Characterization of hemoglobin, water, and NIR scattering in breast tissue: Analysis of intersubject variability and menstrual cycle changes,” J. Biomed. Opt9, 541–552 (2004).
[CrossRef] [PubMed]

Delpy, D. T.

diFlorio Alexander, R. M.

J. Wang, S. D. Jiang, Z. Z. Li, R. M. diFlorio Alexander, R. J. Barth, P. A. Kaufman, B. W. Pogue, and K. D. Paulsen, “In vivo quantitative imaging of normal and cancerous breast tissue using broadband diffuse optical tomography,” Med. Phys.37, 3715–3724 (2010).
[CrossRef] [PubMed]

DiMarzio, C. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag.18, 57–75 (2001).
[CrossRef]

Dogdas, B.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: A 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577–587 (2007).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Sparsely distributed perturbations in optical properties for simultaneous DOT reconstruction.

Fig. 2
Fig. 2

Simulation geometry for simple targets.

Fig. 3
Fig. 3

Simulation geometry of the mouse phantom.

Fig. 4
Fig. 4

The cross-sections of the reconstructed images using various methods for simple target simulations (CASE A and CASE B).

Fig. 5
Fig. 5

The cross-sections of the reconstructed images using various methods for simple target simulations (CASE C and CASE D).

Fig. 6
Fig. 6

The 3-D simultaneous reconstruction results of tumors with optical parameter variations δμa and δμ′s embedded in the mouse phantom.

Tables (9)

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Algorithm 1 Pseudocode implementation of C-SALSA for DOT problem in Eq. (19).

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Table 1 Parameters used in C-SALSA algorithm.

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Table 2 Optical properties for various cases.

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Table 3 Optical properties of the mouse [43, 44].

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Table 4 Optical properties of the targets.

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Table 5 The MSE values of the reconstruction using various methods.

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Table 6 Average run time of the reconstruction using various methods.

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Table 7 MSE for the reconstruction results of the mouse phantom using various methods.

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Table 8 Hausdorff distance in mm for various reconstruction methods.

Equations (21)

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{ D ( r ) u ( r ) μ a ( r ) u ( r ) = S ( r ) , r Ω u ( r ) + n ^ u ( r ) = 0 , r Ω
{ ( D 0 ( r ) k ( r ) 2 ) u ( r ) = [ δ μ a ( r ) δ D ( r ) ] u ( r ) S ( r ) , r Ω u ( r ) + n ^ u ( r ) = 0 , r Ω ,
u ( r ) = u 0 ( r ) Ω G 0 ( r , r ) [ δ μ a ( r ) δ D ( r ) ] u ( r ) d r , r Ω ,
{ ( D 0 ( r ) k ( r ) 2 ) G 0 ( r , r ) = δ ( r r ) , r Ω G 0 ( r , r ) + n ^ G 0 ( r , r ) = 0 , r Ω ,
u 0 ( r ) = Ω G 0 ( r , r ) S ( r ) d r .
0 = Ω n ^ [ δ D ( r ) G 0 ( r , r ) u ( r ) ] d r = Ω [ δ D ( r ) G 0 ( r , r ) u ( r ) ] d r = Ω δ D ( r ) G 0 ( r , r ) u ( r ) d r + Ω G 0 ( r , r ) δ D ( r ) u ( r ) d r ,
u ( r ) = u 0 ( r ) Ω G 0 ( r , r ) u ( r ) δ μ a ( r ) d r Ω G 0 ( r , r ) u ( r ) δ D ( r ) d r , r Ω .
u ( r ; x ) u ( r ; x ( k ) ) Ω G 0 ( r , r ; x ( k ) ) u ( r ; x ( k ) ) δ μ a ( r ) d r Ω G 0 ( r , r ; x ( k ) ) u ( r ; x ( k ) ) δ D ( r ) d r ,
u ( r ; l ) = u 0 ( r ; l ) Ω G 0 ( r , r ) X ( r ; l ) d r = u 0 ( r ; l ) Ω t G 0 ( r , r ) X ( r ; l ) d r , l = 1 , 2 , , N ,
δ μ a ( r ) = 0 , δ D ( r ) = 0 , δ D ( r ) = 0 , r Ω t .
u ^ ( r ; l ) = u 0 ( r ; l ) Ω t G 0 ( r , r ) X ^ ( r ; l ) d r , r Ω t .
u ( r ; l ) = u 0 ( r ; l ) Ω t G 0 ( r , r ) u ^ ( r ; l ) δ μ a ( r ) d r Ω t G 0 ( r , r ) u ^ ( r ; l ) δ D ( r ) d r , l = 1 , 2 , , N .
X ( r ; l ) = i = 1 k X ( r ( i ) ; l ) b ( r , r ( i ) ) , l = 1 , 2 , , N ,
u ( r ; l ) = u 0 ( r ; l ) Ω t G 0 ( r , r ) X ( r ; l ) d r , l = 1 , 2 , , N , = u 0 ( r ; l ) i = 1 k G ˜ 0 ( r , r ( i ) ) X ( r ( i ) ; l ) ,
G ˜ 0 ( r , r ( i ) ) = Ω t G 0 ( r , r ) b ( r , r ( i ) ) d r .
Y = A X .
min X X 0 , subject to Y A X F ε .
y = [ y ( 1 ) y ( N ) ] = [ A μ a ( 1 ) A D ( 1 ) A μ a ( N ) A D ( N ) ] [ δ μ a , 1 δ μ a , k ^ δ D 1 δ D k ^ ] = A x .
arg min x δ μ a 1 + δ D 1 + λ δ μ a T V + λ δ D T V subject to A x y 2 ε , a min δ μ a i a max , D min δ D i D max .
MSE ( x ) = x true x recon 2 x true 2 .
δ μ s ( r ) = ( δ D ( r ) 3 D 0 ( r ) [ δ D ( r ) + D 0 ( r ) ] + δ μ a ( r ) ) .

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