Abstract

Fluorescence tomography is an imaging modality that seeks to reconstruct the distribution of fluorescent dyes inside a highly scattering sample from light measurements on the boundary. Using common inversion methods with L2 penalties typically leads to smooth reconstructions, which degrades the obtainable resolution. The use of total variation (TV) regularization for the inverse model is investigated. To solve the inverse problem efficiently, an augmented Lagrange method is utilized that allows separating the Gauss–Newton minimization from the TV minimization. Results on noisy simulation data provide evidence that the reconstructed inclusions are much better localized and that their half-width measure decreases by at least 25% compared to ordinary L2 reconstructions.

© 2010 Optical Society of America

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2010 (2)

C. Clason, B. Jin, and K. Kunisch, “A duality-based splitting method for ℓ1-TV image restoration with automatic regularization parameter choice,” SIAM J. Sci. Comput. 32, 1484–1505 (2010).
[CrossRef]

C. Clason, B. Jin, and K. Kunisch, “A semismooth Newton method for L1 data fitting with automatic choice of regularization parameters and noise calibration,” SIAM J. Imaging Sci. 3, 199–231 (2010).
[CrossRef]

2009 (4)

T. Goldstein and S. Osher, “The split Bregman method for L1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343(2009).
[CrossRef]

P. Weiss, L. Blanc-Féraud, and G. Aubert, “Efficient schemes for total variation minimization under constraints in image processing,” SIAM J. Sci. Comput. 31, 2047–2080 (2009).
[CrossRef]

J.-F. Aujol, “Some first-order algorithms for total variation based image restoration,” J. Math. Imaging Vision 34, 307–327 (2009).
[CrossRef]

Y. Y. Chen and A. W. Wood, “Application of a temperature-dependent fluorescent dye (Rhodamine B) to the measurement of radiofrequency radiation-induced temperature changes in biological samples,” Bioelectromagnetics (N.Y.) 30, 583–590 (2009).
[CrossRef]

2007 (1)

G. Stadler, “Path-following and augmented Lagrangian methods for contact problems in linear elasticity,” J. Comput. Appl. Math. 203, 533–547 (2007).
[CrossRef]

2006 (1)

M. Hintermüller and K. Kunisch, “Path-following methods for a class of constrained minimization problems in function space,” SIAM J. Optim. 17, 159–187 (2006).
[CrossRef]

2005 (2)

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]

Y. Nesterov, “Smooth minimization of non-smooth functions,” Math. Program. 103, 127–152 (2005).
[CrossRef]

2004 (4)

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med. 35, 342–348 (2004).
[CrossRef] [PubMed]

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

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

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 73 (2004).
[CrossRef]

2003 (1)

2002 (1)

2001 (1)

1999 (1)

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

1998 (1)

K. Kunisch and J. Zou, “Iterative choices of regularization parameters in linear inverse problems,” Inverse Probl. 14, 1247–1264 (1998).
[CrossRef]

1997 (3)

B. Blaschke, A. Neubauer, and O. Scherzer, “On convergence rates for the iteratively regularized Gauss-Newton method,” IMA J. Numer. Anal. 17, 421–436 (1997).
[CrossRef]

T. Hohage, “Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem,” Inverse Probl. 13, 1279–1299 (1997).
[CrossRef]

J. S. Reynolds, T. L. Troy, and E. M. Sevick-Muraca, “Multipixel techniques for frequency-domain photon migration imaging,” Biotechnol. Prog. 13, 669–680 (1997).
[CrossRef] [PubMed]

1996 (1)

1992 (2)

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B 13, 307–314(1992).
[CrossRef] [PubMed]

K. Ito and K. Kunisch, “On the choice of the regularization parameter in nonlinear inverse problems,” SIAM J. Optim. 2, 376–404 (1992).
[CrossRef]

1988 (1)

1976 (2)

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Comput. Math. Appl. 2, 17–40 (1976).
[CrossRef]

I. S. Longmuir and J. A. Knopp, “Measurement of tissue oxygen with a fluorescent probe,” J. Appl. Physiol. 41, 598–602(1976).
[PubMed]

1974 (1)

R. Glowinski and A. Marrocco, “Sur l’approximation, par éléments finis d’ordre 1, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires,” C. R. Acad. Sci. Paris Ser. A 278, 1649–1652 (1974).

1969 (1)

M. R. Hestenes, “Multiplier and gradient methods,” J. Optimization Theory Appl. 4, 303–320 (1969).
[CrossRef]

1966 (1)

V. A. Morozov, “On the solution of functional equations by the method of regularization,” Sov. Math. Dokl. 7, 414–417(1966).

Abbas, Y.

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B 13, 307–314(1992).
[CrossRef] [PubMed]

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. R.

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

Aubert, G.

P. Weiss, L. Blanc-Féraud, and G. Aubert, “Efficient schemes for total variation minimization under constraints in image processing,” SIAM J. Sci. Comput. 31, 2047–2080 (2009).
[CrossRef]

Aujol, J.-F.

J.-F. Aujol, “Some first-order algorithms for total variation based image restoration,” J. Math. Imaging Vision 34, 307–327 (2009).
[CrossRef]

Bakushinsky, A. B.

A. B. Bakushinsky and M. Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, Vol. 577 of Mathematics and Its Applications (Springer, 2004).

Bangerth, W.

Bischof, H.

T. Pock, M. Unger, D. Cremers, and H. Bischof, “Fast and exact solution of total variation models on the GPU,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, 2008. CVPRW ’08 (2008), pp. 1–8.
[CrossRef]

Blanc-Féraud, L.

P. Weiss, L. Blanc-Féraud, and G. Aubert, “Efficient schemes for total variation minimization under constraints in image processing,” SIAM J. Sci. Comput. 31, 2047–2080 (2009).
[CrossRef]

Blaschke, B.

B. Blaschke, A. Neubauer, and O. Scherzer, “On convergence rates for the iteratively regularized Gauss-Newton method,” IMA J. Numer. Anal. 17, 421–436 (1997).
[CrossRef]

Boas, D. A.

Bouman, C. A.

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 73 (2004).
[CrossRef]

A. Chambolle, “Total variation minimization and a class of binary MRF models,” in Proceedings of International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (2005), pp. 136–152.
[CrossRef]

Chance, B.

Chatziioannou, A. F.

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, Y. Y.

Y. Y. Chen and A. W. Wood, “Application of a temperature-dependent fluorescent dye (Rhodamine B) to the measurement of radiofrequency radiation-induced temperature changes in biological samples,” Bioelectromagnetics (N.Y.) 30, 583–590 (2009).
[CrossRef]

Chernomordik, V.

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med. 35, 342–348 (2004).
[CrossRef] [PubMed]

Clason, C.

C. Clason, B. Jin, and K. Kunisch, “A duality-based splitting method for ℓ1-TV image restoration with automatic regularization parameter choice,” SIAM J. Sci. Comput. 32, 1484–1505 (2010).
[CrossRef]

C. Clason, B. Jin, and K. Kunisch, “A semismooth Newton method for L1 data fitting with automatic choice of regularization parameters and noise calibration,” SIAM J. Imaging Sci. 3, 199–231 (2010).
[CrossRef]

Coustaud, D.

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B 13, 307–314(1992).
[CrossRef] [PubMed]

Cremers, D.

T. Pock, M. Unger, D. Cremers, and H. Bischof, “Fast and exact solution of total variation models on the GPU,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, 2008. CVPRW ’08 (2008), pp. 1–8.
[CrossRef]

Devoisselle, J. M.

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B 13, 307–314(1992).
[CrossRef] [PubMed]

Dunsby, C.

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Dymoke-Bradshaw, A.

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Egger, H.

H. Egger, M. Freiberger, and M. Schlottbom, “Analysis of forward and inverse models in fluorescence optical tomography,” Tech. Rep. SFB-2009-075 (SFB Research Center “Mathematical Optimization and Applications in Biomedical Sciences,” 2009).

Ekeland, I.

I. Ekeland and R. Témam, Convex Analysis and Variational Problems (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Elson, D. S.

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Engl, H. W.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).
[CrossRef]

Freiberger, M.

H. Egger, M. Freiberger, and M. Schlottbom, “Analysis of forward and inverse models in fluorescence optical tomography,” Tech. Rep. SFB-2009-075 (SFB Research Center “Mathematical Optimization and Applications in Biomedical Sciences,” 2009).

French, P. M. W.

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Gabay, D.

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Comput. Math. Appl. 2, 17–40 (1976).
[CrossRef]

Galletly, N.

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Gandjbakhche, A.

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med. 35, 342–348 (2004).
[CrossRef] [PubMed]

Gannot, I.

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med. 35, 342–348 (2004).
[CrossRef] [PubMed]

Glowinski, R.

R. Glowinski and A. Marrocco, “Sur l’approximation, par éléments finis d’ordre 1, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires,” C. R. Acad. Sci. Paris Ser. A 278, 1649–1652 (1974).

R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Scientific Computation (Springer-Verlag, 2008). Reprint of the 1984 original.

Goldstein, T.

T. Goldstein and S. Osher, “The split Bregman method for L1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343(2009).
[CrossRef]

Hanke, M.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).
[CrossRef]

Hares, J.

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Hekmat, F.

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med. 35, 342–348 (2004).
[CrossRef] [PubMed]

Hestenes, M. R.

M. R. Hestenes, “Multiplier and gradient methods,” J. Optimization Theory Appl. 4, 303–320 (1969).
[CrossRef]

Hintermüller, M.

M. Hintermüller and K. Kunisch, “Path-following methods for a class of constrained minimization problems in function space,” SIAM J. Optim. 17, 159–187 (2006).
[CrossRef]

Hohage, T.

T. Hohage, “Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem,” Inverse Probl. 13, 1279–1299 (1997).
[CrossRef]

Ito, K.

K. Ito and K. Kunisch, “On the choice of the regularization parameter in nonlinear inverse problems,” SIAM J. Optim. 2, 376–404 (1992).
[CrossRef]

Jiang, H.

Jin, B.

C. Clason, B. Jin, and K. Kunisch, “A semismooth Newton method for L1 data fitting with automatic choice of regularization parameters and noise calibration,” SIAM J. Imaging Sci. 3, 199–231 (2010).
[CrossRef]

C. Clason, B. Jin, and K. Kunisch, “A duality-based splitting method for ℓ1-TV image restoration with automatic regularization parameter choice,” SIAM J. Sci. Comput. 32, 1484–1505 (2010).
[CrossRef]

Joshi, A.

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

A. Joshi, “Adaptive finite element methods for fluorescence enhanced optical tomography,” Ph.D. dissertation (Texas A&M University2005).

Keijzer, M.

Kellett, P. A.

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Knopp, J. A.

I. S. Longmuir and J. A. Knopp, “Measurement of tissue oxygen with a fluorescent probe,” J. Appl. Physiol. 41, 598–602(1976).
[PubMed]

Kokurin, M. Y.

A. B. Bakushinsky and M. Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, Vol. 577 of Mathematics and Its Applications (Springer, 2004).

Kunisch, K.

C. Clason, B. Jin, and K. Kunisch, “A duality-based splitting method for ℓ1-TV image restoration with automatic regularization parameter choice,” SIAM J. Sci. Comput. 32, 1484–1505 (2010).
[CrossRef]

C. Clason, B. Jin, and K. Kunisch, “A semismooth Newton method for L1 data fitting with automatic choice of regularization parameters and noise calibration,” SIAM J. Imaging Sci. 3, 199–231 (2010).
[CrossRef]

M. Hintermüller and K. Kunisch, “Path-following methods for a class of constrained minimization problems in function space,” SIAM J. Optim. 17, 159–187 (2006).
[CrossRef]

K. Kunisch and J. Zou, “Iterative choices of regularization parameters in linear inverse problems,” Inverse Probl. 14, 1247–1264 (1998).
[CrossRef]

K. Ito and K. Kunisch, “On the choice of the regularization parameter in nonlinear inverse problems,” SIAM J. Optim. 2, 376–404 (1992).
[CrossRef]

Lever, M. J.

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Li, X. D.

Longmuir, I. S.

I. S. Longmuir and J. A. Knopp, “Measurement of tissue oxygen with a fluorescent probe,” J. Appl. Physiol. 41, 598–602(1976).
[PubMed]

Marrocco, A.

R. Glowinski and A. Marrocco, “Sur l’approximation, par éléments finis d’ordre 1, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires,” C. R. Acad. Sci. Paris Ser. A 278, 1649–1652 (1974).

Maunoury, V.

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B 13, 307–314(1992).
[CrossRef] [PubMed]

McGinty, J.

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Mercier, B.

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Comput. Math. Appl. 2, 17–40 (1976).
[CrossRef]

Millane, R. P.

Milstein, A. B.

Mordon, S.

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B 13, 307–314(1992).
[CrossRef] [PubMed]

Morozov, V. A.

V. A. Morozov, “On the solution of functional equations by the method of regularization,” Sov. Math. Dokl. 7, 414–417(1966).

Munro, I.

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Neil, M. A. A.

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Nesterov, Y.

Y. Nesterov, “Smooth minimization of non-smooth functions,” Math. Program. 103, 127–152 (2005).
[CrossRef]

Neubauer, A.

B. Blaschke, A. Neubauer, and O. Scherzer, “On convergence rates for the iteratively regularized Gauss-Newton method,” IMA J. Numer. Anal. 17, 421–436 (1997).
[CrossRef]

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).
[CrossRef]

Ntziachristos, V.

O’Leary, M. A.

Oh, S.

Osher, S.

T. Goldstein and S. Osher, “The split Bregman method for L1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343(2009).
[CrossRef]

Pock, T.

T. Pock, M. Unger, D. Cremers, and H. Bischof, “Fast and exact solution of total variation models on the GPU,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, 2008. CVPRW ’08 (2008), pp. 1–8.
[CrossRef]

Powell, M. J. D.

M. J. D. Powell, “A method for nonlinear constraints in minimization problems,” in Optimization (Academic, 1969), pp. 283–298.

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] [PubMed]

Requejo-Isidro, J.

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Reynolds, J. S.

J. S. Reynolds, T. L. Troy, and E. M. Sevick-Muraca, “Multipixel techniques for frequency-domain photon migration imaging,” Biotechnol. Prog. 13, 669–680 (1997).
[CrossRef] [PubMed]

Ron, I.

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med. 35, 342–348 (2004).
[CrossRef] [PubMed]

Scherzer, O.

B. Blaschke, A. Neubauer, and O. Scherzer, “On convergence rates for the iteratively regularized Gauss-Newton method,” IMA J. Numer. Anal. 17, 421–436 (1997).
[CrossRef]

Schlottbom, M.

H. Egger, M. Freiberger, and M. Schlottbom, “Analysis of forward and inverse models in fluorescence optical tomography,” Tech. Rep. SFB-2009-075 (SFB Research Center “Mathematical Optimization and Applications in Biomedical Sciences,” 2009).

Sevick-Muraca, E. M.

J. S. Reynolds, T. L. Troy, and E. M. Sevick-Muraca, “Multipixel techniques for frequency-domain photon migration imaging,” Biotechnol. Prog. 13, 669–680 (1997).
[CrossRef] [PubMed]

Sevick-Muraca, W. M.

Shives, E.

Stadler, G.

G. Stadler, “Path-following and augmented Lagrangian methods for contact problems in linear elasticity,” J. Comput. Appl. Math. 203, 533–547 (2007).
[CrossRef]

Stamp, G. W.

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Star, W. M.

Storchi, P. R. M.

Tao, M.

M. Tao and J. Yang, “Alternating direction algorithms for total variation deconvolution in image reconstruction,” TR0918, Department of Mathematics, Nanjing University, 2009.

Témam, R.

I. Ekeland and R. Témam, Convex Analysis and Variational Problems (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Troy, T. L.

J. S. Reynolds, T. L. Troy, and E. M. Sevick-Muraca, “Multipixel techniques for frequency-domain photon migration imaging,” Biotechnol. Prog. 13, 669–680 (1997).
[CrossRef] [PubMed]

Unger, M.

T. Pock, M. Unger, D. Cremers, and H. Bischof, “Fast and exact solution of total variation models on the GPU,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, 2008. CVPRW ’08 (2008), pp. 1–8.
[CrossRef]

Webb, K. J.

Weiss, P.

P. Weiss, L. Blanc-Féraud, and G. Aubert, “Efficient schemes for total variation minimization under constraints in image processing,” SIAM J. Sci. Comput. 31, 2047–2080 (2009).
[CrossRef]

Weissleder, R.

Wood, A. W.

Y. Y. Chen and A. W. Wood, “Application of a temperature-dependent fluorescent dye (Rhodamine B) to the measurement of radiofrequency radiation-induced temperature changes in biological samples,” Bioelectromagnetics (N.Y.) 30, 583–590 (2009).
[CrossRef]

Xu, Y.

Yang, J.

M. Tao and J. Yang, “Alternating direction algorithms for total variation deconvolution in image reconstruction,” TR0918, Department of Mathematics, Nanjing University, 2009.

Yodh, Y. G.

Zhang, Q.

Zou, J.

K. Kunisch and J. Zou, “Iterative choices of regularization parameters in linear inverse problems,” Inverse Probl. 14, 1247–1264 (1998).
[CrossRef]

Appl. Opt. (2)

Bioelectromagnetics (N.Y.) (1)

Y. Y. Chen and A. W. Wood, “Application of a temperature-dependent fluorescent dye (Rhodamine B) to the measurement of radiofrequency radiation-induced temperature changes in biological samples,” Bioelectromagnetics (N.Y.) 30, 583–590 (2009).
[CrossRef]

Biotechnol. Prog. (1)

J. S. Reynolds, T. L. Troy, and E. M. Sevick-Muraca, “Multipixel techniques for frequency-domain photon migration imaging,” Biotechnol. Prog. 13, 669–680 (1997).
[CrossRef] [PubMed]

C. R. Acad. Sci. Paris Ser. A (1)

R. Glowinski and A. Marrocco, “Sur l’approximation, par éléments finis d’ordre 1, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires,” C. R. Acad. Sci. Paris Ser. A 278, 1649–1652 (1974).

Comput. Math. Appl. (1)

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Comput. Math. Appl. 2, 17–40 (1976).
[CrossRef]

IMA J. Numer. Anal. (1)

B. Blaschke, A. Neubauer, and O. Scherzer, “On convergence rates for the iteratively regularized Gauss-Newton method,” IMA J. Numer. Anal. 17, 421–436 (1997).
[CrossRef]

Inverse Probl. (3)

T. Hohage, “Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem,” Inverse Probl. 13, 1279–1299 (1997).
[CrossRef]

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

K. Kunisch and J. Zou, “Iterative choices of regularization parameters in linear inverse problems,” Inverse Probl. 14, 1247–1264 (1998).
[CrossRef]

J. Appl. Physiol. (1)

I. S. Longmuir and J. A. Knopp, “Measurement of tissue oxygen with a fluorescent probe,” J. Appl. Physiol. 41, 598–602(1976).
[PubMed]

J. Comput. Appl. Math. (1)

G. Stadler, “Path-following and augmented Lagrangian methods for contact problems in linear elasticity,” J. Comput. Appl. Math. 203, 533–547 (2007).
[CrossRef]

J. Math. Imaging Vision (2)

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 73 (2004).
[CrossRef]

J.-F. Aujol, “Some first-order algorithms for total variation based image restoration,” J. Math. Imaging Vision 34, 307–327 (2009).
[CrossRef]

J. Optimization Theory Appl. (1)

M. R. Hestenes, “Multiplier and gradient methods,” J. Optimization Theory Appl. 4, 303–320 (1969).
[CrossRef]

J. Photochem. Photobiol. B (1)

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B 13, 307–314(1992).
[CrossRef] [PubMed]

Lasers Surg. Med. (1)

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med. 35, 342–348 (2004).
[CrossRef] [PubMed]

Math. Program. (1)

Y. Nesterov, “Smooth minimization of non-smooth functions,” Math. Program. 103, 127–152 (2005).
[CrossRef]

New J. Phys. (1)

D. S. Elson, I. Munro, J. Requejo-Isidro, J. McGinty, C. Dunsby, N. Galletly, G. W. Stamp, M. A. A. Neil, M. J. Lever, P. A. Kellett, A. Dymoke-Bradshaw, J. Hares, and P. M. W. French, “Real-time time-domain fluorescence lifetime imaging including single-shot acquisition with a segmented optical image intensifier,” New J. Phys. 6, 180 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

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, 4225–4241 (2005).
[CrossRef] [PubMed]

SIAM J. Imaging Sci. (2)

T. Goldstein and S. Osher, “The split Bregman method for L1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343(2009).
[CrossRef]

C. Clason, B. Jin, and K. Kunisch, “A semismooth Newton method for L1 data fitting with automatic choice of regularization parameters and noise calibration,” SIAM J. Imaging Sci. 3, 199–231 (2010).
[CrossRef]

SIAM J. Optim. (2)

K. Ito and K. Kunisch, “On the choice of the regularization parameter in nonlinear inverse problems,” SIAM J. Optim. 2, 376–404 (1992).
[CrossRef]

M. Hintermüller and K. Kunisch, “Path-following methods for a class of constrained minimization problems in function space,” SIAM J. Optim. 17, 159–187 (2006).
[CrossRef]

SIAM J. Sci. Comput. (2)

P. Weiss, L. Blanc-Féraud, and G. Aubert, “Efficient schemes for total variation minimization under constraints in image processing,” SIAM J. Sci. Comput. 31, 2047–2080 (2009).
[CrossRef]

C. Clason, B. Jin, and K. Kunisch, “A duality-based splitting method for ℓ1-TV image restoration with automatic regularization parameter choice,” SIAM J. Sci. Comput. 32, 1484–1505 (2010).
[CrossRef]

Sov. Math. Dokl. (1)

V. A. Morozov, “On the solution of functional equations by the method of regularization,” Sov. Math. Dokl. 7, 414–417(1966).

Other (10)

I. Ekeland and R. Témam, Convex Analysis and Variational Problems (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

M. Tao and J. Yang, “Alternating direction algorithms for total variation deconvolution in image reconstruction,” TR0918, Department of Mathematics, Nanjing University, 2009.

R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Scientific Computation (Springer-Verlag, 2008). Reprint of the 1984 original.

M. J. D. Powell, “A method for nonlinear constraints in minimization problems,” in Optimization (Academic, 1969), pp. 283–298.

A. Chambolle, “Total variation minimization and a class of binary MRF models,” in Proceedings of International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (2005), pp. 136–152.
[CrossRef]

A. Joshi, “Adaptive finite element methods for fluorescence enhanced optical tomography,” Ph.D. dissertation (Texas A&M University2005).

H. Egger, M. Freiberger, and M. Schlottbom, “Analysis of forward and inverse models in fluorescence optical tomography,” Tech. Rep. SFB-2009-075 (SFB Research Center “Mathematical Optimization and Applications in Biomedical Sciences,” 2009).

A. B. Bakushinsky and M. Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, Vol. 577 of Mathematics and Its Applications (Springer, 2004).

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).
[CrossRef]

T. Pock, M. Unger, D. Cremers, and H. Bischof, “Fast and exact solution of total variation models on the GPU,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, 2008. CVPRW ’08 (2008), pp. 1–8.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Geometry and source and detector placement used for the simulations and (b), (c) two simulation phantoms with spherical inclusions (concentrations in μM )

Fig. 2
Fig. 2

(a) Reconstruction results of the phantom shown in Fig. 1b with L 2 regularization and two TV reconstructions using (b) a lower penalty and (c) a higher one.

Fig. 3
Fig. 3

L 2 regularized (upper row) and total variation reconstructions (lower row) for different noise levels of 0% (left), 5% (middle), and 10% (right) data noise. The white border marks the FWHM.

Tables (5)

Tables Icon

Table 1 List of Optical Tissue Parameters Used for the Computation of Measurement Data [13, 33, 34]

Tables Icon

Table 2 Maximum Concentration and FWHM of the Reconstructed Fluorescent Inclusions a

Tables Icon

Table 3 Number of Gauss–Newton Iterations until Convergence

Tables Icon

Table 4 Algorithm 1: ALM-ADM Total Variation Gauss–Newton Algorithm

Tables Icon

Table 5 Algorithm 2: Nesterov’s Method for Total Variation Minimization

Equations (17)

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

· ( κ ex ( x ) φ ex ( x ) ) + ( μ a , i , ex ( x ) + c ( x ) ε ex + i ω v ex ) φ ex ( x ) = q ( x ) , · ( κ em ( x ) φ em ( x ) ) + ( μ a , i , em ( x ) + c ( x ) ε em + i ω v em ) φ em ( x ) = Q 1 i ω τ c ( x ) ε ex φ ex ( x ) ,
φ i ( x ) + 2 R i κ i ( x ) φ i ( x ) n = 0 , on Ω ,
m = Γ D κ i ( s ) φ i ( s ) n d s .
F c y , where c ( x ) R + and y C s × d .
c ˜ = min c F ( c ) y δ 2 + α R ( c ) .
min δ c 1 2 F ( c k ) + F ( c k ) δ c y δ 2 + α k R ( c k + δ c ) ,
TV ( c ) = | c | d x .
α k R ( c ) = β TV ( c ) + α k 2 c c p 2 .
min δ c , c ¯ max λ 1 2 F ( c k ) δ c + F ( c k ) y δ 2 + α k 2 c k + δ c c p 2 + β TV ( c ¯ ) + λ , c k + δ c c ¯ + μ 2 c k + δ c c ¯ 2 ,
min δ c 1 2 F ( c k ) δ c + F ( c k ) y δ 2 + α k 2 c k + δ c c p 2 + λ , δ c + μ 2 c k + δ c c ¯ 2 ,
[ F ( c k ) H F ( c k ) + ( α k + μ ) I ] δ c = F ( c k ) ( y δ F ( c k ) ) α k ( c k c p ) μ ( c k c ¯ ) λ ,
min c ¯ β TV ( c ¯ ) λ , c ¯ + μ 2 c k + 1 c ¯ 2 ,
min p β 1 2 μ div p + λ 2 + div p , c k + 1 ,
( 1 μ ( div p + λ ) + c k + 1 ) ,
P K ( v ) = β v max ( β , | v | ) ,
min δ c 1 2 F ( c k ) δ c + F ( c k ) y δ 2 + α k 2 c k + δ c c p 2 + λ j , δ c + μ 2 c k + δ c c ¯ j 2
min c ¯ β TV ( c ¯ ) λ j , c ¯ + μ 2 c k + δ c j + 1 c ¯ 2

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