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The Moreau envelope approach for the L1/TV image denoising model
Solving inverse source problems by the Orthogonal Solution and Kernel Correction Algorithm (OSKCA) with applications in fluorescence tomography
1. | School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, United States, United States, United States |
2. | School of Electrical and Computer Engineering, Georgia Institute of Technology, 777 Atlantic Drive NW Atlanta, GA 30332, United States |
References:
[1] |
S. R. Arridge, Optical tomography in medical imaging,, Inverse Problems, 15 (1999), 0266.
doi: 10.1088/0266-5611/15/2/022. |
[2] |
S. R. Arridge and J. C. Schotland, Optical tomography: Forward and inverse problems,, Inverse Problems, 25 (2009), 0266.
doi: 10.1088/0266-5611/25/12/123010. |
[3] |
G. Bao, S. Hou and P. Li, Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm,, Journal of Computational Physics, 227 (2007), 755.
doi: 10.1016/j.jcp.2007.08.020. |
[4] |
A. Behrooz, H. Zhou, A. Eftekhar and A. Adibi, Total variation regularization for 3d reconstruction in fluorescence tomography: experimental phantom studies,, Applied Optics, 51 (2012), 8216.
doi: 10.1364/AO.51.008216. |
[5] |
C. Bremer, V. Ntziachristos and R. Weissleder, Optical-based molecular imaging: contrast agents and potential medical applications,, European Radiology, 13 (2003), 231. Google Scholar |
[6] |
S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods,, vol. 15, (2007). Google Scholar |
[7] |
J. Cai, E. Candès and Z. Shen, A singular value thresholding algorithm for matrix completion,, SIAM Journal on Optimization, 20 (2010), 1956.
doi: 10.1137/080738970. |
[8] |
Y. Censor and E. Tom, Convergence of string-averaging projection schemes for inconsistent convex feasibility problems,, Optimization Methods and Software, 18 (2003), 543.
doi: 10.1080/10556780310001610484. |
[9] |
Y. Chen, W. Hager, M. Yashtini, X. Ye and H. Zhang, Bregman operator splitting with variable stepsize for total variation image reconstruction,, Comput. Optim. Appl., 54 (2013).
doi: 10.1007/s10589-012-9519-2. |
[10] |
Y. Dai and R. Fletcher, Projected barzilai-borwein methods for large-scale box-constrained quadratic programming,, Numerische Mathematik, 100 (2005), 21.
doi: 10.1007/s00211-004-0569-y. |
[11] |
Y. Dai and L. Liao, R-linear convergence of the barzilai and borwein gradient method,, IMA Journal of Numerical Analysis, 22 (2002), 1.
doi: 10.1093/imanum/22.1.1. |
[12] |
J. Dutta, S. Ahn, A. Joshi and R. Leahy, Optimal illumination patterns for fluorescence tomography,, in Biomedical Imaging: From Nano to Macro, (2009), 1275.
doi: 10.1109/ISBI.2009.5193295. |
[13] |
J. Dutta, S. Ahn, C. Li, S. R. Cherry and R. M. Leahy, Joint l1 and total variation regularization for fluorescence molecular tomography,, Physics in Medicine and Biology, 57 (2012), 1459. Google Scholar |
[14] |
H. Egger, M. Freiberger and M. Schlottbom, On forward and inverse models in fluorescence diffuse optical tomography,, Inverse Problems and Imaging, 4 (2010), 411.
doi: 10.3934/ipi.2010.4.411. |
[15] |
H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, vol. 375, (1996).
doi: 10.1007/978-94-009-1740-8. |
[16] |
E. Esser, Applications of lagrangian-based alternating direction methods and connections to split bregman,, CAM Report, 9 (2009). Google Scholar |
[17] |
T. J. Farrell, M. S. Patterson and B. Wilson, A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,, Medical Physics, 19 (1992), 879.
doi: 10.1118/1.596777. |
[18] |
H. Gao and H. Zhao, Multilevel bioluminescence tomography based on radiative transfer equation part 1: l1 regularization,, Optics Express, 18 (2010), 1854.
doi: 10.1364/OE.18.001854. |
[19] |
H. Gao and H. Zhao, Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity,, Optics Express, 18 (2010), 2894.
doi: 10.1364/OE.18.002894. |
[20] |
T. Goldstein and S. Osher, The split bregman method for l1-regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323.
doi: 10.1137/080725891. |
[21] |
G. Golub and C. Van Loan, Matrix Computations,, vol. 3, (1996).
|
[22] |
P. Hansen and D. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problems,, SIAM Journal on Scientific Computing, 14 (1993), 1487.
doi: 10.1137/0914086. |
[23] |
R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams and B. J. Tromberg, Boundary conditions for the diffusion equation in radiative transfer,, Journal of the Optical Society of America. A, 11 (1994), 2727.
doi: 10.1364/JOSAA.11.002727. |
[24] |
D. Hawrysz and E. Sevick-Muraca, Developments toward diagnostic breast cancer imaging using near-infrared optical measurements and fluorescent contrast agents,, Neoplasia (New York, 2 (2000).
doi: 10.1038/sj.neo.7900118. |
[25] |
J. Hebden, S. Arridge and D. Delpy, Optical imaging in medicine: I. experimental techniques,, Physics in Medicine and Biology, 42 (1999).
doi: 10.1088/0031-9155/42/5/007. |
[26] |
G. Herman, A. Lent and S. Rowland, ART: Mathematics and applications: A report on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques,, Journal of theoretical biology, 42 (1973), 1.
doi: 10.1016/0022-5193(74)90135-0. |
[27] |
A. Ishimaru, Wave Propagation and Scattering in Random Media,, vol. 2, (1997).
|
[28] |
J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems,, vol. 160, (2004).
|
[29] |
O. A. Ladyzhenskaia and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations,, Academic Press, (1968). Google Scholar |
[30] |
V. A. Markel and J. C. Schotland, Inverse problem in optical diffusion tomography. II. role of boundary conditions,, Journal of the Optical Society of America. A, 19 (2002), 558.
doi: 10.1364/JOSAA.19.000558. |
[31] |
A. Neumaier, Solving ill-conditioned and singular linear systems: A tutorial on regularization,, SIAM Review, 40 (1998), 636.
doi: 10.1137/S0036144597321909. |
[32] |
Y. Notay, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123.
|
[33] |
L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259.
doi: 10.1016/0167-2789(92)90242-F. |
[34] |
W. Rudin, Functional Analysis,, McGraw-Hill, (1991).
doi: 10.1007/978-3-642-84455-3_77. |
[35] |
J. C. Schotland and V. A. Markel, Inverse scattering with diffusing waves,, Journal of the Optical Society of America. A, 18 (2001), 2767.
doi: 10.1364/JOSAA.18.002767. |
[36] |
M. Schweiger, S. R. Arridge, M. Hiraoka and D. Delpy, The finite element method for the propagation of light in scattering media: Boundary and source conditions,, Medical Physics, 22 (1995).
doi: 10.1118/1.597634. |
[37] |
G. Wang, Y. Li and M. Jiang, Uniqueness theorems in bioluminescence tomography,, Medical Physics, 31 (2004).
doi: 10.1118/1.1766420. |
[38] |
Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction,, SIAM Journal on Imaging Sciences, 1 (2008).
doi: 10.1137/080724265. |
[39] |
R. Weissleder, C. Tung, U. Mahmood and A. Bogdanov, In vivo imaging of tumors with protease-activated near-infrared fluorescent probes,, Nature Biotechnology, 17 (1999), 375. Google Scholar |
[40] |
C. Wu and X. Tai, Augmented lagrangian method, dual methods, and split bregman iteration for rof, vectorial tv, and high order models,, SIAM J. Imaging Sci., 3 (2010), 300.
doi: 10.1137/090767558. |
show all references
References:
[1] |
S. R. Arridge, Optical tomography in medical imaging,, Inverse Problems, 15 (1999), 0266.
doi: 10.1088/0266-5611/15/2/022. |
[2] |
S. R. Arridge and J. C. Schotland, Optical tomography: Forward and inverse problems,, Inverse Problems, 25 (2009), 0266.
doi: 10.1088/0266-5611/25/12/123010. |
[3] |
G. Bao, S. Hou and P. Li, Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm,, Journal of Computational Physics, 227 (2007), 755.
doi: 10.1016/j.jcp.2007.08.020. |
[4] |
A. Behrooz, H. Zhou, A. Eftekhar and A. Adibi, Total variation regularization for 3d reconstruction in fluorescence tomography: experimental phantom studies,, Applied Optics, 51 (2012), 8216.
doi: 10.1364/AO.51.008216. |
[5] |
C. Bremer, V. Ntziachristos and R. Weissleder, Optical-based molecular imaging: contrast agents and potential medical applications,, European Radiology, 13 (2003), 231. Google Scholar |
[6] |
S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods,, vol. 15, (2007). Google Scholar |
[7] |
J. Cai, E. Candès and Z. Shen, A singular value thresholding algorithm for matrix completion,, SIAM Journal on Optimization, 20 (2010), 1956.
doi: 10.1137/080738970. |
[8] |
Y. Censor and E. Tom, Convergence of string-averaging projection schemes for inconsistent convex feasibility problems,, Optimization Methods and Software, 18 (2003), 543.
doi: 10.1080/10556780310001610484. |
[9] |
Y. Chen, W. Hager, M. Yashtini, X. Ye and H. Zhang, Bregman operator splitting with variable stepsize for total variation image reconstruction,, Comput. Optim. Appl., 54 (2013).
doi: 10.1007/s10589-012-9519-2. |
[10] |
Y. Dai and R. Fletcher, Projected barzilai-borwein methods for large-scale box-constrained quadratic programming,, Numerische Mathematik, 100 (2005), 21.
doi: 10.1007/s00211-004-0569-y. |
[11] |
Y. Dai and L. Liao, R-linear convergence of the barzilai and borwein gradient method,, IMA Journal of Numerical Analysis, 22 (2002), 1.
doi: 10.1093/imanum/22.1.1. |
[12] |
J. Dutta, S. Ahn, A. Joshi and R. Leahy, Optimal illumination patterns for fluorescence tomography,, in Biomedical Imaging: From Nano to Macro, (2009), 1275.
doi: 10.1109/ISBI.2009.5193295. |
[13] |
J. Dutta, S. Ahn, C. Li, S. R. Cherry and R. M. Leahy, Joint l1 and total variation regularization for fluorescence molecular tomography,, Physics in Medicine and Biology, 57 (2012), 1459. Google Scholar |
[14] |
H. Egger, M. Freiberger and M. Schlottbom, On forward and inverse models in fluorescence diffuse optical tomography,, Inverse Problems and Imaging, 4 (2010), 411.
doi: 10.3934/ipi.2010.4.411. |
[15] |
H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, vol. 375, (1996).
doi: 10.1007/978-94-009-1740-8. |
[16] |
E. Esser, Applications of lagrangian-based alternating direction methods and connections to split bregman,, CAM Report, 9 (2009). Google Scholar |
[17] |
T. J. Farrell, M. S. Patterson and B. Wilson, A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,, Medical Physics, 19 (1992), 879.
doi: 10.1118/1.596777. |
[18] |
H. Gao and H. Zhao, Multilevel bioluminescence tomography based on radiative transfer equation part 1: l1 regularization,, Optics Express, 18 (2010), 1854.
doi: 10.1364/OE.18.001854. |
[19] |
H. Gao and H. Zhao, Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity,, Optics Express, 18 (2010), 2894.
doi: 10.1364/OE.18.002894. |
[20] |
T. Goldstein and S. Osher, The split bregman method for l1-regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323.
doi: 10.1137/080725891. |
[21] |
G. Golub and C. Van Loan, Matrix Computations,, vol. 3, (1996).
|
[22] |
P. Hansen and D. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problems,, SIAM Journal on Scientific Computing, 14 (1993), 1487.
doi: 10.1137/0914086. |
[23] |
R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams and B. J. Tromberg, Boundary conditions for the diffusion equation in radiative transfer,, Journal of the Optical Society of America. A, 11 (1994), 2727.
doi: 10.1364/JOSAA.11.002727. |
[24] |
D. Hawrysz and E. Sevick-Muraca, Developments toward diagnostic breast cancer imaging using near-infrared optical measurements and fluorescent contrast agents,, Neoplasia (New York, 2 (2000).
doi: 10.1038/sj.neo.7900118. |
[25] |
J. Hebden, S. Arridge and D. Delpy, Optical imaging in medicine: I. experimental techniques,, Physics in Medicine and Biology, 42 (1999).
doi: 10.1088/0031-9155/42/5/007. |
[26] |
G. Herman, A. Lent and S. Rowland, ART: Mathematics and applications: A report on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques,, Journal of theoretical biology, 42 (1973), 1.
doi: 10.1016/0022-5193(74)90135-0. |
[27] |
A. Ishimaru, Wave Propagation and Scattering in Random Media,, vol. 2, (1997).
|
[28] |
J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems,, vol. 160, (2004).
|
[29] |
O. A. Ladyzhenskaia and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations,, Academic Press, (1968). Google Scholar |
[30] |
V. A. Markel and J. C. Schotland, Inverse problem in optical diffusion tomography. II. role of boundary conditions,, Journal of the Optical Society of America. A, 19 (2002), 558.
doi: 10.1364/JOSAA.19.000558. |
[31] |
A. Neumaier, Solving ill-conditioned and singular linear systems: A tutorial on regularization,, SIAM Review, 40 (1998), 636.
doi: 10.1137/S0036144597321909. |
[32] |
Y. Notay, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123.
|
[33] |
L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259.
doi: 10.1016/0167-2789(92)90242-F. |
[34] |
W. Rudin, Functional Analysis,, McGraw-Hill, (1991).
doi: 10.1007/978-3-642-84455-3_77. |
[35] |
J. C. Schotland and V. A. Markel, Inverse scattering with diffusing waves,, Journal of the Optical Society of America. A, 18 (2001), 2767.
doi: 10.1364/JOSAA.18.002767. |
[36] |
M. Schweiger, S. R. Arridge, M. Hiraoka and D. Delpy, The finite element method for the propagation of light in scattering media: Boundary and source conditions,, Medical Physics, 22 (1995).
doi: 10.1118/1.597634. |
[37] |
G. Wang, Y. Li and M. Jiang, Uniqueness theorems in bioluminescence tomography,, Medical Physics, 31 (2004).
doi: 10.1118/1.1766420. |
[38] |
Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction,, SIAM Journal on Imaging Sciences, 1 (2008).
doi: 10.1137/080724265. |
[39] |
R. Weissleder, C. Tung, U. Mahmood and A. Bogdanov, In vivo imaging of tumors with protease-activated near-infrared fluorescent probes,, Nature Biotechnology, 17 (1999), 375. Google Scholar |
[40] |
C. Wu and X. Tai, Augmented lagrangian method, dual methods, and split bregman iteration for rof, vectorial tv, and high order models,, SIAM J. Imaging Sci., 3 (2010), 300.
doi: 10.1137/090767558. |
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