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Bilevel optimization for calibrating point spread functions in blind deconvolution
Iterative choice of the optimal regularization parameter in TV image restoration
1. | CREATIS, CNRS UMR 5220; INSERM U1044; INSA de Lyon; Université de Lyon 1, Université de Lyon, 69621, Villeurbanne Cedex, France, France, France |
References:
[1] |
M. Afonso, J. Bioucas-Dias and M. Figueiredo, Fast image recovery using variable splitting and constrained optimization, IEEE Trans. Image Process., 19 (2010), 2345-2356.
doi: 10.1109/TIP.2010.2047910. |
[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 (2011), 681-695.
doi: 10.1109/TIP.2010.2076294. |
[3] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press Oxford, 2000. |
[4] |
J. F. Aujol and G. Gilboa, Constrained and SNR-based solutions for TV-Hibert space image denoising, J. Math. Imaging Vis., 26 (2006), 217-237.
doi: 10.1007/s10851-006-7801-6. |
[5] |
S. D. Babacan, R. Molina and A. K. Katsaggelos, Parameter estimation in TV image restoration using variational distribution approximation, IEEE Trans. Image Process., 17 (2008), 326-339.
doi: 10.1109/TIP.2007.916051. |
[6] |
S. D. Babacan, R. Molina and A. K. Katsaggelos, Total Variation Super Resolution Using A Variational Approach, Proc. Int.Conf. Image Process., (2008), 641-644. |
[7] |
S. Becker, J. Bobin and E. Candes, NESTA: A fast and accurate first-order method for sparse recovery, SIAM Journal on Imaging Sciences, 4 (2011), 1-39.
doi: 10.1137/090756855. |
[8] |
T. Blu and F. Luisier, The SURE-LET approach to image denoising, IEEE Trans. Image Process., 16 (2007), 2778-2786.
doi: 10.1109/TIP.2007.906002. |
[9] |
L. M. Bregman, The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math+, 7 (1967), 200-217. |
[10] |
M. Burger and S. Osher, Convergence rates of convex variational regularization, Inverse Probl., 20 (2004), 1411-1421.
doi: 10.1088/0266-5611/20/5/005. |
[11] |
C. A. Deledalle, S. Vaiter, J. Fadili and G. Peyre, Stein Unbiased Gradient estimator for the risk (SUGAR) for multiple parameter selection, SIAM Journal on Imaging Sciences, 7 (2014), 2448-2487.
doi: 10.1137/140968045. |
[12] |
Y. C. Eldar, Generalized SURE for exponential families: Application to regularization, IEEE Trans. Signal Process., 57 (2009), 471-481.
doi: 10.1109/TSP.2008.2008212. |
[13] |
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.
doi: 10.1007/978-94-009-1740-8. |
[14] |
E. Esser, Applications of Lagrangian-based alternating direction methods and connections to split Bregman, CAM report, 2009. |
[15] |
J. M. Fadili, G. Peyre, S. Vaiter, C. Deledalle and J. Salmon, Stable recovery with analysis decomposable priors,, preprint, ().
|
[16] |
K. Frick, D. A. Lorenz and E. Resmerita, Morozov's principle for the augmented Lagrangian method applied to linear inverse problems, Multiscale Model Simul., 9 (2011), 1528-1548.
doi: 10.1137/100812835. |
[17] |
G. H. Golub, M. T. Heath and C. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.
doi: 10.2307/1268518. |
[18] |
M. Grasmair, Linear convergence rates for Tikhonov regularization with positively homogeneous functionals, Inverse Problems, 27 (2011), 075014, 16pp.
doi: 10.1088/0266-5611/27/7/075014. |
[19] |
P. C. Hansen and D. P. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problem, SIAM Journal on Scientific Computing, 14 (1993), 1487-1503.
doi: 10.1137/0914086. |
[20] |
M. F. Hutchinson, A stochastic estimator of the trace of the influence matrix for laplacian smooting splines, Commun. Stat. Simulat., 19 (1990), 433-450.
doi: 10.1080/03610919008812864. |
[21] |
K. C. Kiwiel, Proximal minimization methods with generalized Bregman functions, SIAM J. Control Optim., 35 (1997), 1142-1168.
doi: 10.1137/S0363012995281742. |
[22] |
K. Kunisch and J. Zou, Iterative choices of regularization parameters in linear inverse problems, Inverse Probl., 14 (1998), 1247-1264.
doi: 10.1088/0266-5611/14/5/010. |
[23] |
K. Kunisch, On a class of damped Morozov principles, Computing, 50 (1993), 185-198.
doi: 10.1007/BF02243810. |
[24] |
H. Liao, F. Li and M. K. Ng, Selection of regularization parameter in total variation restoration, JOSA A, 26 (2009), 2311-2320.
doi: 10.1364/JOSAA.26.002311. |
[25] |
Y.Lin, B.Wohlberg and H.Guo, UPRE method for total variation parameter selection, Signal Processing, 90 (2010), 2546-2551. |
[26] |
V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5280-1. |
[27] |
M. K. Ng, P. Weiss and X. Yuan, Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods, SIAM Journal on Scientific Computing, 32 (2010), 2710-2736.
doi: 10.1137/090774823. |
[28] |
S. Ramani, T. Blu and M. Unser, Monte-Carlo SURE: A black-box optimization of regularization parameters for general denoising algorithms, IEEE Trans. Image Process., 17 (2008), 1540-1554.
doi: 10.1109/TIP.2008.2001404. |
[29] |
S. Ramani, Z. Liu, J. Rosen, J. Nielsen and J. A. Fessler, Regularization parameter selection for nonlinear iterative image restoration and MRI reconstruction using GCV and SURE-based methods, IEEE Transactions on Image Processing, 21 (2012), 3659-3672.
doi: 10.1109/TIP.2012.2195015. |
[30] |
Z. Ren, C. He and Q. Zhang, Fractional order total variation regularization for image super-resolution, Signal Process., 93 (2013), 2408-2421. |
[31] |
E. Resmerita, Regularization of ill-posed problems in Banach spaces: Convergence rates, Inverse Probl., 21 (2005), 1303-1314.
doi: 10.1088/0266-5611/21/4/007. |
[32] |
L. I. Rudin , S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268. |
[33] |
M. Salome, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Barucheland and P. Spanne, A synchrotron radiation microtomography system for the analysis of trabecular bone samples, Medical Physics, 26 (1999), 2194-2204. |
[34] |
O. Scherzer, M. Grassmair, H. Grossauer, M. Haltmaier and F. Lenzen, Variational Methods in Imaging, Springer Verlag, New York, 2008. |
[35] |
A. N. Tikhonov and V. Y. Arsenin, Solutions to ill-posed problems, Winston-Wiley, New York, 1977. |
[36] |
E. Van den Berg E and M. P. Friedlander, Probing the Pareto frontier for basis pursuit solutions, SIAM Journal on Scientific Computing, 31 (2008), 890-912.
doi: 10.1137/080714488. |
[37] |
S. Vaiter, A. Deledalle, G. Peyre, C. Dossal and J. Fadili, Local behavior of sparse analysis regularization: Applications to risk estimation, Applied and Computational Harmonic Analysis, 35 (2013), 433-451.
doi: 10.1016/j.acha.2012.11.006. |
[38] |
C. R. Vogel, Computational Methods for Inverse Problems, SIAM, Philadelphia, PA, 2002.
doi: 10.1137/1.9780898717570. |
[39] |
C. Vonesch, S. Ramani and M. Unser, Risk estimation for non-linear image deconvolution with a wavelet-domain sparsity constraint, Inverse Problems, 27 (2011), 075014. |
[40] |
C. Vonesch, S. Ramani and M. Unser, Recursive risk estimation for non-linear image deconvolution with a wavelet-domain sparsity constraint, ICIP 2008, 15th IEEE International Conference on Image Processing, (2008), 665-668. |
[41] |
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), 248-272.
doi: 10.1137/080724265. |
[42] |
Y. W. Wen and R. H. Chan, Parameter selection for total-variation-based image restoration using discrepancy principle, IEEE Trans. Image Process., 21 (2012), 1770-1781.
doi: 10.1109/TIP.2011.2181401. |
[43] |
J. Yang, W. Yin, Y. Zhang and Y. Wang, A fast algorithm for edge-preserving variational multichannel image restoration, SIAM Journal on Imaging Sciences, 2 (2009), 569-592.
doi: 10.1137/080730421. |
show all references
References:
[1] |
M. Afonso, J. Bioucas-Dias and M. Figueiredo, Fast image recovery using variable splitting and constrained optimization, IEEE Trans. Image Process., 19 (2010), 2345-2356.
doi: 10.1109/TIP.2010.2047910. |
[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 (2011), 681-695.
doi: 10.1109/TIP.2010.2076294. |
[3] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press Oxford, 2000. |
[4] |
J. F. Aujol and G. Gilboa, Constrained and SNR-based solutions for TV-Hibert space image denoising, J. Math. Imaging Vis., 26 (2006), 217-237.
doi: 10.1007/s10851-006-7801-6. |
[5] |
S. D. Babacan, R. Molina and A. K. Katsaggelos, Parameter estimation in TV image restoration using variational distribution approximation, IEEE Trans. Image Process., 17 (2008), 326-339.
doi: 10.1109/TIP.2007.916051. |
[6] |
S. D. Babacan, R. Molina and A. K. Katsaggelos, Total Variation Super Resolution Using A Variational Approach, Proc. Int.Conf. Image Process., (2008), 641-644. |
[7] |
S. Becker, J. Bobin and E. Candes, NESTA: A fast and accurate first-order method for sparse recovery, SIAM Journal on Imaging Sciences, 4 (2011), 1-39.
doi: 10.1137/090756855. |
[8] |
T. Blu and F. Luisier, The SURE-LET approach to image denoising, IEEE Trans. Image Process., 16 (2007), 2778-2786.
doi: 10.1109/TIP.2007.906002. |
[9] |
L. M. Bregman, The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math+, 7 (1967), 200-217. |
[10] |
M. Burger and S. Osher, Convergence rates of convex variational regularization, Inverse Probl., 20 (2004), 1411-1421.
doi: 10.1088/0266-5611/20/5/005. |
[11] |
C. A. Deledalle, S. Vaiter, J. Fadili and G. Peyre, Stein Unbiased Gradient estimator for the risk (SUGAR) for multiple parameter selection, SIAM Journal on Imaging Sciences, 7 (2014), 2448-2487.
doi: 10.1137/140968045. |
[12] |
Y. C. Eldar, Generalized SURE for exponential families: Application to regularization, IEEE Trans. Signal Process., 57 (2009), 471-481.
doi: 10.1109/TSP.2008.2008212. |
[13] |
H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.
doi: 10.1007/978-94-009-1740-8. |
[14] |
E. Esser, Applications of Lagrangian-based alternating direction methods and connections to split Bregman, CAM report, 2009. |
[15] |
J. M. Fadili, G. Peyre, S. Vaiter, C. Deledalle and J. Salmon, Stable recovery with analysis decomposable priors,, preprint, ().
|
[16] |
K. Frick, D. A. Lorenz and E. Resmerita, Morozov's principle for the augmented Lagrangian method applied to linear inverse problems, Multiscale Model Simul., 9 (2011), 1528-1548.
doi: 10.1137/100812835. |
[17] |
G. H. Golub, M. T. Heath and C. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.
doi: 10.2307/1268518. |
[18] |
M. Grasmair, Linear convergence rates for Tikhonov regularization with positively homogeneous functionals, Inverse Problems, 27 (2011), 075014, 16pp.
doi: 10.1088/0266-5611/27/7/075014. |
[19] |
P. C. Hansen and D. P. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problem, SIAM Journal on Scientific Computing, 14 (1993), 1487-1503.
doi: 10.1137/0914086. |
[20] |
M. F. Hutchinson, A stochastic estimator of the trace of the influence matrix for laplacian smooting splines, Commun. Stat. Simulat., 19 (1990), 433-450.
doi: 10.1080/03610919008812864. |
[21] |
K. C. Kiwiel, Proximal minimization methods with generalized Bregman functions, SIAM J. Control Optim., 35 (1997), 1142-1168.
doi: 10.1137/S0363012995281742. |
[22] |
K. Kunisch and J. Zou, Iterative choices of regularization parameters in linear inverse problems, Inverse Probl., 14 (1998), 1247-1264.
doi: 10.1088/0266-5611/14/5/010. |
[23] |
K. Kunisch, On a class of damped Morozov principles, Computing, 50 (1993), 185-198.
doi: 10.1007/BF02243810. |
[24] |
H. Liao, F. Li and M. K. Ng, Selection of regularization parameter in total variation restoration, JOSA A, 26 (2009), 2311-2320.
doi: 10.1364/JOSAA.26.002311. |
[25] |
Y.Lin, B.Wohlberg and H.Guo, UPRE method for total variation parameter selection, Signal Processing, 90 (2010), 2546-2551. |
[26] |
V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5280-1. |
[27] |
M. K. Ng, P. Weiss and X. Yuan, Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods, SIAM Journal on Scientific Computing, 32 (2010), 2710-2736.
doi: 10.1137/090774823. |
[28] |
S. Ramani, T. Blu and M. Unser, Monte-Carlo SURE: A black-box optimization of regularization parameters for general denoising algorithms, IEEE Trans. Image Process., 17 (2008), 1540-1554.
doi: 10.1109/TIP.2008.2001404. |
[29] |
S. Ramani, Z. Liu, J. Rosen, J. Nielsen and J. A. Fessler, Regularization parameter selection for nonlinear iterative image restoration and MRI reconstruction using GCV and SURE-based methods, IEEE Transactions on Image Processing, 21 (2012), 3659-3672.
doi: 10.1109/TIP.2012.2195015. |
[30] |
Z. Ren, C. He and Q. Zhang, Fractional order total variation regularization for image super-resolution, Signal Process., 93 (2013), 2408-2421. |
[31] |
E. Resmerita, Regularization of ill-posed problems in Banach spaces: Convergence rates, Inverse Probl., 21 (2005), 1303-1314.
doi: 10.1088/0266-5611/21/4/007. |
[32] |
L. I. Rudin , S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268. |
[33] |
M. Salome, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Barucheland and P. Spanne, A synchrotron radiation microtomography system for the analysis of trabecular bone samples, Medical Physics, 26 (1999), 2194-2204. |
[34] |
O. Scherzer, M. Grassmair, H. Grossauer, M. Haltmaier and F. Lenzen, Variational Methods in Imaging, Springer Verlag, New York, 2008. |
[35] |
A. N. Tikhonov and V. Y. Arsenin, Solutions to ill-posed problems, Winston-Wiley, New York, 1977. |
[36] |
E. Van den Berg E and M. P. Friedlander, Probing the Pareto frontier for basis pursuit solutions, SIAM Journal on Scientific Computing, 31 (2008), 890-912.
doi: 10.1137/080714488. |
[37] |
S. Vaiter, A. Deledalle, G. Peyre, C. Dossal and J. Fadili, Local behavior of sparse analysis regularization: Applications to risk estimation, Applied and Computational Harmonic Analysis, 35 (2013), 433-451.
doi: 10.1016/j.acha.2012.11.006. |
[38] |
C. R. Vogel, Computational Methods for Inverse Problems, SIAM, Philadelphia, PA, 2002.
doi: 10.1137/1.9780898717570. |
[39] |
C. Vonesch, S. Ramani and M. Unser, Risk estimation for non-linear image deconvolution with a wavelet-domain sparsity constraint, Inverse Problems, 27 (2011), 075014. |
[40] |
C. Vonesch, S. Ramani and M. Unser, Recursive risk estimation for non-linear image deconvolution with a wavelet-domain sparsity constraint, ICIP 2008, 15th IEEE International Conference on Image Processing, (2008), 665-668. |
[41] |
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), 248-272.
doi: 10.1137/080724265. |
[42] |
Y. W. Wen and R. H. Chan, Parameter selection for total-variation-based image restoration using discrepancy principle, IEEE Trans. Image Process., 21 (2012), 1770-1781.
doi: 10.1109/TIP.2011.2181401. |
[43] |
J. Yang, W. Yin, Y. Zhang and Y. Wang, A fast algorithm for edge-preserving variational multichannel image restoration, SIAM Journal on Imaging Sciences, 2 (2009), 569-592.
doi: 10.1137/080730421. |
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