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A stable method solving the total variation dictionary model with $L^\infty$ constraints
1. | Institute of Microelectronics, Chinese Academy of Sciences, Beijing, China |
2. | MAP5, Université Paris Descartes, Paris, 75006, France |
3. | School of Computer and Information Technology, Beijing Jiaotong University, Beijing, China |
4. | Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China |
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.
doi: 10.1109/TIP.2010.2047910. |
[2] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problem,, Oxford, (2000).
|
[3] |
G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations,, Applied Mathematical Sciences, (2006).
|
[4] |
A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems,, SIAM J. Imag. Sci., 2 (2009), 183.
doi: 10.1137/080716542. |
[5] |
M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting,, in Proc. SIGGRAPH, (2000), 417.
doi: 10.1145/344779.344972. |
[6] |
S. Boyd and L. Vandenberghe, Convex Optimization,, Cambridge University Press, (2004).
doi: 10.1017/CBO9780511804441. |
[7] |
J. P. Boyle and R. L. Dykstra, A method for finding projections onto the intersection of convex sets in Hilbert spaces,, Lecture Notes in Statistics, 37 (1986), 28.
doi: 10.1007/978-1-4613-9940-7_3. |
[8] |
X. Bresson, A Short Note for Nonlocal TV Minimization,, Technical Report, (2009). Google Scholar |
[9] |
X. Bresson and T. F. Chan, Fast dual minimization of the vectorial total variation norm and applications to color image processing,, Inverse Problems and Imaging, 2 (2008), 455.
doi: 10.3934/ipi.2008.2.455. |
[10] |
A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms, with a new one,, Multiscale Model. Simul., 4 (2005), 490.
doi: 10.1137/040616024. |
[11] |
A. Bugeau, M. Bertalmio, V. Caselles and G. Sapiro, A Comprehensive Framework for Image Inpainting,, IEEE Trans. Image Process., 19 (2010), 2634.
doi: 10.1109/TIP.2010.2049240. |
[12] |
E. Candes and F. Guo, A new multiscale transform, minimum total variation synthesis: Application to edge-preserving image reconstruction,, Signal Processing, 82 (2002), 1519. Google Scholar |
[13] |
A. Chambolle, An algorithm for total variation minimization and applications,, J. Math. Imag. Vis., 20 (2004), 89.
doi: 10.1023/B:JMIV.0000011321.19549.88. |
[14] |
A. Chambolle, R. DeVore, N.-Y. Lee and B. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage,, IEEE Trans. Image Process., 7 (1998), 319.
doi: 10.1109/83.661182. |
[15] |
A. Chambolle and T. Pock, A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging,, J. Math. Imaging Vis., 40 (2011), 120.
doi: 10.1007/s10851-010-0251-1. |
[16] |
T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration,, SIAM J. Sci. Comp., 20 (1999), 1964.
doi: 10.1137/S1064827596299767. |
[17] |
T. F. Chan, A. M. Yip and F. E. Park, Simultaneous total variation image inpainting and blind deconvolution,, Int. J. of Imaging Systems and Technology, 15 (2005), 92.
doi: 10.1002/ima.20041. |
[18] |
T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods,, Society for Industrial and Applied Mathematics (SIAM), (2005).
doi: 10.1137/1.9780898717877. |
[19] |
C. Chaux, J. C. Pesquet and N. Pustelnik, Nested iterative algorithms for convex constrained image recovery problems,, SIAM J. Imag. Sci., 2 (2009), 730.
doi: 10.1137/080727749. |
[20] |
P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward backward splitting,, Multiscale Model. Simul., 4 (2005), 1168.
doi: 10.1137/050626090. |
[21] |
I. Daubechies, Ten Lectures on Wavelets,, SIAM Publ., (1992).
doi: 10.1137/1.9781611970104. |
[22] |
I. Daubechies, M. Defriese and C. D. Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,, Commun.Pure Appl. Math., 57 (2004), 1413.
doi: 10.1002/cpa.20042. |
[23] |
D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage,, Biometrika, 81 (1994), 425.
doi: 10.1093/biomet/81.3.425. |
[24] |
D. L. Donoho and I. M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage,, J. Amer. Statist. Assoc., 90 (1995), 1200.
doi: 10.1080/01621459.1995.10476626. |
[25] |
I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Studies Math. Appl., (1976).
|
[26] |
D. Gabay, Applications of the method of multipliers to variational inequalities,, in Augmented Lagrangian Methods: Applications to the Solution of Boundary-Valued Problems, (1983), 299. Google Scholar |
[27] |
G. Gilboa and S. Osher, Nonlocal operators with applications to image processing,, Multiscale Model. Simul., 7 (2008), 1005.
doi: 10.1137/070698592. |
[28] |
T. Goldstein and S. Osher, The split Bregman method for l1 regularized problems,, SIAM J. Imag. Sci., 2 (2009), 323.
doi: 10.1137/080725891. |
[29] |
S. Lintner and F. Malgouyres, Solving a variational image restoration model which involves contraints,, Inverse. Probl., 20 (2004), 815.
doi: 10.1088/0266-5611/20/3/010. |
[30] |
J. Liu, X-C. Tai, H. Huang and Z. Huan, A weighted dictionary learning model for denoising images corrupted by mixed noise,, IEEE Trans. on Image Process, 22 (2013), 1108.
doi: 10.1109/TIP.2012.2227766. |
[31] |
F. Malgouyres, A framework for image deblurring using wavelet packet bases,, Appl. and Comp. Harmonic Analysis, 12 (2002), 309.
doi: 10.1006/acha.2002.0379. |
[32] |
F. Malgouyres, Mathematical analysis of a model which combines total variation and wavelets for image restoration,, Journal of Information Processes, 2 (2002), 1. Google Scholar |
[33] |
F. Malgouyres, Minimizing the total variation under a general convex constraint for image restoration,, IEEE Trans. on Image Process, 11 (2002), 1450.
doi: 10.1109/TIP.2002.806241. |
[34] |
S. Masnou and J.-M. Morel, Level lines based disocclusion,, Int. Conf. on Image Processing, 3 (1998), 259.
doi: 10.1109/ICIP.1998.999016. |
[35] |
C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denoising,, Inverse Probl., 27 (2011), 45009.
doi: 10.1088/0266-5611/27/4/045009. |
[36] |
J.-J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien,, C.R. Acad. Sci. Paris Ser. A Math, 255 (1962), 2897.
|
[37] |
J.-J. Moreau, Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273.
|
[38] |
Y. Nesterov, A method of solving a convex programming problem with convergence rate $O(1/k^2)$, (Russian),, Dokl. Akad. Nauk SSSR, 269 (1983), 543.
|
[39] |
M. Ng, W. Fan and X. Yuan, Inexact alternating direction methods for image recovery,, SIAM J. Sci. Comp., 33 (2011), 1643.
doi: 10.1137/100807697. |
[40] |
G. Peyre, S. Bougleux and L. Cohen, Non-local regularization of inverse problems,, Inverse Problems and Imaging, 5 (2011), 511.
doi: 10.3934/ipi.2011.5.511. |
[41] |
L. Rudin S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259. Google Scholar |
[42] |
S. Setzer, Operator splittings, Bregman methods and frame shrinkage in image processing,, Int. J. Comput. Vis., 92 (2011), 265.
doi: 10.1007/s11263-010-0357-3. |
[43] |
G. Steidl, J.Weickert, T. Brox, P. Mrázek and M. Welk, On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and sides,, SIAM J. Numer. Anal., 42 (2004), 686.
doi: 10.1137/S0036142903422429. |
[44] |
X.-C. Tai and C. Wu, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model,, SSVM 2009, 42 (2009), 502. Google Scholar |
[45] |
A. Tikhonov and V. Arsenin, Solution of Ill-Posed Problems,, Winston and Sons, (1977).
|
[46] |
C. Wu, J. Zhang and X.-C. Tai, Augmented Lagrangian method for total variation restoration with non-quadratic fidelity,, Inverse Problems and Imaging, 5 (2011), 237.
doi: 10.3934/ipi.2011.5.237. |
[47] |
C. Zalinescu, Convex Analysis in General Vector Spaces,, Singapore: World Scientific, (2002).
doi: 10.1142/9789812777096. |
[48] |
T. Zeng, Incorporating known features into a total variation dictionary model for source separation,, Int. Conf. on Image Processing, (2008), 577.
doi: 10.1109/ICIP.2008.4711820. |
[49] |
T. Zeng and F. Malgouyres, Using Gabor dictionaries in a TV-$L^\infty$ model for denoising,, Int. Conf. on Acoust. Speech and Signal Proc., (2006), 865. Google Scholar |
[50] |
T. Zeng and M. K. Ng, On the total variation dictionary model,, IEEE trans. on Image Process., 19 (2010), 821.
doi: 10.1109/TIP.2009.2034701. |
[51] |
X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized Nonlocal Regularization for Deconvolution and Sparse Reconstruction,, SIAM J. Imag. Sci., 3 (2010), 253.
doi: 10.1137/090746379. |
[52] |
M. Zhu and T. Chan, An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration,, UCLA CAM Report 08-34 (2008)., (2008), 08. Google Scholar |
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.
doi: 10.1109/TIP.2010.2047910. |
[2] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problem,, Oxford, (2000).
|
[3] |
G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations,, Applied Mathematical Sciences, (2006).
|
[4] |
A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems,, SIAM J. Imag. Sci., 2 (2009), 183.
doi: 10.1137/080716542. |
[5] |
M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting,, in Proc. SIGGRAPH, (2000), 417.
doi: 10.1145/344779.344972. |
[6] |
S. Boyd and L. Vandenberghe, Convex Optimization,, Cambridge University Press, (2004).
doi: 10.1017/CBO9780511804441. |
[7] |
J. P. Boyle and R. L. Dykstra, A method for finding projections onto the intersection of convex sets in Hilbert spaces,, Lecture Notes in Statistics, 37 (1986), 28.
doi: 10.1007/978-1-4613-9940-7_3. |
[8] |
X. Bresson, A Short Note for Nonlocal TV Minimization,, Technical Report, (2009). Google Scholar |
[9] |
X. Bresson and T. F. Chan, Fast dual minimization of the vectorial total variation norm and applications to color image processing,, Inverse Problems and Imaging, 2 (2008), 455.
doi: 10.3934/ipi.2008.2.455. |
[10] |
A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms, with a new one,, Multiscale Model. Simul., 4 (2005), 490.
doi: 10.1137/040616024. |
[11] |
A. Bugeau, M. Bertalmio, V. Caselles and G. Sapiro, A Comprehensive Framework for Image Inpainting,, IEEE Trans. Image Process., 19 (2010), 2634.
doi: 10.1109/TIP.2010.2049240. |
[12] |
E. Candes and F. Guo, A new multiscale transform, minimum total variation synthesis: Application to edge-preserving image reconstruction,, Signal Processing, 82 (2002), 1519. Google Scholar |
[13] |
A. Chambolle, An algorithm for total variation minimization and applications,, J. Math. Imag. Vis., 20 (2004), 89.
doi: 10.1023/B:JMIV.0000011321.19549.88. |
[14] |
A. Chambolle, R. DeVore, N.-Y. Lee and B. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage,, IEEE Trans. Image Process., 7 (1998), 319.
doi: 10.1109/83.661182. |
[15] |
A. Chambolle and T. Pock, A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging,, J. Math. Imaging Vis., 40 (2011), 120.
doi: 10.1007/s10851-010-0251-1. |
[16] |
T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration,, SIAM J. Sci. Comp., 20 (1999), 1964.
doi: 10.1137/S1064827596299767. |
[17] |
T. F. Chan, A. M. Yip and F. E. Park, Simultaneous total variation image inpainting and blind deconvolution,, Int. J. of Imaging Systems and Technology, 15 (2005), 92.
doi: 10.1002/ima.20041. |
[18] |
T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods,, Society for Industrial and Applied Mathematics (SIAM), (2005).
doi: 10.1137/1.9780898717877. |
[19] |
C. Chaux, J. C. Pesquet and N. Pustelnik, Nested iterative algorithms for convex constrained image recovery problems,, SIAM J. Imag. Sci., 2 (2009), 730.
doi: 10.1137/080727749. |
[20] |
P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward backward splitting,, Multiscale Model. Simul., 4 (2005), 1168.
doi: 10.1137/050626090. |
[21] |
I. Daubechies, Ten Lectures on Wavelets,, SIAM Publ., (1992).
doi: 10.1137/1.9781611970104. |
[22] |
I. Daubechies, M. Defriese and C. D. Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,, Commun.Pure Appl. Math., 57 (2004), 1413.
doi: 10.1002/cpa.20042. |
[23] |
D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage,, Biometrika, 81 (1994), 425.
doi: 10.1093/biomet/81.3.425. |
[24] |
D. L. Donoho and I. M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage,, J. Amer. Statist. Assoc., 90 (1995), 1200.
doi: 10.1080/01621459.1995.10476626. |
[25] |
I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Studies Math. Appl., (1976).
|
[26] |
D. Gabay, Applications of the method of multipliers to variational inequalities,, in Augmented Lagrangian Methods: Applications to the Solution of Boundary-Valued Problems, (1983), 299. Google Scholar |
[27] |
G. Gilboa and S. Osher, Nonlocal operators with applications to image processing,, Multiscale Model. Simul., 7 (2008), 1005.
doi: 10.1137/070698592. |
[28] |
T. Goldstein and S. Osher, The split Bregman method for l1 regularized problems,, SIAM J. Imag. Sci., 2 (2009), 323.
doi: 10.1137/080725891. |
[29] |
S. Lintner and F. Malgouyres, Solving a variational image restoration model which involves contraints,, Inverse. Probl., 20 (2004), 815.
doi: 10.1088/0266-5611/20/3/010. |
[30] |
J. Liu, X-C. Tai, H. Huang and Z. Huan, A weighted dictionary learning model for denoising images corrupted by mixed noise,, IEEE Trans. on Image Process, 22 (2013), 1108.
doi: 10.1109/TIP.2012.2227766. |
[31] |
F. Malgouyres, A framework for image deblurring using wavelet packet bases,, Appl. and Comp. Harmonic Analysis, 12 (2002), 309.
doi: 10.1006/acha.2002.0379. |
[32] |
F. Malgouyres, Mathematical analysis of a model which combines total variation and wavelets for image restoration,, Journal of Information Processes, 2 (2002), 1. Google Scholar |
[33] |
F. Malgouyres, Minimizing the total variation under a general convex constraint for image restoration,, IEEE Trans. on Image Process, 11 (2002), 1450.
doi: 10.1109/TIP.2002.806241. |
[34] |
S. Masnou and J.-M. Morel, Level lines based disocclusion,, Int. Conf. on Image Processing, 3 (1998), 259.
doi: 10.1109/ICIP.1998.999016. |
[35] |
C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denoising,, Inverse Probl., 27 (2011), 45009.
doi: 10.1088/0266-5611/27/4/045009. |
[36] |
J.-J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien,, C.R. Acad. Sci. Paris Ser. A Math, 255 (1962), 2897.
|
[37] |
J.-J. Moreau, Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273.
|
[38] |
Y. Nesterov, A method of solving a convex programming problem with convergence rate $O(1/k^2)$, (Russian),, Dokl. Akad. Nauk SSSR, 269 (1983), 543.
|
[39] |
M. Ng, W. Fan and X. Yuan, Inexact alternating direction methods for image recovery,, SIAM J. Sci. Comp., 33 (2011), 1643.
doi: 10.1137/100807697. |
[40] |
G. Peyre, S. Bougleux and L. Cohen, Non-local regularization of inverse problems,, Inverse Problems and Imaging, 5 (2011), 511.
doi: 10.3934/ipi.2011.5.511. |
[41] |
L. Rudin S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259. Google Scholar |
[42] |
S. Setzer, Operator splittings, Bregman methods and frame shrinkage in image processing,, Int. J. Comput. Vis., 92 (2011), 265.
doi: 10.1007/s11263-010-0357-3. |
[43] |
G. Steidl, J.Weickert, T. Brox, P. Mrázek and M. Welk, On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and sides,, SIAM J. Numer. Anal., 42 (2004), 686.
doi: 10.1137/S0036142903422429. |
[44] |
X.-C. Tai and C. Wu, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model,, SSVM 2009, 42 (2009), 502. Google Scholar |
[45] |
A. Tikhonov and V. Arsenin, Solution of Ill-Posed Problems,, Winston and Sons, (1977).
|
[46] |
C. Wu, J. Zhang and X.-C. Tai, Augmented Lagrangian method for total variation restoration with non-quadratic fidelity,, Inverse Problems and Imaging, 5 (2011), 237.
doi: 10.3934/ipi.2011.5.237. |
[47] |
C. Zalinescu, Convex Analysis in General Vector Spaces,, Singapore: World Scientific, (2002).
doi: 10.1142/9789812777096. |
[48] |
T. Zeng, Incorporating known features into a total variation dictionary model for source separation,, Int. Conf. on Image Processing, (2008), 577.
doi: 10.1109/ICIP.2008.4711820. |
[49] |
T. Zeng and F. Malgouyres, Using Gabor dictionaries in a TV-$L^\infty$ model for denoising,, Int. Conf. on Acoust. Speech and Signal Proc., (2006), 865. Google Scholar |
[50] |
T. Zeng and M. K. Ng, On the total variation dictionary model,, IEEE trans. on Image Process., 19 (2010), 821.
doi: 10.1109/TIP.2009.2034701. |
[51] |
X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized Nonlocal Regularization for Deconvolution and Sparse Reconstruction,, SIAM J. Imag. Sci., 3 (2010), 253.
doi: 10.1137/090746379. |
[52] |
M. Zhu and T. Chan, An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration,, UCLA CAM Report 08-34 (2008)., (2008), 08. Google Scholar |
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