# American Institute of Mathematical Sciences

May  2014, 8(2): 507-535. doi: 10.3934/ipi.2014.8.507

## 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

Received  November 2011 Revised  May 2013 Published  May 2014

Image restoration plays an important role in image processing, and numerous approaches have been proposed to tackle this problem. This paper presents a modified model for image restoration, that is based on a combination of Total Variation and Dictionary approaches. Since the well-known TV regularization is non-differentiable, the proposed method utilizes its dual formulation instead of its approximation in order to exactly preserve its properties. The data-fidelity term combines the one commonly used in image restoration and a wavelet thresholding based term. Then, the resulting optimization problem is solved via a first-order primal-dual algorithm. Numerical experiments demonstrate the good performance of the proposed model. In a last variant, we replace the classical TV by the nonlocal TV regularization, which results in a much higher quality of restoration.
Citation: Liyan Ma, Lionel Moisan, Jian Yu, Tieyong Zeng. A stable method solving the total variation dictionary model with $L^\infty$ constraints. Inverse Problems & Imaging, 2014, 8 (2) : 507-535. doi: 10.3934/ipi.2014.8.507
##### References:
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Image Process., 19 (2010), 2634. doi: 10.1109/TIP.2010.2049240. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] I. 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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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [37] J.-J. Moreau, Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [47] C. Zalinescu, Convex Analysis in General Vector Spaces,, Singapore: World Scientific, (2002). doi: 10.1142/9789812777096. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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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##### 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. Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problem,, Oxford, (2000). Google Scholar [3] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations,, Applied Mathematical Sciences, (2006). Google Scholar [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. Google Scholar [5] M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting,, in Proc. SIGGRAPH, (2000), 417. doi: 10.1145/344779.344972. Google Scholar [6] S. Boyd and L. Vandenberghe, Convex Optimization,, Cambridge University Press, (2004). doi: 10.1017/CBO9780511804441. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] I. Daubechies, Ten Lectures on Wavelets,, SIAM Publ., (1992). doi: 10.1137/1.9781611970104. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [25] I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Studies Math. Appl., (1976). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [37] J.-J. Moreau, Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [47] C. Zalinescu, Convex Analysis in General Vector Spaces,, Singapore: World Scientific, (2002). doi: 10.1142/9789812777096. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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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