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February  2014, 8(1): 53-77. doi: 10.3934/ipi.2014.8.53

## The Moreau envelope approach for the L1/TV image denoising model

 1 Department of Mathematics, Syracuse University, Syracuse, NY 13244, United States, United States 2 Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150 3 School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

Received  October 2012 Revised  August 2013 Published  March 2014

This paper presents the Moreau envelope viewpoint for the L1/TV image denoising model. The main algorithmic difficulty for the numerical treatment of the L1/TV model lies in the non-differentiability of both the fidelity and regularization terms of the model. To overcome this difficulty, we propose five modified L1/TV models by replacing one or two non-differentiable functions in the L1/TV model with their corresponding Moreau envelopes. We prove that several existing approaches for the L1/TV model essentially solve some of the modified models, but not the original L1/TV model. Algorithms for the L1/TV model and its five variants are proposed under a unified framework based on fixed-point equations (via the proximity operator) which characterize the solutions of the models. Depending upon whether we smooth the regularization term or not, two different types of proximity algorithms are presented. The convergence rates of both types of the algorithms are improved significantly by exploring either the strategy of the Gauss-Seidel iteration, or the FISTA, or both. We compare the performance of various modified L1/TV models for the problem of impulse noise removal, and make recommendations based on our numerical experiments for using these models in applications.
Citation: Feishe Chen, Lixin Shen, Yuesheng Xu, Xueying Zeng. The Moreau envelope approach for the L1/TV image denoising model. Inverse Problems & Imaging, 2014, 8 (1) : 53-77. doi: 10.3934/ipi.2014.8.53
##### References:
 [1] J. E. Aujol, G. Gilboa, T. Chan and S. Osher, Structure-texture image decomposition-modeling, algorithms, and parameter selection,, International Journal of Computer Vision, 67 (2006), 111.  doi: 10.1007/s11263-006-4331-z.  Google Scholar [2] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Space,, Springer Press, (2011).  doi: 10.1007/978-1-4419-9467-7.  Google Scholar [3] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems,, SIAM Journal on Imaging Science, 2 (2009), 183.  doi: 10.1137/080716542.  Google Scholar [4] P. Bloomfield and W. Steiger, Least Absolute Deviations: Theory, Applications, and Algorithms,, Birkhäuser Boston, (1983).   Google Scholar [5] A. Bovik, Handbook of Image and Video Processing,, Academic Press, (2000).   Google Scholar [6] A. Chambolle, An algorithm for total variation minimization and applications,, Special issue on mathematics and image analysis, 20 (2004), 89.  doi: 10.1023/B:JMIV.0000011321.19549.88.  Google Scholar [7] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging,, Journal of Mathematical Imaging and Vision, 40 (2011), 120.  doi: 10.1007/s10851-010-0251-1.  Google Scholar [8] T. Chan and S. Esedoglu, Aspects of total variation regularized l1 function approximation,, SIAM Journal on Applied Mathematics, 65 (2005), 1817.  doi: 10.1137/040604297.  Google Scholar [9] R. Chartrand and V. Staneva, Total variation regularization of images corrupted by non-gaussian noise using a quasi-newton method,, IET Image Processing, 2 (2008), 295.  doi: 10.1049/iet-ipr:20080017.  Google Scholar [10] T. Chen, W. Yin, X. Zhou, D. Comaniciu and T. Huang, Total variation models for variable lighting face recognition,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 28 (2006), 1519.  doi: 10.1109/TPAMI.2006.195.  Google Scholar [11] C. Clason, B. Jin and K. Kunisch, A duality-based splitting method for l1-tv image restoration with automatic regularization parameter choice,, SIAM Journal on Scientific Computing, 32 (2010), 1484.  doi: 10.1137/090768217.  Google Scholar [12] P. Combettes and V. Wajs, Signal recovery by proximal forward-backward splitting,, Multiscale Model and Simulation, 4 (2005), 1168.  doi: 10.1137/050626090.  Google Scholar [13] Y. Dong, M. Hintermuller and M. Neri, An Efficient Primal-Dual Method for $l^1$-TV image Restoration,, SIAM Journal on Imaging Sciences, 2 (2009), 1168.  doi: 10.1137/090758490.  Google Scholar [14] T. Goldstein and S. Osher, The split bregman method for l1-regularized problems,, SIAM Journal on Imaging Science, 2 (2009), 323.  doi: 10.1137/080725891.  Google Scholar [15] Y. Gousseau and J.-M. Morel, Are natural images of bounded variation?,, SIAM Journal on Mathematical Analysis, 33 (2001), 634.  doi: 10.1137/S0036141000371150.  Google Scholar [16] X. Guo, F. Li and M. Ng, A fast $l^1$-TV algorithm for image restoration,, SIAM Journal of Scientific Computing, 31 (2009), 2322.  doi: 10.1137/080724435.  Google Scholar [17] B. He and X. Yuan, Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective,, SIAM Journal on Imaging Sciences, 5 (2012), 119.  doi: 10.1137/100814494.  Google Scholar [18] Q. Li, C. Micchelli, L. Shen and Y. Xu, A proximity algorithm accelerated by Gauss-Seidel iterations for L1/TV denoising models,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/9/095003.  Google Scholar [19] C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denosing,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/4/045009.  Google Scholar [20] C. A. Micchelli, L. Shen, Y. Xu and X. Zeng, Proximity algorithms for the $l_1$/$tv$ image denosing models,, Advances in Computational Mathematics, 38 (2013), 401.  doi: 10.1007/s10444-011-9243-y.  Google Scholar [21] J.-J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien,, C. R. Acad. Sci. Paris Sér. A Math., 255 (1962), 1897.   Google Scholar [22] M. Nikolova, A variational approach to remove outliers and impulse noise,, Journal of Mathematical Imaging and Vision, 20 (2004), 99.  doi: 10.1023/B:JMIV.0000011920.58935.9c.  Google Scholar [23] R. Rockafellar, Augmented lagrangians and applications of the proximal point algorithm in convex programming,, Mathematics of Operations Research, 8 (1976), 421.  doi: 10.1287/moor.1.2.97.  Google Scholar [24] R. T. Rockafellar and R. Wets, Variational Analysis,, Springer, (1998).  doi: 10.1007/978-3-642-02431-3.  Google Scholar [25] C. Wu, J. Zhang and X. 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 [26] J. Yang, Y. Zhang and W. Yin, An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise,, SIAM Journal on Scientific Computing, 31 (2009), 2842.  doi: 10.1137/080732894.  Google Scholar [27] W. Yin, T. Chen, S. Zhou and A. Chakraborty, Background correction for cdna microarray images using the tv+l1 model,, Bioinformatics, 21 (2005), 2410.  doi: 10.1093/bioinformatics/bti341.  Google Scholar [28] W. Yin, D. Goldfard and S. Osher, The total variation l1 model for multiscale decomposition,, Multiscale Modelling and Simulation, 6 (2007), 190.  doi: 10.1137/060663027.  Google Scholar [29] C. Zach, T. Pock and H. Bischof, A duality based approach for realtime tv-l1 optical flow,, Pattern Recognition, 4713 (2007), 214.   Google Scholar

show all references

##### References:
 [1] J. E. Aujol, G. Gilboa, T. Chan and S. Osher, Structure-texture image decomposition-modeling, algorithms, and parameter selection,, International Journal of Computer Vision, 67 (2006), 111.  doi: 10.1007/s11263-006-4331-z.  Google Scholar [2] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Space,, Springer Press, (2011).  doi: 10.1007/978-1-4419-9467-7.  Google Scholar [3] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems,, SIAM Journal on Imaging Science, 2 (2009), 183.  doi: 10.1137/080716542.  Google Scholar [4] P. Bloomfield and W. Steiger, Least Absolute Deviations: Theory, Applications, and Algorithms,, Birkhäuser Boston, (1983).   Google Scholar [5] A. Bovik, Handbook of Image and Video Processing,, Academic Press, (2000).   Google Scholar [6] A. Chambolle, An algorithm for total variation minimization and applications,, Special issue on mathematics and image analysis, 20 (2004), 89.  doi: 10.1023/B:JMIV.0000011321.19549.88.  Google Scholar [7] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging,, Journal of Mathematical Imaging and Vision, 40 (2011), 120.  doi: 10.1007/s10851-010-0251-1.  Google Scholar [8] T. Chan and S. Esedoglu, Aspects of total variation regularized l1 function approximation,, SIAM Journal on Applied Mathematics, 65 (2005), 1817.  doi: 10.1137/040604297.  Google Scholar [9] R. Chartrand and V. Staneva, Total variation regularization of images corrupted by non-gaussian noise using a quasi-newton method,, IET Image Processing, 2 (2008), 295.  doi: 10.1049/iet-ipr:20080017.  Google Scholar [10] T. Chen, W. Yin, X. Zhou, D. Comaniciu and T. Huang, Total variation models for variable lighting face recognition,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 28 (2006), 1519.  doi: 10.1109/TPAMI.2006.195.  Google Scholar [11] C. Clason, B. Jin and K. Kunisch, A duality-based splitting method for l1-tv image restoration with automatic regularization parameter choice,, SIAM Journal on Scientific Computing, 32 (2010), 1484.  doi: 10.1137/090768217.  Google Scholar [12] P. Combettes and V. Wajs, Signal recovery by proximal forward-backward splitting,, Multiscale Model and Simulation, 4 (2005), 1168.  doi: 10.1137/050626090.  Google Scholar [13] Y. Dong, M. Hintermuller and M. Neri, An Efficient Primal-Dual Method for $l^1$-TV image Restoration,, SIAM Journal on Imaging Sciences, 2 (2009), 1168.  doi: 10.1137/090758490.  Google Scholar [14] T. Goldstein and S. Osher, The split bregman method for l1-regularized problems,, SIAM Journal on Imaging Science, 2 (2009), 323.  doi: 10.1137/080725891.  Google Scholar [15] Y. Gousseau and J.-M. Morel, Are natural images of bounded variation?,, SIAM Journal on Mathematical Analysis, 33 (2001), 634.  doi: 10.1137/S0036141000371150.  Google Scholar [16] X. Guo, F. Li and M. Ng, A fast $l^1$-TV algorithm for image restoration,, SIAM Journal of Scientific Computing, 31 (2009), 2322.  doi: 10.1137/080724435.  Google Scholar [17] B. He and X. Yuan, Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective,, SIAM Journal on Imaging Sciences, 5 (2012), 119.  doi: 10.1137/100814494.  Google Scholar [18] Q. Li, C. Micchelli, L. Shen and Y. Xu, A proximity algorithm accelerated by Gauss-Seidel iterations for L1/TV denoising models,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/9/095003.  Google Scholar [19] C. A. Micchelli, L. Shen and Y. Xu, Proximity algorithms for image models: Denosing,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/4/045009.  Google Scholar [20] C. A. Micchelli, L. Shen, Y. Xu and X. Zeng, Proximity algorithms for the $l_1$/$tv$ image denosing models,, Advances in Computational Mathematics, 38 (2013), 401.  doi: 10.1007/s10444-011-9243-y.  Google Scholar [21] J.-J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien,, C. R. Acad. Sci. Paris Sér. A Math., 255 (1962), 1897.   Google Scholar [22] M. Nikolova, A variational approach to remove outliers and impulse noise,, Journal of Mathematical Imaging and Vision, 20 (2004), 99.  doi: 10.1023/B:JMIV.0000011920.58935.9c.  Google Scholar [23] R. Rockafellar, Augmented lagrangians and applications of the proximal point algorithm in convex programming,, Mathematics of Operations Research, 8 (1976), 421.  doi: 10.1287/moor.1.2.97.  Google Scholar [24] R. T. Rockafellar and R. Wets, Variational Analysis,, Springer, (1998).  doi: 10.1007/978-3-642-02431-3.  Google Scholar [25] C. Wu, J. Zhang and X. 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 [26] J. Yang, Y. Zhang and W. Yin, An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise,, SIAM Journal on Scientific Computing, 31 (2009), 2842.  doi: 10.1137/080732894.  Google Scholar [27] W. Yin, T. Chen, S. Zhou and A. Chakraborty, Background correction for cdna microarray images using the tv+l1 model,, Bioinformatics, 21 (2005), 2410.  doi: 10.1093/bioinformatics/bti341.  Google Scholar [28] W. Yin, D. Goldfard and S. Osher, The total variation l1 model for multiscale decomposition,, Multiscale Modelling and Simulation, 6 (2007), 190.  doi: 10.1137/060663027.  Google Scholar [29] C. Zach, T. Pock and H. Bischof, A duality based approach for realtime tv-l1 optical flow,, Pattern Recognition, 4713 (2007), 214.   Google Scholar
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