Abstract
We introduce in this paper a class of multi-step fixed-point proximity algorithms for solving optimization problems in the context of image processing. The objective functions of such optimization problems are the sum of two convex functions having one composed with an affine transformation which is often the regularization term. We are particularly interested in the scenario when the convex functions involved in the objective function have low regularity (not differentiable) since many practical problems encountered in image processing have this nature. We characterize the solutions of the optimization problem as fixed-points of a mapping defined in terms of the proximity operators of the two convex functions. The algorithmic and mathematical challenges come from the fact that the mapping is a composition of a firmly non-expansive operator with an expansive affine transform. A class of multi-step iterative schemes are developed based on the fixed-point equations that characterize the solutions. For the purpose of studying convergence of the proposed algorithms, we introduce a notion of weakly firmly non-expansive mappings and establish under certain conditions that the sequence generated from a weakly firmly non-expansive mapping is convergent. We use this general convergence result to conclude that the proposed multi-step algorithms converge. We in particular design a class of two-step algorithms for solving the optimization problem which include several existing algorithms as special examples and at the same time offer novel algorithms. Moreover, we identify the well-known alternating split Bregman iteration method as a special case of the proposed algorithm and modify it to improve its convergence result. A class of two-step algorithms for the total variation based image restoration models are presented.
Similar content being viewed by others
References
Aubert, G., Vese, L.: A variational method in image recovery. SIAM J. Numer. Anal. 34, 1948–1979 (1997)
Bauschke, H. L., Combettes, P. L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, AMS Books in Mathematics. Springer, New York (2011)
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)
Cai, J.-F., Chan, R., Shen, L., Shen, Z.: Simultaneously inpainting in image and transformed domains. Numer. Math. 112, 509–533 (2009)
Cai, J.-F., Chan, R. H., Shen, L., Shen, Z.: Convergence analysis of tight framelet approach for missing data recovery. Adv. Comput. Math. 31, 87–113 (2009)
Cai, J.-F., Osher, S., Shen, Z.: Linearized Bregman iteration for frame based image deblurring. SIAM J. Imaging Sci. 2, 226–252 (2009)
Split Bregman methods and frame based image restoration. Multiscale Model. Simul.:SIAM Interdiscip. J. 2, 337–369 (2009)
Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004)
Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011)
Chan, R., Chan, T., Shen, L., Shen, Z.: Wavelet algorithms for high-resolution image reconstruction. SIAM J. Sci. Comput. 24, 1408–1432 (2003)
Chan, R., Riemenschneider, S. D., Shen, L., Shen, Z.: Tight frame: The efficient way for high-resolution image reconstruction. Appl. Comput. Harmon. Anal. 17, 91–115 (2004)
Chan, T., Esedoglu, S.: Aspects of total variation regularized l 1 function approximation. SIAM J. Appl. Math. 65, 1817–1837 (2005)
Chan, T., Golub, G. H., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 20, 1964–1977 (1999)
Chen, F., Shen, L., Suter, B. W., Xu, Y.: Nesterov’s Algorithm Solving Dual Formulation for Compressed Sensing. J. Comput. Appl. Math. 260, 1–17 (2014)
Chen, F., Shen, L., Xu, Y., Zeng, X.: The Moreau Envelope Approach for the L1/TV Image Denoising Model. Inverse Probl. Imaging 8(1), 53–77 (2014)
Clason, C., Jin, B., Kunisch, K.: A duality-based splitting method for L1-TV image restoration with automatic regularization parameter choice. SIAM J. Sci. Comput. 32, 1484–1505 (2010)
Combettes, P., Wajs, V.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul.:SIAM Interdiscip. J. 4, 1168–1200 (2005)
Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation part i: Fast and exact minimization. J. Math. Imaging Vis. 26, 261–276 (2006)
Daubechies, I., Defrise, M., Mol, C. D.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pur. Appl. Math. 57, 1413–1541 (2004)
Dong, Y., Hintermuller, M., Neri, M.: A primal-dual method for l 1-TV image denoising. SIAM J. Imaging Sci. 2, 1168–1189 (2009)
Figueiredo, M., Nowak, R. D.: An EM algorithm for wavelet-based image restoration. IEEE Trans. Image Process. 12, 906–916 (2003)
Goldstein, T., Osher, S.: The split Bregman method for ℓ 1 regularization problems. SIAM J. Imaging Sci. 2, 323–343 (2009)
He, B., Yuan, X.: Convergence analysis of primal-dual algorithms for a saddle-point problem: From contraction perspective. SIAM J. Imaging Sci. 5, 119–149 (2012)
Hiriart-Urruty, J., Lemarechal, C.: Convex Analysis and Minimization Algorithms: Part 1: Fundamentals, vol. 1. Springer, New York (1996)
Hintermuller, M., Stadler, G.: An feasible primal-dual algorithm for TV-based inf-convolution-type image restoration. SIAM J. Sci. Comput. 28, 1–23 (2006)
Krol, A., Li, S., Shen, L., Xu, Y.: Preconditioned alternating projection algorithms for maximum a Posteriori ECT reconstruction. Inverse Probl. 28, 115005 (35pp) (2012)
Li, Q., Micchelli, C. A., Shen, L., Xu, Y.: A proximity algorithm accelerated by Gauss-Seidel iterations for L1/TV denoising models. Inverse Probl. 28, 095003 (20pp) (2012)
Li, Q., Shen, L., Yang, L.: Split-Bregman iteration for framelet based image inpainting. Appl. Comput. Harmon. Anal. 32, 145–154 (2012)
Micchelli, C. A., Shen, L., Xu, Y.: Proximity algorithms for image models: Denoising. Inverse Probl. 27, 045009 (30pp) (2011)
Micchelli, C. A., Shen, L., Xu, Y., Zeng, X.: Proximity algorithms for image models II: L1/TV denosing. Adv. Comput. Math. 38, 401–426 (2013)
Moreau, J.-J.: Fonctions convexes duales et points proximaux dans un espace hilbertien. C.R. Acad. Sci. Paris Sér. A Math. 255, 1897–2899 (1962)
Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Program. 103, 127–152 (2005)
Nikolova, M.: Local strong homogeneity of a regularized estimator. SIAM J. Appl. Math 61, 633–658 (2000)
Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20, 99–120 (2004)
Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul.: SIAM Interdiscip. J. 4, 460–489 (2005)
Pock, T., Chambolle, A.: Diagonal preconditioning for first order primal-dual algorithms in convex optimization (2011)
Rockafellar, R. T.: Convex Analysis. Princeton University Press, Princeton (1970)
Rockafellar, R. T.: Monotone operators and the proximal point algorithm. SIAM J. Control. Optim. 14, 877–898 (1976)
Rudin, L., Osher, S.: Total variation based image restoration with free local constraints. IEEE Int. Conf. Image Process., 31–35 (1994)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Setzer, S.: Split Bregman Algorithm, Douglas-Rachford Splitting and Frame Shrinkage. In: Morken, K., Lysaker, M., Lie, K.-A., Tai, X.-C. (eds.) Scale Space and Variational Methods in Computer Vision, LNCS, vol. 5567, pp 464–476. Springer, Berlin (2009)
Vogel, C., Oman, M.: Iterative methods for total variation denoising. SIAM 17, 227–238 (1996)
Fast, robust total variation-based reconstruction of noisy, blurring images. IEEE Trans. Image Process. 7, 813–824 (1998)
Yang, J., Zhang, Y., Yin, W.: An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise. SIAM J. Sci. Comput. 31, 2842–2865 (2009)
Zhang, X., Burger, M., Osher, S.: A unified primal-dual algorithm framework based on Bregman iteration. J. Sci. Comput. 46, 20–46 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Raymond H. Chan
This research is supported in part by the US National Science Foundation under grant DMS-1115523, by Guangdong Provincial Government of China through the “Computational Science Innovative Research Team” program and by the Natural Science Foundation of China under grants 11071286 and 91130009.
Rights and permissions
About this article
Cite this article
Li, Q., Shen, L., Xu, Y. et al. Multi-step fixed-point proximity algorithms for solving a class of optimization problems arising from image processing. Adv Comput Math 41, 387–422 (2015). https://doi.org/10.1007/s10444-014-9363-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-014-9363-2