Split Bregman iteration algorithm for total bounded variation regularization based image deblurring. (English) Zbl 1202.94062

A novel total bounded variation regularization based image deblurring model is presented in the paper. The necessary definitions and preliminaries about the proposed model are described. A brief overview of some related iterative algorithms is given. Existence and uniqueness of the proposed extended split Bregman iteration is proven. Based on this, a rigorous convergence analysis of the corresponded iterative algorithm is provided. Numerical experiments intended for demonstrating the proposed method are accomplished. Three numerical results are presented to illustrate the efficiency and feasibility of the proposed algorithm. It is compared with the standard total variation based regularization scheme of L. I. Rudin, S. Osher and E. Fatemi [Physica D 60, No. 1–4, 259–268 (1992; Zbl 0780.49028)] (the ROF model). The computations are performed in MATLAB. The obtained results are very promising particularly from the preserved details point of view compared with the standard scheme of the ROF method. It is believed that the proposed model and algorithm can be extended to further application in image processing and computer vision.


94A08 Image processing (compression, reconstruction, etc.) in information and communication theory


Zbl 0780.49028


Matlab; RecPF
Full Text: DOI


[1] Acar, R.; Vogel, C.R., Analysis of total variation penalty methods for ill-posed problems, Inverse problems, 10, 1217-1229, (1994) · Zbl 0809.35151
[2] X. Bresson, T.F. Chan, Fast dual minimization of the vectorial total variation norm and applications to color image processing, UCLA, CAM Report 07-25, 2007.
[3] Cai, J.F.; Osher, S.; Shen, Z., Linearized Bregman iterations for frame-based image deblurring, SIAM J. imaging sci., 2, 226-252, (2009) · Zbl 1175.94010
[4] J.F. Cai, S. Osher, Z. Shen, Split Bregman methods and frame based image restoration, UCLA, CAM Report 09-28, 2009. · Zbl 1189.94014
[5] Cai, J.F.; Osher, S.; Shen, Z., Linearized Bregman iterations for compressed sensing, Math. comp., 78, 1515-1536, (2009) · Zbl 1198.65102
[6] Cai, J.F.; Osher, S.; Shen, Z.W., Convergence of the linearized Bregman iteration for l1-norm minimization, Math. comp., 78, 2127-2136, (2009) · Zbl 1198.65103
[7] Chambolle, A.; Lions, P.L., Image recovery via total variational minimization and related problems, Numer. math., 76, 167-188, (1997) · Zbl 0874.68299
[8] Chan, T.F.; Golub, G.H.; Mulet, P., A nonlinear primal-dual method for total variation-based image restoration, SIAM J. sci. comput., 20, 1964-1977, (1999) · Zbl 0929.68118
[9] Chan, T.F.; Mulet, P., On the convergence of the lagged diffusivity fixed point method in total variation image restoration, SIAM J. numer. anal., 36, 354-367, (1999) · Zbl 0923.65037
[10] Chavent, G.; Kunisch, K., Regularization of linear least squares problems by total bounded variation, ESAIM control optim. calc. var., 2, 359-376, (1997) · Zbl 0890.49010
[11] Combettes, P.L.; Wajs, V.R., Signal recovery by proximal forward-backward splitting, SIAM J. multiscale model. simul., 4, 1168-1200, (2005) · Zbl 1179.94031
[12] J. Darbon, S. Osher, Fast discrete optimization for sparse approximations and deconvolutions, UCLA, preprint, 2007.
[13] Ekeland, I.; Temam, R., Convex analysis and variational problems, Classics appl. math., vol. 28, (1999), SIAM Philadelphia · Zbl 0939.49002
[14] Goldstein, T.; Osher, S., The split Bregman algorithm for l1 regularized problems, SIAM J. imaging sci., 2, 323-343, (2009) · Zbl 1177.65088
[15] Hintermüller, M.; Kunisch, K., Total bounded variation regularization as a bilaterally constrained optimization problem, SIAM J. appl. math., 64, 1311-1333, (2004) · Zbl 1055.94504
[16] Y. Li, F. Santosa, An affine scaling algorithm for minimizing total variation in image enhancement, Tech. Report, Cornell University, USA, TR94-1470, 1994.
[17] Osher, S.; Burger, M.; Goldfarb, D.; Xu, J.; Yin, W., An iterative regularization method for total variation-based image restoration, Multiscale model. simul., 4, 460-489, (2005) · Zbl 1090.94003
[18] A. Osher, Y. Mao, B. Dong, W. Yin, Fast linearized Bregman iterations for compressed sensing and sparse denoising, UCLA, CAM Report 08-37, 2008.
[19] Rudin, L.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Phys. D, 60, 259-268, (1992) · Zbl 0780.49028
[20] Schopfer, F.; Louis, A.K.; Schuster, T., Nonlinear iterative methods for linear ill-posed problems in Banach space, Inverse problems, 22, 311-329, (2006) · Zbl 1088.65052
[21] Tikhonov, A.N.; Arsenin, V.Y., Solutions of ill-posed problems, (1977), Winston and Sons Washington, DC · Zbl 0354.65028
[22] Tseng, P., Applications of splitting algorithm to decomposition in convex programming and variational inequalities, SIAM J. control optim., 29, 119-138, (1991) · Zbl 0737.90048
[23] Vogel, C.; Oman, M., Iteration methods for total variation denoising, SIAM J. sci. comput., 17, 227-238, (1996) · Zbl 0847.65083
[24] Wang, Y.; Yang, J.; Yin, W.; Zhang, Y., A new alternating minimization algorithm for total variation image reconstruction, SIAM J. imaging sci., 1, 248-272, (2008) · Zbl 1187.68665
[25] Y. Wang, W. Yin, Y. Zhang, A fast algorithm for image deblurring with total variation regularization, CAAM Technical Report, 2007.
[26] C. Wu, X. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, UCLA, CAM Report 09-76, 2009.
[27] Yin, W.; Osher, S.; Goldfarb, D.; Darbon, J., Bregman iterative algorithms for l1-minimization with applications to compressend sensing, SIAM J. imaging sci., 1, 143-168, (2008) · Zbl 1203.90153
[28] W. Yin, Analysis and generalizations of the linearized Bregman method, UCLA, CAM Report 09-42, 2009.
[29] X.Q. Zhang, M. Burger, X. Bresson, S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, UCLA, CAM Report 09-03, 2009. · Zbl 1191.94030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.