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Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. (English) Zbl 1206.90245
Summary: In image processing, the Rudin-Osher-Fatemi (ROF) model [{\it L. I. Rudin, S. Osher} and {\it E. Fatemi}, Physica D 60, No. 1--4, 259--268 (1992; Zbl 0780.49028)] based on total variation (TV) minimization has proven to be very useful. So far, many researchers have contributed to designing fast numerical schemes and overcoming the nondifferentiability of the model. Methods considered to be particularly efficient for the ROF model include the Chan-Golub-Mulet (CGM) primal-dual method [{\it T. F. Chan, G. H. Golub} and {\it P. Mulet}, SIAM J. Sci. Comput. 20, No. 6, 1964--1977 (1999; Zbl 0929.68118)], Chambolle’s dual method [{\it A. Chambolle}, J. Math. Imaging Vis. 20, 89--97 (2004)], the splitting and quadratic penalty-based method [{\it Y. Wang, J. Yang, W. Yin} and {\it Y. Zhang}, SIAM J. Imaging Sci. 1, No. 3, 248--272 (2008; Zbl 1187.68665)], and the split Bregman iteration [{\it T. Goldstein} and {\it S. Osher}, SIAM J. Imaging Sci. 2, No. 2, 323--343 (2009; Zbl 1177.65088)], as well as the augmented Lagrangian method [{\it X. C. Tai} and {\it C. Wu}, Lect. Notes Comput. Sci. 5567, Berlin: Springer, 502--513 (2009)]. In this paper, we first review the augmented Lagrangian method for the ROF model and then provide some convergence analysis and extensions to vectorial TV and high order models. All the algorithms and analysis will be presented in the discrete setting, which is much clearer for practical implementation than the continuous setting as Tai and Wu [loc. cit.]. We also present, in the discrete setting, the connections between the augmented Lagrangian method, the dual methods, and the split Bregman iteration. Using our extensions and observations, we can easily figure out CGM and the split Bregman iteration for vectorial TV and high order models, which, to the best of our knowledge, have not been presented in the literature. Numerical examples demonstrate the efficiency and accuracy of our method, especially in the image deblurring case.

90C90Applications of mathematical programming
68U10Image processing (computing aspects)
65K05Mathematical programming (numerical methods)
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