Augmented Lagrangian Method, Dual Methods, and Split Bregman Iteration for ROF, Vectorial TV, and High Order Models

Abstract

In image processing, the Rudin–Osher–Fatemi (ROF) model [L. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259–268] 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 [T.F. Chan, G.H. Golub, and P. Mulet, SIAM J. Sci. Comput., 20 (1999), pp. 1964–1977], Chambolle's dual method [A. Chambolle, J. Math. Imaging Vis., 20 (2004), pp. 89–97], the splitting and quadratic penalty-based method [Y. Wang, J. Yang, W. Yin, and Y. Zhang, SIAM J. Imaging Sci., 1 (2008), pp. 248–272], and the split Bregman iteration [T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2 (2009), pp. 323–343], as well as the augmented Lagrangian method [X.C. Tai and C. Wu, Lecture Notes in Comput. Sci. 5567, Springer-Verlag, Berlin, 2009, pp. 502–513]. 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 in Tai and Wu, above. 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.