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

In image processing, the Rudin-Osher-Fatemi (ROF) model [L. Rudin, S. Osher, and E. Fatemi, Physica D, 60(1992), pp. 259–268] based on total variation (TV) minimization has proven to be very useful. A lot of efforts have been devoted to obtain fast numerical schemes and overcome the non-differentiability of the model. Methods considered to be particularly efficient for the ROF model include the dual methods of Chan-Golub-Mulet (CGM) [T.F. Chan, G.H. Golub, and P. Mulet, SIAM J. Sci. Comput., 20(1999), pp. 1964–1977] and Chambolle [A. Chambolle, J. Math. Imaging Vis., 20(2004), pp. 89–97], and splitting and penalty based method [Y. Wang, J. Yang, W. Yin, and Y. Zhang, SIAM J. Imaging Sciences, 1(2008), pp. 248–272], as well as split Bregman iteration [T. Goldstein, and S. Osher, SIAM J. Imaging Sciences, 2(2009), pp. 323–343]. In this paper, we propose to use augmented Lagrangian method to solve the model. Convergence analysis will be given for the method. In addition, we observe close connections between the method proposed here and some of the existing methods. We show that the augmented Lagrangian method, dual methods, and split Bregman iteration are different iterative procedures to solve the same system. Moreover, the proposed method is extended to vectorial TV and high order models. Using the approach here, we can easily obtain the CGM dual method and split Bregman iteration for vectorial TV and high order models, which, to our best of knowledge, have not been presented in the literature. Numerical examples demonstrate the efficiency and accuracy of our method.