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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 [L. I. Rudin, S. Osher and 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 [T. F. Chan, G. H. Golub and P. Mulet, SIAM J. Sci. Comput. 20, No. 6, 1964–1977 (1999; Zbl 0929.68118)], Chambolle’s dual method [A. Chambolle, J. Math. Imaging Vis. 20, No. 1–2, 89–97 (2004; Zbl 1366.94048)], the splitting and quadratic penalty-based method [Y. Wang, J. Yang, W. Yin and Y. Zhang, SIAM J. Imaging Sci. 1, No. 3, 248–272 (2008; Zbl 1187.68665)], and the split Bregman iteration [T. Goldstein and S. Osher, SIAM J. Imaging Sci. 2, No. 2, 323–343 (2009; Zbl 1177.65088)], as well as the augmented Lagrangian method [X. C. Tai and C. Wu, SSVM 2009, Lect. Notes Comput. Sci. 5567, 502–513 (2009; Zbl 1371.94362)].

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.

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.

### MSC:

90C90 | Applications of mathematical programming |

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

68U10 | Computing methodologies for image processing |

65K05 | Numerical mathematical programming methods |