Augmented Lagrangian Method, Dual Methods and Split Bregman Iteration for ROF Model

  • Authors:

  • Xue-Cheng Tai;Chunlin Wu

  • Affiliations:

  • Division of Mathematical Science, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore Department of Mathematics,University of Bergen, Bergen, Norway N-5008;Division of Mathematical Science, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore

  • Venue:
  • SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
  • Year:
  • 2009

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Abstract

In the recent decades the ROF model (total variation (TV) minimization) has made great successes in image restoration due to its good edge-preserving property. However, the non-differentiability of the minimization problem brings computational difficulties. Different techniques have been proposed to overcome this difficulty. Therein methods regarded to be particularly efficient include dual methods of CGM (Chan, Golub, and Mulet) [7] Chambolle [6] and split Bregman iteration [14], as well as splitting-and-penalty based method [28] [29]. In this paper, we show that most of these methods can be classified under the same framework. The dual methods and split Bregman iteration are just different iterative procedures to solve the same system resulted from a Lagrangian and penalty approach. We only show this relationship for the ROF model. However, it provides a uniform framework to understand these methods for other models. In addition, we provide some examples to illustrate the accuracy and efficiency of the proposed algorithm.