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Domain Decomposition Methods for Nonlocal Total Variation Image Restoration

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Nonlocal total variation (TV) regularization (Gilboa and Osher in Multiscale Model Simulat 7(3): 1005–1028, 2008; Zhou and Schölkopf in Pattern recognition, proceedings of the 27th DAGM symposium. Springer, Berlin, pp 361–368, 2005) has been widely used for the natural image processing, since it is able to preserve repetitive textures and details of images. However, its applications have been limited in practice, due to the high computational cost for large scale problems. In this paper, we apply domain decomposition methods (DDMs) (Xu et al. in Inverse Probl Imag 4(3):523–545, 2010) to the nonlocal TV image restoration. By DDMs, the original problem is decomposed into much smaller subproblems defined on subdomains. Each subproblem can be efficiently solved by the split Bregman algorithm and Bregmanized operator splitting algorithm in Zhang et al. (SIAM J Imaging Sci 3(3):253–276, 2010). Furthermore, by using coloring technique on the domain decomposition, all subproblems defined on subdomains with same colors can be computed in parallel. Our numerical examples demonstrate that the proposed methods can efficiently solve the nonlocal TV based image restoration problems, such as denoising, deblurring and inpainting. It can be computed in parallel with a considerable speedup ratio and speedup efficiency.

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Acknowledgments

The first two authors are grateful for the visit in MAS, SPMS, Nanyang Technological University, Singapore invited by Prof. Xue-Cheng Tai and Prof. Li-Lian Wang, which initiates this project. We also thank the reviewers for the comments, which have greatly improved the paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tianjin Normal University, Tianjin , 300387, People’s Republic of China

    Huibin Chang

  2. Department of Mathematics, MOE-LSC and Institute of Natural Science, Shanghai Jiao Tong University, Shanghai , 200240, People’s Republic of China

    Xiaoqun Zhang

  3. Department of Mathematics, University of Bergen, 125007 , Bergen, Norway

    Xue-Cheng Tai

  4. Department of Mathematics, East China Normal University, Shanghai , 200241, People’s Republic of China

    Danping Yang

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  1. Huibin Chang
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  2. Xiaoqun Zhang
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Corresponding author

Correspondence to Huibin Chang.

Additional information

The authors’ research was supported by MOE IDM project NRF2007IDM-IDM002-010, Singapore. The second author was partially supported by NSFC11101277, NSFC11161130004, and by the Shanghai Pujiang Talent Program under Grant Number 11PJ1405900. The last author were partially supported by NSFC11071080.

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Chang, H., Zhang, X., Tai, XC. et al. Domain Decomposition Methods for Nonlocal Total Variation Image Restoration. J Sci Comput 60, 79–100 (2014). https://doi.org/10.1007/s10915-013-9786-9

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