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A Nonconvex Model with Minimax Concave Penalty for Image Restoration

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Abstract

A natural image u is often sparse under a given transformation W, one can use \(L_0\) norm of Wu as a regularisation term in image reconstructions. Since minimizing the \(L_0\) norm is a NP hard problem, the \(L_1\) norm is widely used as an replacement. However, recent studies show that nonconvex penalties, e.g., MCP, enjoy better performance for sparse signal recovery. In this paper, we propose a nonconvex model for image restoration with a minimax concave penalty (MCP). First we establish the existence of a global minimizer for the nonconvex model. Then we solve this model by using the alternating direction method of multipliers algorithm. The convergence of the proposed algorithm is analysed with properly chosen parameters. Numerical experiments show that the MCP model outperforms TV model in terms of efficiency and accuracy.

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Acknowledgements

We thank the reviewers and editor for providing very useful comments and suggestions. We also thank Dr. Yifei Lou for providing the code of \(L_1-0.5L_2\). The research of Y. Jiao is partially supported by National Science Foundation of China No. 11501579, X. Lu is supported by National Science Foundation of China Nos. 11471253 and 91630313, and T. Zeng is supported in part by National Science Foundation of China No. 11671002, CUHK start-up and CUHK DAG 4053296.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, People’s Republic of China

    Juntao You & Xiliang Lu

  2. School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, 430072, Hubei, People’s Republic of China

    Yuling Jiao

  3. Hubei Key Laboratory of Computational Science (Wuhan University), Wuhan, 430072, Hubei, People’s Republic of China

    Xiliang Lu

  4. Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

    Tieyong Zeng

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  1. Juntao You
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  2. Yuling Jiao
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  4. Tieyong Zeng
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Correspondence to Tieyong Zeng.

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You, J., Jiao, Y., Lu, X. et al. A Nonconvex Model with Minimax Concave Penalty for Image Restoration. J Sci Comput 78, 1063–1086 (2019). https://doi.org/10.1007/s10915-018-0801-z

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  • DOI: https://doi.org/10.1007/s10915-018-0801-z

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