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Wavelet Frame Based Image Restoration via Combined Sparsity and Nonlocal Prior of Coefficients

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Abstract

Owing to the good ability of sparsely approximating piece-wise smooth functions like images, the (tight) wavelet frame has been widely investigated and applied for image restoration and other image processing problems over the past few years. Most of the variational models based on wavelet frame proposed in the past utilize the \(l_{1}\) norm of frame coefficients as a sparsity prior. Very recently, the variational model which penalizes the \(l_{0}\) “norm” of frame coefficients was proposed for image restoration, and proved to outperform the commonly used \(l_{1}\) minimization methods in the quality of restored images. Though the \(l_{0}\) “norm” has the ability of preserving sharp edges and smooth regions, textures and small details may be mistakenly removed at the same time. Therefore, we introduce a \(l_0-l_2\) regularization model which contains a nonlocal prior of frame coefficients to avoid this issue in this paper. Meanwhile, a narrow-band technique is introduced to further improve the computational efficiency of the proposed algorithm. Numerical experiments demonstrate that the propose algorithm is superior to the recently proposed algorithm for \(l_{0}\) “norm” minimization in both iterative time and recovery quality.

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Acknowledgments

The work was supported in part by the National Natural Science Foundation of China under Grant 61401473 and 61271014. We appreciate the constructive comments of the anonymous reviewers, which led to great improvements in this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Third Military Medical University, Chongqing, 400038, People’s Republic of China

    Dai-Qiang Chen & Yan Zhou

  2. School of Automation, Chongqing University, Chongqing, 401331, People’s Republic of China

    Yan Zhou

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  1. Dai-Qiang Chen
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Correspondence to Dai-Qiang Chen.

Additional information

The work was supported in part by the National Natural Science Foundation of China under Grant 61271014 and 61401473.

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Chen, DQ., Zhou, Y. Wavelet Frame Based Image Restoration via Combined Sparsity and Nonlocal Prior of Coefficients. J Sci Comput 66, 196–224 (2016). https://doi.org/10.1007/s10915-015-0018-3

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