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Removal of Mixed Gaussian and Impulse Noise Using Directional Tensor Product Complex Tight Framelets

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Abstract

In this paper, we propose a frame-based iterative algorithm to restore images which are corrupted by mixed Gaussian and impulse noise, under the assumption that the image region corrupted by impulse noise is unknown. The removal of mixed Gaussian and impulse noise by our proposed algorithm is split into two subproblems which are solved alternatively and iteratively. With an initial guessed region of location for impulse noise, the first subproblem is to inpaint a corrupted image by solving a frame-based convex minimization scheme using the balanced approach, where sparse and redundant directional representations play a key role. Motivated by our recent work on frame-based image denoising and image inpainting, we shall employ the tight frame generated from the directional tensor product complex tight framelets in our balanced approach to remove the mixed Gaussian and impulse noise. Such tensor product complex tight framelets provide sparse directional representations for natural images and can capture the cartoon and texture parts of images very well. The second subproblem is to estimate the image region of locations where the pixels are corrupted by impulse noise. We solve the second subproblem using an \(l_0\)-minimization scheme. We consider both salt-and-pepper impulse noise and random-valued impulse noise. Numerical experiments show that our proposed algorithm compares favorably or often outperforms three well-known recent image-restoration methods employed for removing the mixed Gaussian and impulse noise.

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Notes

  1. http://perso.telecom-paristech.fr/~delon/Demos/Impulse/

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Acknowledgments

The authors would like to thank Yi Wang and Ming Yan for providing us their source codes for their image-restoration algorithms in [23, 24]. The authors also thank Zhenpeng Zhao for providing us his source Matlab code for implementing the discrete framelet transform using \({\hbox {TP-}{\mathbb {C}}\hbox {TF}}_6\). The authors are also grateful to the editor and the reviewers for their constructive suggestions and comments that improved the paper. Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grants 05865 and 261351, and the Pacific Institute for the Mathematical Sciences (PIMS) CRG grant. Research of Yi Shen was also supported in part by a PIMS postdoctoral fellowship, the National Natural Science Foundation of China under Grant 11101359 and the Zhejiang Provincial Natural Science Foundation of China under Grant LY15A010020.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Sci–Tech University, Hangzhou, 310028, China

    Yi Shen

  2. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Yi Shen & Bin Han

  3. Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, AB, T2N 1N4, Canada

    Yi Shen & Elena Braverman

Authors
  1. Yi Shen
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  2. Bin Han
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  3. Elena Braverman
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Correspondence to Yi Shen.

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Shen, Y., Han, B. & Braverman, E. Removal of Mixed Gaussian and Impulse Noise Using Directional Tensor Product Complex Tight Framelets. J Math Imaging Vis 54, 64–77 (2016). https://doi.org/10.1007/s10851-015-0589-5

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  • DOI: https://doi.org/10.1007/s10851-015-0589-5

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