Signal, Image and Video Processing

, Volume 12, Issue 6, pp 1027–1034 | Cite as

Minmax-concave total variation denoising

  • Huiqian DuEmail author
  • Yilin Liu
Original Paper


Total variation (TV) denoising is a commonly used method for recovering 1-D signal or 2-D image from additive white Gaussian noise observation. In this paper, we define the Moreau enhanced function of \(L_1\) norm as \({\varPhi }_\alpha (x)\) and introduce the minmax-concave TV (MCTV) in the form of \({\varPhi }_\alpha (Dx)\), where D is the finite difference operator. We present that MCTV approaches \(\Vert Dx\Vert _0\) if the non-convexity parameter \(\alpha \) is chosen properly and apply it to denoising problem. MCTV can strongly induce the signal sparsity in gradient domain, and moreover, its form allows us to develop corresponding fast optimization algorithms. We also prove that although this regularization term is non-convex, the cost function can maintain convexity by specifying \(\alpha \) in a proper range. Experimental results demonstrate the effectiveness of MCTV for both 1-D signal and 2-D image denoising.


Total variation Signal denoising Non-convex regularization \(L_1\) norm 


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Information and ElectronicsBeijing Institute of TechnologyBeijingChina

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