# American Institute of Mathematical Sciences

May  2014, 8(2): 507-535. doi: 10.3934/ipi.2014.8.507

## A stable method solving the total variation dictionary model with $L^\infty$ constraints

 1 Institute of Microelectronics, Chinese Academy of Sciences, Beijing, China 2 MAP5, Université Paris Descartes, Paris, 75006, France 3 School of Computer and Information Technology, Beijing Jiaotong University, Beijing, China 4 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

Received  November 2011 Revised  May 2013 Published  May 2014

Image restoration plays an important role in image processing, and numerous approaches have been proposed to tackle this problem. This paper presents a modified model for image restoration, that is based on a combination of Total Variation and Dictionary approaches. Since the well-known TV regularization is non-differentiable, the proposed method utilizes its dual formulation instead of its approximation in order to exactly preserve its properties. The data-fidelity term combines the one commonly used in image restoration and a wavelet thresholding based term. Then, the resulting optimization problem is solved via a first-order primal-dual algorithm. Numerical experiments demonstrate the good performance of the proposed model. In a last variant, we replace the classical TV by the nonlocal TV regularization, which results in a much higher quality of restoration.
Citation: Liyan Ma, Lionel Moisan, Jian Yu, Tieyong Zeng. A stable method solving the total variation dictionary model with $L^\infty$ constraints. Inverse Problems & Imaging, 2014, 8 (2) : 507-535. doi: 10.3934/ipi.2014.8.507
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