Fixed-point algorithms for a TVL1 image restoration model

ABSTRACT

In this paper, we study fixed point proximity algorithms for a TVL1 restoration model recovering blurred images with impulsive noise, and image inpainting. The model that minimizes the sum of a data fidelity term in the ${\ell }^1$-norm, a term in ${\ell }^2$-norm and total-variation regularization term is strictly convex. We obtain the solution of the model through finding a fixed point of a nonlinear mapping expressed in terms of the proximity operator of the ${\ell }^1$-norm or the ${\ell }^2$-norm, each of which is explicitly given. The non-expansivity of the mapping is also analysed theoretically. This formulation naturally leads to fixed-point algorithms for numerical treatment of the model. Numerical experiments demonstrate that the proposed algorithms perform favourably.

KEYWORDS:

• Total variation
• proximity operator
• deblurring
• impulsive noise
• inpainting

2010 AMS SUBJECT CLASSIFICATIONS:

• 68U10
• 94A08
• 49J40
• 35A15

Acknowledgments

The authors would like to thank the anonymous referees for their careful reading, helpful comments, and suggestions that led to a significant improvement of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported in part by the US National Science Foundation # DMS-1522332, by the National Natural Science Foundation of China #61373087, #11201312, #61272252, #61472257, by the China Scholarship Council, by the Foundation for Distinguished Young Teachers in Guangdong, China #Yq2013144, by the Guangdong Natural Science Foundation of China #2015A030313557, #2015A030313550, by the HD Video R & D Platform for Intelligent Analysis and Processing in Guangdong Engineering Technology Research Centre of Colleges and Universities #GCZX-A1409, and by the Shenzhen Key Laboratory of Media Security.