Non-Lipschitz Models for Image Restoration with Impulse Noise Removal

Abstract

Impulse noise is a special type of noise which corrupts a portion of the image pixels while keeping the others unaffected. L$_1$TV, L$_1$Nonconvex, and NonconvexTV are three common variational models for impulse noise removal. However, non-Lipschitz cases were ruled out for L$_1$Nonconvex and NonconvexTV in former works. In this paper, we consider L$_1$non-Lipschitz and non-LipschitzTV. Uniform lower bound theories for these two models are obtained, which yield the models' characteristics. Based on the lower bound theories, we propose two new algorithms for the two models. Instead of introducing auxiliary parameters to convert the non-Lipschitz models into Lipschitz ones in usual approximation methods for non-Lipschitz optimization, the new algorithms overcome the non-Lipschitzian by iteratively adding constraints on the supports of non-Lipschitz terms. The new algorithms can be easily implemented, and the global convergence is also established. Numerical examples are given to show good performance of the algorithm and the rationality of the theories. The advantageous models for difference cases are also discussed.