Image Restoration: A General Wavelet Frame Based Model and Its Asymptotic Analysis

TL;DR

This paper introduces a comprehensive wavelet frame based image restoration model that encompasses existing models and new variants, providing asymptotic analysis linking discrete and continuum formulations, along with an algorithm and a numerical example.

Contribution

The paper presents a unified wavelet frame based model for image restoration that generalizes existing models and introduces new variants, with asymptotic analysis connecting discrete and continuum approaches.

Findings

The general model includes most existing wavelet frame models as special cases.

Asymptotic analysis links the discrete model to a new continuum variational model.

An algorithm for solving the general model is proposed and demonstrated with a numerical example.

Abstract

Image restoration is one of the most important areas in imaging science. Mathematical tools have been widely used in image restoration, where wavelet frame based approach is one of the successful examples. In this paper, we introduce a generic wavelet frame based image restoration model, called the "general model", which includes most of the existing wavelet frame based models as special cases. Moreover, the general model also includes examples that are new to the literature. Motivated by our earlier studies [1-3], We provide an asymptotic analysis of the general model as image resolution goes to infinity, which establishes a connection between the general model in discrete setting and a new variatonal model in continuum setting. The variational model also includes some of the existing variational models as special cases, such as the total generalized variational model proposed by [4]. In the end, we introduce an algorithm solving the general model and present one numerical simulation as an example.