A New Accelerated Viscosity Iterative Method for an Infinite Family of Nonexpansive Mappings with Applications to Image Restoration Problems

Jenwit Puangpee,Suthep Suantai

doi:10.3390/math8040615

Abstract

The image restoration problem is one of the popular topics in image processing which is extensively studied by many authors because of its applications in various areas of science, engineering and medical image. The main aim of this paper is to introduce a new accelerated fixed algorithm using viscosity approximation technique with inertial effect for finding a common fixed point of an infinite family of nonexpansive mappings in a Hilbert space and prove a strong convergence result of the proposed method under some suitable control conditions. As an application, we apply our algorithm to solving image restoration problem and compare the efficiency of our algorithm with FISTA method which is a popular algorithm for image restoration. By numerical experiments, it is shown that our algorithm has more efficiency than that of FISTA.

Highlights

Let us first consider a simple linear inverse problem as the following form: Ax = b + w, (1)where x ∈ Rn×1 is the solution of the problem to be approximated, A ∈ Rm×n and b ∈ Rm×1 are known and w ∈ Rm×1 is an additive noise vector
Motivated and inspired by all of these researches going on in this direction, in this paper, we introduce a new accelerated algorithm for finding a common fixed point of a family of nonexpansive mappings { Tn } in Hilbert spaces based on the concept of inertial forward-backward, of Mann and of viscosity algorithms
We prove its strong convergence under some suitable conditions

Summary

Introduction

Where x ∈ Rn×1 is the solution of the problem to be approximated, A ∈ Rm×n and b ∈ Rm×1 are known and w ∈ Rm×1 is an additive noise vector. Such problems (1) arise in various applications such as the image and signal processing problems, astrophysical problems and data classification problems. The purpose of the image restoration problem is to minimize the additive noise in which the classical estimator is the least squares (LS) given as follows:. In 1977, Tikhonov and Arsenin [1]

Objectives

Methods

Results

Conclusion