A Nonmonotone Gradient Algorithm for Total Variation Image Denoising Problems

Peng Wang,Shifang Yuan,Xiangyun Xie,Shengwu Xiong

doi:10.1155/2016/4310612

Abstract

The total variation (TV) model has been studied extensively because it is able to preserve sharp attributes and capture some sparsely critical information in images. However, TV denoising problem is usually ill-conditioned that the classical monotone projected gradient method cannot solve the problem efficiently. Therefore, a new strategy based on nonmonotone approach is digged out as accelerated spectral project gradient (ASPG) for solving TV. Furthermore, traditional TV is handled by vectorizing, which makes the scheme far more complicated for designing algorithms. In order to simplify the computing process, a new technique is developed in view of matrix rather than traditional vector. Numerical results proved that our ASPG algorithm is better than some state-of-the-art algorithms in both accuracy and convergence speed.

Highlights

Blur and noise are unavoidable during the acquisition and transmission of images, and this may be introduced by the poor lighting conditions, faulty camera or unperfect transmission channels, and so on
Similar techniques were tactfully managed to different traditional algorithms [8, 10, 11, 16, 17] and corresponding optimal complexity results were obtained: Beck and Teboulle [10] proposed a fast duality-based gradient projection methods to solve total variation (TV); Chambolle and Pock [11] and Goldfarb et al [16] accelerated the classical primal-dual algorithm and alternating direction augmented Lagrangian methods, respectively; more recently, Ouyang et al [17] adopted the same technique to enhance the convergence of alternating direction method of multipliers (ADMM)
By combining Barzilai-Borwein nonmonotone [19] ideas with traditional projected gradient algorithm, Birgin et al [13, 14] developed a new gradient algorithm that is spectral projected gradient (SPG), which is suitable for convex optimization problems with projection easy to be computed on feasible set

Summary

A Nonmonotone Gradient Algorithm for Total Variation Image Denoising Problems

The total variation (TV) model has been studied extensively because it is able to preserve sharp attributes and capture some sparsely critical information in images. TV denoising problem is usually ill-conditioned that the classical monotone projected gradient method cannot solve the problem efficiently. A new strategy based on nonmonotone approach is digged out as accelerated spectral project gradient (ASPG) for solving TV. Traditional TV is handled by vectorizing, which makes the scheme far more complicated for designing algorithms. In order to simplify the computing process, a new technique is developed in view of matrix rather than traditional vector. Numerical results proved that our ASPG algorithm is better than some state-ofthe-art algorithms in both accuracy and convergence speed

Introduction

Primal-Dual Framework of Total Variation Denoising

Accelerated Spectral Projected Gradient Based TV Denoising

Numerical Experiments

Conclusions