Fixed Point Algorithm Based on Quasi-Newton Method for Convex Minimization Problem with Application to Image Deblurring

TL;DR

This paper introduces a fast fixed point algorithm based on quasi-Newton methods for convex optimization problems involving non-differentiable functions, specifically applied to TV-based image deblurring, demonstrating improved efficiency.

Contribution

The paper proposes a novel quasi-Newton based fixed point algorithm for convex problems with non-differentiable functions, applied to image deblurring, with proven global convergence.

Findings

The algorithm outperforms existing fixed-point methods in computational efficiency.

Numerical results show effective deblurring with additive and multiplicative noise.

The method converges globally due to the non-expansion property of the proximity operator.

Abstract

Solving an optimization problem whose objective function is the sum of two convex functions has received considerable interests in the context of image processing recently. In particular, we are interested in the scenario when a non-differentiable convex function such as the total variation (TV) norm is included in the objective function due to many variational models established in image processing have this nature. In this paper, we propose a fast fixed point algorithm based on the quasi-Newton method for solving this class of problem, and apply it in the field of TV-based image deblurring. The novel method is derived from the idea of the quasi-Newton method, and the fixed-point algorithms based on the proximity operator, which were widely investigated very recently. Utilizing the non-expansion property of the proximity operator we further investigate the global convergence of the proposed algorithm. Numerical experiments on image deblurring problem with additive or multiplicative noise are presented to demonstrate that the proposed algorithm is superior to the recently developed fixed-point algorithm in the computational efficiency.