An inexact primal-dual method with correction step for a saddle point problem in image debluring

TL;DR

This paper introduces an inexact primal-dual method with a correction step for saddle point problems, demonstrating convergence and applying it effectively to TV-L1 image deblurring with promising numerical results.

Contribution

It proposes a novel inexact primal-dual algorithm with correction steps that relaxes step size constraints and proves convergence with an $O(1/N)$ rate, applied successfully to image deblurring.

Findings

Convergence of the proposed method is proven.

The method achieves an $O(1/N)$ convergence rate.

Numerical results show high efficiency in image deblurring.

Abstract

In this paper,we present an inexact primal-dual method with correction step for a saddle point problem by introducing the notations of inexact extended proximal operators with symmetric positive definite matrix $D$. Relaxing requirement on primal-dual step sizes, we prove the convergence of the proposed method. We also establish the $O(1/N)$ convergence rate of our method in the ergodic sense. Moreover, we apply our method to solve TV-L$_1$ image deblurring problems. Numerical simulation results illustrate the efficiency of our method.