Accelerated Proximal Iterative re-Weighted $\ell_1$ Alternating Minimization for Image Deblurring

TL;DR

This paper introduces accelerated algorithms based on proximal iterative re-weighted $ ext{l}_1$ minimization for nonconvex $ ext{l}_p$ TV image deblurring, improving convergence speed and edge preservation.

Contribution

It proposes two novel algorithms, PIRL1-AM and APIRL1-AM, specifically designed for nonconvex $ ext{l}_p$ TV deblurring, extending existing methods to nonconvex problems with enhanced efficiency.

Findings

APIRL1-AM accelerates convergence compared to PIRL1-AM.

Both algorithms effectively preserve image edges during deblurring.

Numerical results demonstrate improved computational efficiency and image quality.

Abstract

The quadratic penalty alternating minimization (AM) method is widely used for solving the convex $\ell_1$ total variation (TV) image deblurring problem. However, quadratic penalty AM for solving the nonconvex nonsmooth $\ell_p$, $0 < p < 1$ TV image deblurring problems is less studied. In this paper, we propose two algorithms, namely proximal iterative re-weighted $\ell_1$ AM (PIRL1-AM) and its accelerated version, accelerated proximal iterative re-weighted $\ell_1$ AM (APIRL1-AM) for solving the nonconvex nonsmooth $\ell_p$ TV image deblurring problem. The proposed algorithms are derived from the proximal iterative re-weighted $\ell_1$ (IRL1) algorithm and the proximal gradient algorithm. Numerical results show that PIRL1-AM is effective in retaining sharp edges in image deblurring while APIRL1-AM can further provide convergence speed up in terms of the number of algorithm iterations and computational time.