  The goal of this paper is to develop a novel numerical method for efficient
multiplicative noise removal. The nonlocal self-similarity of natural images
implies that the matrices formed by their nonlocal similar patches are
low-rank. By exploiting this low-rank prior with application to multiplicative
noise removal, we propose a nonlocal low-rank model for this task and develop a
proximal alternating reweighted minimization (PARM) algorithm to solve the
optimization problem resulting from the model. Specifically, we utilize a
generalized nonconvex surrogate of the rank function to regularize the patch
matrices and develop a new nonlocal low-rank model, which is a nonconvex
nonsmooth optimization problem having a patchwise data fidelity and a
generalized nonlocal low-rank regularization term. To solve this optimization
problem, we propose the PARM algorithm, which has a proximal alternating scheme
with a reweighted approximation of its subproblem. A theoretical analysis of
the proposed PARM algorithm is conducted to guarantee its global convergence to
a critical point. Numerical experiments demonstrate that the proposed method
for multiplicative noise removal significantly outperforms existing methods
such as the benchmark SAR-BM3D method in terms of the visual quality of the
denoised images, and the PSNR (the peak-signal-to-noise ratio) and SSIM (the
structural similarity index measure) values.
