A Fast Algorithm Based on a Sylvester-like Equation for LS Regression with GMRF Prior

TL;DR

This paper introduces a fast algorithm for penalized least squares regression with GMRF prior, utilizing a Sylvester-like equation solved analytically to improve computational efficiency in image processing tasks.

Contribution

It formulates the GMRF-regularized LS regression as a Sylvester-like equation and solves it efficiently, enabling faster algorithms for image processing applications.

Findings

Efficient solution of GMRF-regularized LS regression via Sylvester-like equations.

The proposed method accelerates convergence in multichannel image processing.

Experimental results show improved performance with ADMM in constrained LS problems.

Abstract

This paper presents a fast approach for penalized least squares (LS) regression problems using a 2D Gaussian Markov random field (GMRF) prior. More precisely, the computation of the proximity operator of the LS criterion regularized by different GMRF potentials is formulated as solving a Sylvester-like matrix equation. By exploiting the structural properties of GMRFs, this matrix equation is solved columnwise in an analytical way. The proposed algorithm can be embedded into a wide range of proximal algorithms to solve LS regression problems including a convex penalty. Experiments carried out in the case of a constrained LS regression problem arising in a multichannel image processing application, provide evidence that an alternating direction method of multipliers performs quite efficiently in this context.