Greedy randomized block Kaczmarz method for matrix equation AXB=C and its applications in color image restoration

TL;DR

This paper introduces a greedy randomized block Kaczmarz method for solving matrix equations like AXB=C, demonstrating faster convergence and applying it effectively to color image restoration.

Contribution

It develops new variants of the block Kaczmarz method for matrix equations and proves their convergence properties, improving upon existing methods.

Findings

Algorithms converge to the least-norm solution when consistent.

Convergence rate is faster than the existing randomized block Kaczmarz method.

Effective application in color image restoration.

Abstract

In view of the advantages of simplicity and effectiveness of the Kaczmarz method, which was originally employed to solve the large-scale system of linear equations $Ax=b$, we study the greedy randomized block Kaczmarz method (ME-GRBK) and its relaxation and deterministic versions to solve the matrix equation $AXB=C$, which is commonly encountered in the applications of engineering sciences. It is demonstrated that our algorithms converge to the unique least-norm solution of the matrix equation when it is consistent and their convergence rate is faster than that of the randomized block Kaczmarz method (ME-RBK). Moreover, the block Kaczmarz method (ME-BK) for solving the matrix equation $AXB=C$ is investigated and it is found that the ME-BK method converges to the solution $A^{+}CB^{+}+X^{0}-A^{+}AX^{0}BB^{+}$ when it is consistent. The numerical tests verify the theoretical results and the methods presented in this paper are applied to the color image restoration problem to obtain satisfactory restored images.