Matrix equation representation of the convolution equation and its unique solvability

TL;DR

This paper explores representing the convolution equation as a generalized Sylvester equation, analyzing its unique solvability, especially in image processing contexts, and proposes reductions to simpler forms for efficient solutions.

Contribution

It introduces a novel matrix equation representation of convolution equations and analyzes conditions for their unique solvability, including reductions for practical applications.

Findings

Convolution equations can be represented as generalized Sylvester equations.

Reduction to simpler forms is possible for certain image processing examples.

Conditions for unique solvability are established.

Abstract

We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\times n}$ is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and analyze the unique solvability of the convolution equation.