Matrix equation representation of convolution equation and its unique solvability

06/27/2023

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by   Yuki Satake, et al.

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