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We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\timesn}$ is to be determined. The convolution equation can be regarded as a linearsystem with a coefficient matrix of special structure. This fact has led tomany studies including efficient numerical algorithms for solving theconvolution equation. In this study, we show that the convolution equation canbe represented as a generalized Sylvester equation. Furthermore, for somerealistic examples arising from image processing, we show that the generalizedSylvester equation can be reduced to a simpler form, and analyze the uniquesolvability of the convolution equation.

Matrix equation representation of the convolution equation and its unique solvability