  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.
