Matrix equation representation of the convolution equation and its unique solvability (Q6590687)

The authors consider the convolution equation appearing in image restoration problems arising from areas such as astronomy and medical imaging. Mathematically, a filter matrix \(F\) is applied to an original image \(X\), yielding an observed degraded image \(B\):\N\[\NF * X = B.\N\]\NHere both \(X\) and \(B\) are of sizes \(m\times n\), while the filter matrix \(F\) is \(3\times 3\), and the convolution operation \(*\) is defined as follows:\N\[\N[F*X]_{ij} = \sum_{l_1=1}^{3} \sum_{l_2=1}^{3} f_{l_1 l_2} x_{i-l_1+2,j-l_2+2},\N\]\Nwhere small letters \(f\) and \(x\) denote matrix entries of \(F\) and \(X\), respectively.\N\NThis convolution equation is considered for three kinds of boundary conditions, and seven kinds of more or less realistic filters \(F\). It is shown that, for all cases, the resulting equation fits into a (quite large) framework of ``generalized Sylvester equations''. More importantly, the authors are able to provide, in each specific case except one (emboss filter \(F\) and reflexive boundary conditions), algebraic reasonings showing whether there exists a unique solution to the corresponding equation. For the mentioned more complicated case, the authors put forward a reasonable conjecture based on concrete examples.\N\NAs for actual numerical algorithms utilizing the representation of the convolution equation as a generalized Sylvester equation, the authors just express their interest to present these in a future work.