Generalized convolution as a tool for the multi-dimensional filtering tasks

In this paper the so-called generalized convolution, being in fact an adequate adaptation of the well known circular convolution concept to any invertible block-transform, is proposed, developed, and analysed. First the proposed idea is summarized for a one-dimensional case. Then it is extended to multidimensional problems. The presented generalized convolution concept is based on the earlier A-convolution. This idea is recalled at the beginning and a set of techniques for studying the dependence of the respective coefficients on the arbitrary transform, is suggested. The generalized convolution matrix, being an extension of that for the circular convolution, is introduced and adapted to an arbitrary invertible transform. The filtering problem is then defined and presented in the transform domain. The multidimensional analysis starts with two-dimensional problems, then it is continued with formulas for multidimensional filtering tasks. The paper is illustrated with examples computed for twenty carefully selected transforms. Among them are Haar, Hadamard, Hartley, Karhunen-Loeve and a family of 16 discrete trigonometric transforms.