Fast filtering by generalized convolution related to discrete trigonometric transforms

The authors show that for a family of discrete trigonometric transforms (the set of 16 transforms), filters, which are typically defined and computed in the transform domain by means of static but dense multiplications, can be realized directly in the primary domain using the so-called generalized convolution. This concept is based on the generalized convolution matrices, which are distinct for each transform. These matrices are either sparse or can be made sparse at the cost of some slight and usually fully acceptable approximation. It is shown that filtering with the generalized convolution matrices may require less computations than typical filtering procedures performed in the transform domain, even if fast algorithms for the forward and for the inverse transformations are used. The theoretical results are illustrated and verified with examples of half-band low-pass filters.