Design and applications of adjustable 2D digital filters with elliptical and circular symmetry

This work proposes an efficient analytical design procedure for elliptical and circular 2D digital filters with an adjustable response. The design relies on a 1D low-pass prototype filter with a specified bandwidth, to which a particular frequency mapping is applied. The prototype can be scaled in frequency, having adjustable selectivity. This analytical procedure yields 2D filters with accurate shape even near the frequency plane margins, very low distortions and steep transition at a relatively low order, thus being very efficient. The elliptical filters can be tuned to a desired orientation angle and can be used as directional filters. Also an uniform circular filter bank with a specified number of bands is designed. The frequency response results directly factored, and the corresponding filter matrices result as a convolution of small size matrices, which simplifies implementation and allows for a sequential filtering, in several steps. The proposed method is based entirely on accurate approximations and frequency mappings, without using any numerical optimization algorithms. Several examples of filtering on various test images are provided, to illustrate the capabilities of this class of filters.