Design of circularly symmetric two-dimensional recursive filters

The digital filtering of two-dimensional signals offers the many advantages characteristic of digital computers, such as flexibility and accuracy. Applications exist in the processing of images and geophysical data. A technique is presented for designing stable two-dimensional recursive filters whose magnitude response is approximately circularly symmetric. This is achieved by cascading a number of elementary filters which are called rotated filters because they are designed by rotating one-dimensional continuous filters and using the two-dimensional z-transform to obtain the corresponding digital filter. Stability of these filters is considered in detail and the results obtained are stated in two corollaries. In particular it is proved that rotated filters are stable if the angle of rotation is between 270° and 360°. Finally, methods of analysis and design of the shape, circular symmetry, and cutoff frequency of two-dimensional recursive filters are discussed.