Design of multistage separable two-dimensional digital filters with frequency domain specifications

Abstract

The separable decomposition method for 2-D filter design is examined in the continuous frequency domain. In the general case, improved accuracy and better insight can be obtained in comparison with the discrete counterparts as examined in the literature. Substantial simplifications both for the off-line computational load and the subsequent implementation are possible for frequency response functions being constant on their effective domain. This is achieved through a piecewise linear approximation of the domain boundary which reduces the original eigenvalue problem for operators with 2-D kernel into a solution of decoupled simple boundary-value problems in each direction. The latter are solvable through a simple Newton-Raphson procedure combined with a shooting method. This results into 1-D constituent filters with piecewise sinusoidal frequency responses. Completely analytical results are obtained for fan and directional filter design. Error estimates are given and examples illustrate the applicability of the results.