Circular harmonic analysis of PSF's corresponding to separable polar-coordinate frequency responses with emphasis on fan filtering

A method based on radius-dependent angular Fourier coefficients (circular harmonics) is used to analyze the point spread function (PSF) of any 2-D filter whose frequency response is separable in the polar-coordinate variables. The approach relies on the ability to express the PSF as the angular cyclic convolution of two functions related to the angular and radial characteristics of the frequency response, respectively. A general theory is developed and is subsequently specialized to the ideal, but important, case of fan filtering. In the general case, the exponential and Gaussian radial profiles are considered in detail, both because of their mathematical tractability and because of their usefulness for discussing fan filtering (uniform radial profile) through limiting arguments. Besides providing insight into the structure of the PSF for separable and fan filters, the theory leads to a synthesis procedure via FFT's, which is demonstrated in situations of interest.