Fourier spectra of radially periodic images with a non-symmetric radial period

It has been previously shown that a radially periodic image having a symmetric radial period can be decomposed into a circular Fourier series of circular cosine functions with radial frequencies of f=1/T, 2/T, ..., and that its Fourier spectrum consists of a series of half-order derivative impulse rings with radii f=n/T (which are the Fourier transforms of the circular cosines in the sum). In the present paper these results are extended to the general case of radially periodic images, where the radial period does not necessarily have a symmetric profile. Such a general radially periodic function can be decomposed into a circular Fourier series which is a weighted sum of circular cosine and sine functions with radial frequencies of f=1/T, 2/T, .... In terms of the spectral domain, the Fourier spectrum of a general radially periodic function consists of half-order derivative impulse rings with radii f=n/T (which are the Fourier transforms of the weighted circular cosines and sines in the sum)