Fourier spectrum of radially periodic images

Abstract

Although the spectrum of radially periodic images is often expressed in terms of finite or infinite series of Bessel functions, such expressions do not clearly reveal the exact impulsive structure of the spectrum. An alternative Fourier decomposition of radially periodic images, in terms of circular cosine functions, is presented, and its significant advantages are shown. It is shown that the Fourier transform of the circular cosine function, which can be expressed in terms of a half-order derivative of the impulse ring δ(r-f), plays a fundamental role in the spectra of radially periodic functions. Just as any symmetric periodic function p(x) in the one-dimensional case can be represented by a sum of cosines with frequencies of f=1/T, 2/T, … [the Fourier series decomposition of p(x)], a radially periodic function in the two-dimensional case can be decomposed into a circular Fourier series, which is a sum of circular cosine functions with radial frequencies of f=1/T, 2/T, … . This result can also be formulated in terms of the spectral domain: Just as the Fourier transform of a one-dimensional periodic function consists of impulse pairs located at f=n/T (the Fourier transforms of the cosines in the sum), the Fourier spectrum of a radially periodic function in the two-dimensional case consists of half-order derivative impulse rings with radii f=n/T (which are the Fourier transforms of the circular cosines in the sum). The significance of these results is discussed, and it is briefly shown how they can be extended into dimensions other than two.