Fourier spectrum of radially periodic images

I. Amidror

Journal of the Optical Society of America A, Vol. 14, No. 4, 1997, pp. 816-826.

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. In this paper we present an alternative Fourier decomposition of radially periodic images, in terms of circular cosine functions, and we show its significant advantages. We show that the Fourier transform of the circular cosine function, which can be expressed in terms of a half-order derivative of the impulse ring delta(r-f), plays a fundamental role in the spectra of radially periodic functions. Just as any symmetric periodic function p(x) in the 1D 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 2D 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 be also formulated in terms of the spectral domain: Just as the Fourier transform of a 1D 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 2D 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). We discuss the significance of these results, and briefly show how they can be extended into dimensions other than two.