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.