Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group

Fourier spectrum of radially periodic images

Not Accessible

Your library or personal account may give you access

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.

© 1997 Optical Society of America

Full Article  |  PDF Article
More Like This
Fourier spectrum of curvilinear gratings of the second order

Isaac Amidror
J. Opt. Soc. Am. A 15(4) 900-913 (1998)

Fourier spectrum of halftone images

D. Kermisch and P. G. Roetling
J. Opt. Soc. Am. 65(6) 716-723 (1975)

Use of the analyticity of the generalized Fourier spectrum in object reconstruction

Darryl J. Sanchez and J. K. McIver
J. Opt. Soc. Am. A 14(4) 792-798 (1997)

Cited By

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Figures (7)

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Equations (65)

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Select as filters


Select Topics Cancel
© Copyright 2024 | Optica Publishing Group. All Rights Reserved