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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Stephen A. Burns
  • Vol. 26, Iss. 8 — Aug. 1, 2009
  • pp: 1767–1777

Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates

Natalie Baddour  »View Author Affiliations


JOSA A, Vol. 26, Issue 8, pp. 1767-1777 (2009)
http://dx.doi.org/10.1364/JOSAA.26.001767


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Abstract

For functions that are best described in terms of polar coordinates, the two-dimensional Fourier transform can be written in terms of polar coordinates as a combination of Hankel transforms and Fourier series—even if the function does not possess circular symmetry. However, to be as useful as its Cartesian counterpart, a polar version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution, etc. This paper derives the requisite polar version of the standard Fourier operations. In particular, convolution—two dimensional, circular, and radial one dimensional—is discussed in detail. It is shown that standard multiplication/convolution rules do apply as long as the correct definition of convolution is applied.

© 2009 Optical Society of America

OCIS Codes
(070.4790) Fourier optics and signal processing : Spectrum analysis
(070.6020) Fourier optics and signal processing : Continuous optical signal processing
(100.6950) Image processing : Tomographic image processing
(350.6980) Other areas of optics : Transforms

ToC Category:
Fourier Optics and Signal Processing

History
Original Manuscript: February 24, 2009
Revised Manuscript: May 26, 2009
Manuscript Accepted: June 11, 2009
Published: July 10, 2009

Citation
Natalie Baddour, "Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates," J. Opt. Soc. Am. A 26, 1767-1777 (2009)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-26-8-1767

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