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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: Franco Gori
  • Vol. 27, Iss. 10 — Oct. 1, 2010
  • pp: 2144–2155

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

Natalie Baddour  »View Author Affiliations


JOSA A, Vol. 27, Issue 10, pp. 2144-2155 (2010)
http://dx.doi.org/10.1364/JOSAA.27.002144


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Abstract

For functions that are best described with spherical coordinates, the three-dimensional Fourier transform can be written in spherical coordinates as a combination of spherical Hankel transforms and spherical harmonic series. However, to be as useful as its Cartesian counterpart, a spherical version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution, etc. This paper derives the spherical version of the standard Fourier operation toolset. In particular, convolution in various forms is discussed in detail as this has important consequences for filtering. It is shown that standard multiplication and convolution rules do apply as long as the correct definition of convolution is applied.

© 2010 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: July 12, 2010
Manuscript Accepted: August 12, 2010
Published: September 13, 2010

Citation
Natalie Baddour, "Operational and convolution properties of three-dimensional Fourier transforms in spherical polar coordinates," J. Opt. Soc. Am. A 27, 2144-2155 (2010)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-27-10-2144


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