OSA's Digital Library

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


View Full Text Article

Enhanced HTML    Acrobat PDF (212 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1999).
  2. K. Howell, “Fourier transforms,” in The Transforms and Applications Handbook, 2nd ed., A.D.Poularikas, ed. (CRC Press, 2000), pp. 2.1-2.159.
  3. G. Chirikjian and A. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups (CRC Press, 2001).
  4. Y. Xu, M. Xu, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography--II: Cylindrical geometry,” IEEE Trans. Med. Imaging 21, 829-833 (2002). [CrossRef] [PubMed]
  5. A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” Appl. Comput. Harmonic Anal. 21, 145-167 (2006). [CrossRef]
  6. R. Piessens, “The Hankel transform,” in The Transforms and Applications Handbook, 2nd ed., A.D.Poularikas, ed. (CRC Press, 2000), pp. 9.1-9.30.
  7. J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2004).
  8. G. Arfken and H. Weber, Mathematical Methods for Physicists (Elsevier, 2005).
  9. A. Oppenheim and R. Schafer, Discrete-time Signal Processing (Prentice-Hall, 1989).
  10. A. D. Jackson and L. C. Maximon, “Integrals of products of Bessel functions,” SIAM J. Math. Anal. 3, 446-460 (1972). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited