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

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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### References

- R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1999).
- K. Howell, “Fourier transforms,” in The Transforms and Applications Handbook (CRC, 2000), pp. 2.1–2.159.
- G. S. Chirikjian and A. B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis (CRC Press, 2000). [CrossRef]
- M. Xu and L. V. Wang, “Time-domain reconstruction for thermoacoustic tomography in a spherical geometry,” IEEE Trans. Med. Imaging 21, 814–822 (2002). [CrossRef] [PubMed]
- A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” J. Appl. Comput. Harmonic Anal. 21, 145–167 (2006). [CrossRef]
- M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging (SIAM, 1988).
- N. Baddour, “Fourier diffraction theorem for diffusion-based thermal tomography,” J. Phys. A 39, 14379–14395 (2006). [CrossRef]
- N. Baddour, “Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates,” J. Opt. Soc. Am. A 26, 1767–1777 (2009). [CrossRef]
- J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-sphere,” Adv. Appl. Math. 15, 202–250 (1994). [CrossRef]
- R. Piessens, “The Hankel transform,” in The Transforms and Applications Handbook (CRC, 2000), pp. 9.1–9.30. [CrossRef]
- “Spherical harmonics,” Wikipedia, the Free Encyclopedia.
- J. C. Slater, Quantum Theory of Atomic Structure, Vol. I of International Series in Pure and Applied Physics (McGraw-Hill, New York, 1960).
- “Slater integrals,” Wikipedia, the Free Encyclopedia.
- G. Arfken and H. Weber, Mathematical Methods for Physicists (Elsevier Academic, 2005).
- E. W. Weisstein, “Spherical Harmonics,” Wolfram MathWorld.
- R. Basri and D. Jacobs, “Lambertian reflectance and linear subspaces,” in Proceedings of the Eighth IEEE International Conference on Computer Vision, 2001 (ICCV 2001) (2001), Vol. 2, pp. 383–390. [CrossRef]
- R. Mehrem, J. T. Londergan, and M. H. Macfarlane, “Analytic expressions for integrals of products of spherical Bessel functions,” J. Phys. A 24, 1435–1453 (1991). [CrossRef]
- V. Fabrikant, “Computation of infinite integrals involving three Bessel functions by introduction of new formalism,” Z. Angew. Math. Mech. 83, 363–374 (2003). [CrossRef]
- A. D. Jackson and L. C. Maximon, “Integrals of products of Bessel functions,” SIAM J. Math. Anal. 3, 446–460 (1972). [CrossRef]
- R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for reflection,” ACM Trans. Graphics 23, 1004–1042 (2004). [CrossRef]
- T. Inui, Group Theory and Its Applications in Physics (Springer-Verlag, 1990). [CrossRef]
- R. Ramamoorthi and P. Hanrahan, “An efficient representation for irradiance environment maps,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ACM, 2001), pp. 497–500.

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