## Dipole-sheet transform

JOSA, Vol. 72, Issue 4, pp. 468-475 (1982)

http://dx.doi.org/10.1364/JOSA.72.000468

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### Abstract

A new integral transform, derived from the three-dimensional Radon transform, is introduced. The basis functions for this transform, which may be physically interpreted as sheets of dipoles, are shown to be orthonormal and complete. The inverse transform is derived, and an expression for the Fourier transform of the basis functions is found. It is shown that all spherically symmetric functions retain the same functional form under this transform and that it can be used to reduce certain differential equations, such as the Helmholtz equation, to a spherically symmetric form, even if the original problem has no symmetry at all.

© 1982 Optical Society of America

**Citation**

Harrison H. Barrett, "Dipole-sheet transform," J. Opt. Soc. Am. **72**, 468-475 (1982)

http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-72-4-468

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### References

- J. Radon, "Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten," Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Naturwiss. Kl. 69, 262–78 (1917).
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- L. A. Shepp, "Computerized tomography and nuclear magnetic resonance," J. Comput. Assist. Tomog. 4, 94–107 (1980).
- H. H. Barrett, "Three-dimensional image reconstruction from planar projections, with application to optical data processing," presented at SPIE Advanced Institute on Transformations in Optical Signal Processing, Seattle, Washington, February 1981.
- R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 274.
- C. M. Vest and D. G. Steel, "Reconstruction of spherically symmetric objects from slit-imaged emission: application to spatially resolved spectroscopy," Opt. Lett. 3, 54–56 (1978).
- J. E. Greivenkamp, W. Swindell, A. F. Gmitro, and H. H. Barrett, "Incoherent optical processor for x-ray transaxial tomography," Appl. Opt. 20, 264–273 (1981).
- A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, and S. K. Gordon, "Optical computers for reconstructing objects from their x-ray projections," Opt. Eng. 19, 260 (1981).

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