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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

  • Vol. 18, Iss. 9 — Sep. 1, 2001
  • pp: 2132–2137

Relation between computed tomography and diffraction tomography

Greg Gbur and Emil Wolf  »View Author Affiliations


JOSA A, Vol. 18, Issue 9, pp. 2132-2137 (2001)
http://dx.doi.org/10.1364/JOSAA.18.002132


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Abstract

The relationship between computed tomography (CAT) and diffraction tomography (DT) is investigated. A simple condition with a clear physical meaning is derived for the applicability of CAT. Corrections due to scattering are incorporated into CAT, and it is shown that the effect of scattering may be characterized by a two-dimensional fractional Fourier transform. The implications of these results for the three-dimensional imaging of weakly scattering objects are also discussed.

© 2001 Optical Society of America

OCIS Codes
(110.6960) Imaging systems : Tomography
(290.3200) Scattering : Inverse scattering

History
Original Manuscript: November 27, 2000
Revised Manuscript: February 21, 2001
Manuscript Accepted: February 21, 2001
Published: September 1, 2001

Citation
Greg Gbur and Emil Wolf, "Relation between computed tomography and diffraction tomography," J. Opt. Soc. Am. A 18, 2132-2137 (2001)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-18-9-2132


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References

  1. M. Born, E. Wolf, Principles of Optics, 7th, expanded ed. (Cambridge U. Press, Cambridge, UK, 1999).
  2. G. T. Herman, Image Reconstruction from Projections (Academic, Orlando, Fla., 1980).
  3. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
  4. E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), pp. 83–110.
  5. Very recently a method somewhat analogous to DT was proposed that circumvents the need for measurement of the phase. Instead, one measures the power scattered by the object when it is illuminated simultaneously by pairs of plane waves [P. S. Carney, E. Wolf, G. S. Agarwal, “Diffraction tomography using power extinction measurements,” J. Opt. Soc. Am. A 16, 2643–2648 (1999)]. However, as yet the method has not been tested experimentally. [CrossRef]
  6. This question was briefly addressed by A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. 6, 374–376 (1981). [CrossRef] [PubMed]
  7. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  8. Equation (17) is equivalent to Eq. (8) of Sec. 4.11 of Ref. 1.
  9. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960), Chap. 2, Sec. 7.
  10. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980). [CrossRef]
  11. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987). [CrossRef]
  12. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994). [CrossRef] [PubMed]
  13. D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, H. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424–2431 (1995). [CrossRef]
  14. D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996). [CrossRef]
  15. For a thorough discussion of the Rayleigh range see, for instance, J. F. Ramsey, “Tubular beams from radiating apertures,” in Advances in Microwaves, L. F. Young, ed. (Academic, New York, 1968), Vol. 3, pp. 127–221.
  16. We have neglected the contribution of the evanescent waves [p2+q2>1in Eq. (6)], because with our choices of the values of kσ0and kd,the contribution of the evanescent waves will be negligible.
  17. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), pp. 667–669.

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