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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 19 — Sep. 18, 2006
  • pp: 8837–8848

A family of approximations spanning the Born and Rytov scattering series

Daniel L. Marks  »View Author Affiliations

Optics Express, Vol. 14, Issue 19, pp. 8837-8848 (2006)

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A new hybrid scattering series is derived that incorporates as special cases both the Born and Rytov scattering series, and includes a parameter so that the behavior can be continuously varied between the two series. The parameter enables the error to be shifted between the Born and Rytov error terms to improve accuracy. The linearized hybrid approximation is derived as well as its condition of validity. Higher order terms of the hybrid series are also found. Also included is the integral equation that defines the exact solution to the forward scattering problem as well as its Fréchet derivative, which is used for the solution of inverse multiple scattering problems. Finally, the linearized hybrid approximation is demonstrated by simulations of inverse scattering off of uniform circular cylinders, where it is shown that the hybrid approximation achieves smaller error than either the Born or Rytov approximations alone.

© 2006 Optical Society of America

OCIS Codes
(100.3190) Image processing : Inverse problems
(110.2990) Imaging systems : Image formation theory
(290.3200) Scattering : Inverse scattering

ToC Category:

Original Manuscript: June 26, 2006
Revised Manuscript: August 18, 2006
Manuscript Accepted: August 19, 2006
Published: September 18, 2006

Daniel L. Marks, "A family of approximations spanning the Born and Rytov scattering series," Opt. Express 14, 8837-8848 (2006)

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  1. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, NJ, 1995).
  2. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
  3. G. Beylkin and M. L. Oristaglio, "Distorted-wave Born and distorted-wave Rytov approximations," Opt. Commun. 53, 213-216 (1985). [CrossRef]
  4. S. D. Rajan and G. V. Frisk, "A comparison between the Born and Rytov approximations for the inverse backscattering problem," Geophysics 54, 864-871 (1989). [CrossRef]
  5. M. J. Woodward, "Wave-equation tomography," Geophysics 57, 15-26 (1992). [CrossRef]
  6. M. I. Sancer and A. D. Varvatsis, "A comparison of the Born and Rytov methods," Proc. IEEE 58, 140-141 (1970). [CrossRef]
  7. M. Slaney, A. C. Kak, and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. MTT-32, 860-874 (1984). [CrossRef]
  8. F. C. Lin and M. A. Fiddy, "The Born-Rytov controversy: I. Comparing the analytical and approximate expressions for the one-dimensional deterministic case," J. Opt. Soc. Am. A 9, 1102-1110 (1992). [CrossRef]
  9. F. C. Lin and M. A. Fiddy, "Born-Rytov controversy: II Applications to nonlinear and stochastic scattering problems in one-dimensional half-space media," J. Opt. Soc. Am. A 10, 1971-1983 (1993). [CrossRef]
  10. G. Gbur and E. Wolf, "Relation between computed tomography and diffraction tomography," J. Opt. Soc. Am. A 18, 2132-2137 (2001). [CrossRef]
  11. Z. Q. Lu, "Multidimensional structure diffraction tomography for varying object orientation through generalized scattered waves," Inv. Prob. 1, 339-356 (1985). [CrossRef]
  12. Z.-Q. Lu, "JKM Perturbation Theory, Relaxation Perturbation Theory, and their Applications to Inverse Scattering: Theory and Reconstruction Algorithms," IEEE Trans. Ultra. Ferroelectr. Freq. Control UFFC-32, 722-730 (1986).
  13. G. A. Tsihrintzis and A. J. Devaney, "Higher order (nonlinear) diffraction tomography: inversion of the Rytov series," IEEE Trans. Inf. Theory 46, 1748-1761 (2000). [CrossRef]
  14. W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1990). [CrossRef] [PubMed]
  15. R. E. Kleinman and P. M. van der Berg, "A modified gradient method for two-dimensional problems in tomography," J. Comput. Appl. Math 42, 17-35 (1992). [CrossRef]
  16. R. E. Kleinman and P. M. van der Berg, "An extended-range modified gradient technique for profile inversion," Radio Sci. 28, 877-884 (1993). [CrossRef]
  17. K. Belkebir and A. G. Tijhuis, "Modified gradient method and modified Born method for solving a two dimensional inverse scattering problem," Inv. Prob. 17, 1671-1688 (2001). [CrossRef]
  18. G. A. Tsihrintzis and A. J. Devaney, "Higher-order (Nonlinear) Diffraction Tomography: Reconstruction Algorithms and Computer Simulation," IEEE Trans. Image Process. 9, 1560-1572 (2000). [CrossRef]
  19. G. A. Tsihrintzis and A. J. Devaney, "A Volterra series approach to nonlinear traveltime tomography," IEEE Trans. Geosco. Remote Sens. 38, 1733-1742 (2000). [CrossRef]
  20. A. J. Devaney, "A filtered back propagation algorithm for diffraction tomography," Ultrason. Imaging 4, 336-350 (1982). [CrossRef] [PubMed]
  21. V. A. Markel, J. A. O’Sullivan, and J. C. Schotland, "Inverse problem in optical diffusion tomography. IV. Nonlinear inversion formulas," J. Opt. Soc. Am. A 20, 903-912 (2003). [CrossRef]
  22. M. Slaney, "Diffraction Tomography Algorithms from Malcolm Slaney PhD Dissertation," obtained from http://rvl4.ecn.purdue.edu/˜malcolm/purdue/diffract.tar.Z.

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