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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

  • Editor: Stephen A. Burns
  • Vol. 23, Iss. 9 — Sep. 1, 2006
  • pp: 2105–2118

Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function. I. Theory

Angelo Sassaroli, Fabrizio Martelli, and Sergio Fantini  »View Author Affiliations


JOSA A, Vol. 23, Issue 9, pp. 2105-2118 (2006)
http://dx.doi.org/10.1364/JOSAA.23.002105


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Abstract

We approach the perturbative solution to the diffusion equation for the case of absorbing inclusions embedded in a heterogeneous scattering medium by using general properties of the radiative transfer equation and the solution of the Fredholm equation of the second kind given by the Neumann series. The terms of the Neumann series are used to obtain the expression of the moments of the generalized temporal point-spread function derived in transport theory. The moments are calculated independently by using Monte Carlo simulations for validation of the theory. While the mixed moments are correctly derived from the theory by using the solution of the diffusion equation in the geometry of interest, in order to obtain the self moments we should reframe the problem in transport theory and use a suitable solution of the radiative transfer equation for the calculation of the multiple integrals of the corresponding Neumann series. Since the rigorous theory leads to impractical formulas, in order to simplify and speed up the calculation of the self moments, we propose a heuristic method based on the calculation of only a single integral and some scaling parameters. We also propose simple quadrature rules for the calculation of the mixed moments for speeding up the computation of perturbations due to multiple defects. The theory can be developed in the continuous-wave domain, the time domain, and the frequency domain. In a companion paper [ J. Opt. Soc. Am. A 23, 2119 (2006) ] we discuss the conditions of applicability of the theory in practical cases found in diffuse optical imaging of biological tissues.

© 2006 Optical Society of America

OCIS Codes
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(170.5280) Medical optics and biotechnology : Photon migration
(290.7050) Scattering : Turbid media
(300.1030) Spectroscopy : Absorption

ToC Category:
Medical Optics and Biotechnology

History
Original Manuscript: October 25, 2005
Revised Manuscript: March 8, 2006
Manuscript Accepted: March 25, 2006

Virtual Issues
Vol. 1, Iss. 10 Virtual Journal for Biomedical Optics

Citation
Angelo Sassaroli, Fabrizio Martelli, and Sergio Fantini, "Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function. I. Theory," J. Opt. Soc. Am. A 23, 2105-2118 (2006)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-23-9-2105


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