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Inverse problem in optical diffusion tomography. IV. Nonlinear inversion formulas.

Vadim A Markel1, Joseph A O'Sullivan, John C Schotland

  • 1Department of Radiology, Washington University, St. Louis, Missouri 63110, USA. vmarkel@altai.wustl.edu

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
|May 16, 2003
PubMed
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This study addresses the nonlinear inverse scattering problem for diffuse light. Researchers developed a functional series expansion solution, incorporating nonlinear corrections to previous linear inversion methods.

Area of Science:

  • Optics and photonics
  • Applied mathematics
  • Computational physics

Background:

  • Previous work focused on the linear inverse scattering problem for diffuse light.
  • The current study extends this to the more complex nonlinear inverse scattering problem.
  • Understanding light scattering is crucial in various fields, including medical imaging and materials science.

Purpose of the Study:

  • To develop and present a solution for the nonlinear inverse scattering problem for diffuse light.
  • To introduce a functional series expansion method for solving this nonlinear problem.
  • To compare the nonlinear solution with existing linear inversion techniques.

Main Methods:

  • Formulated the nonlinear inverse scattering problem.
  • Derived a solution using a functional series expansion.

Related Experiment Videos

  • The first term of the expansion corresponds to the pseudoinverse of the linearized forward-scattering operator.
  • Higher-order terms in the expansion account for nonlinear corrections.

    • Main Results:

    • A novel solution for the nonlinear inverse scattering problem was obtained.
    • The solution is presented as a functional series expansion.
    • The initial term of the series recovers previously established linear inversion formulas.
    • Subsequent terms provide corrections for nonlinear scattering effects.

    Conclusions:

    • The functional series expansion offers a comprehensive approach to the nonlinear inverse scattering problem.
    • This method effectively incorporates nonlinear scattering phenomena, improving upon linear approximations.
    • Computer simulations in model systems validated the derived theoretical results.