Inverse problem in optical diffusion tomography IV Nonlinear inversion formulas

Abstract

We continue our study of the inverse scattering problem for diffuse light. In contrast to our earlier work, in which we considered the linear inverse problem, we now consider the nonlinear problem. We obtain a solution to this problem in the form of a functional series expansion. The first term in this expansion is the pseudoinverse of the linearized forward-scattering operator and leads to the linear inversion formulas that we have reported previously. The higher-order terms represent nonlinear corrections to this result. We illustrate our results with computer simulations in model systems.

Ultra Wideband Wave-Based Linear Inversion in Lossless Ladder Networks

“…A discussion on the range of validity of the weak scattering solution is given in Appendix B. Note that an exact representation of the higher order terms that extends the range of validity of such weak scattering approximations can be obtained by expanding V s n into a series of higher order perturbed contributions within the perturbation theory framework as was carried out for continuous media in [22][23][24] or, alternatively, as in [25].…”

Inverse Born series for diffuse waves

“…where φ = u i − u. Physically, the scattering data φ(x 1 , x 2 ) is proportional to the change in intensity measured by a point detector at x 2 ∈ ∂Ω due to a point source at x 1 ∈ ∂Ω [3]. Each term in the series is multilinear in η and the operator K j is defined by…”

“…Such an approach gives rise to image reconstruction algorithms that, at present, are not well understood mathematically. In previous work, we have shown that, to some extent, it is possible to fill this gap by employing methods which invert the Born series [3]. The resulting image reconstruction algorithms are fast, direct, and have analyzable convergence, stability and approximation error [4].…”

Nonlinear inverse scattering and three-dimensional near-field optical imaging

“…[9][10][11] Though the algebraic structure of series solutions to different ISPs is similar, their analytic structure and convergence properties are quite different. This is a reflection of the underlying physical difference between the shortrange propagation of evanescent waves in NFT and the longrange propagation of waves in quantum mechanics or acoustics.…”