Estimation of optical absorption in anisotropic background

Abstract

In this paper we present a model for anisotropic light propagation and reconstructions of optical absorption coefficient in the presence of anisotropies. To model the anisotropies, we derive the diffusion equation in an anisotropic case, and present the diffusion matrix as an eigenvalue decomposition. The inverse problem considered in this paper is to estimate the optical absorption when the directions of anisotropy are known, but the strength may vary. To solve this inverse problem, two approaches are taken. First, we assume that the strength of anisotropy is known, and compare maximum a posteriori reconstructions using a fixed value for the strength when the value for the strength is both correct and incorrect. We then extend the solution to allow an uncertainty of the strength of the anisotropy by choosing a prior distribution for the strength and calculating the marginal posterior density. Numerical examples of maximum a posteriori estimates are again presented. The results in this paper suggest that the anisotropy of the body is a property that cannot be ignored in the estimation of the absorption coefficient.

Hybrid model of light propagation in random media based on the time-dependent radiative transfer and diffusion equations

“…condition is reduced to the Robin boundary condition within the DA [22]. For two dimensional cases, it is…”

Finite element model for the coupled radiative transfer equation and diffusion approximation

“…Derivations of the DA have been presented in numerous papers. Here a short review of the derivation is given according to References [16,29].…”

“…where n is a dimension-dependent constant which obtains values 2 = 1/ and 3 = 1/4 and A is a parameter governing the internal reflection at the boundary * [29].…”

Analysis of Optical Tomography With Non-Scattering Regions

“…The diffusion approximation, as it is given in Definition 2.2, is of course very abstract. With a little work [8], however, equation (2.6) can be written out explicitly. The following lemma presents the resulting equation.…”

“…Because the diffusion approximation is only an approximation, one cannot really say which convention is the correct one. The expression given in Lemma 2.5 is, however, carefully conducted from the mathematical model described by Definition 2.2 [8], and so it is one possible choice.…”