Abstract
In this paper, we study the one-dimensional inverse problem for the diffusion equation based optical tomography. The objective of the present work is a mathematical and numerical analysis concerning one-dimensional inverse problem. In the first stage, the forward diffusion equation with boundary conditions is solved using an intermediate elliptic equation. We give the existence and the uniqueness results of the solution. An approximation of the photon density in frequency-domain is proposed using a Splines Galerkin method. In the second stage, we give theoretical results such as the stability and lipschitz-continuity of the forward solution and the Fréchet differentiability of the Dirichlet-to-Neumann nonlinear map with respect to the optical parameters. The Fréchet derivative is used to linearize the considered inverse problem. The Newton method based on the regularization technique will allow us to compute the approximate solutions of the inverse problem. Several test examples are used to verify high accuracy, effectiveness and good resolution properties for smooth and discontinuous optical property solutions.
Highlights
This paper deals with the inverse problem in one-dimensional optical tomography
We present some theoretical results concerning the continuity of the forward solutions and Dirichlet-to-Neumann (DtN) map with respect to the optical parameters, Lipschitz-continuous property of the DtN map, the adjoint diffusion problem and the Frechet derivative
We presented an non-linear inverse problem for one-dimensional diffusion transport equation with Robin boundary conditions
Summary
This paper deals with the inverse problem in one-dimensional optical tomography. This problem consists of the estimation of the optical properties, namely the absorption coefficient μa, the diffusion coefficient D and the refractive index ν, from the flow measurements at the boundary of the computational domain. We present some theoretical results concerning the continuity of the forward solutions and Dirichlet-to-Neumann (DtN) map with respect to the optical parameters, Lipschitz-continuous property of the DtN map, the adjoint diffusion problem and the Frechet derivative. Such derivative allows us to obtain a linearized problem from the nonlinear inverse problem. This guarantees the stability of the solution of the forward diffusion problem in terms of the Robin boundary conditions.
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