An inverse problem for one-dimensional diffusion equation in optical tomography

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.

Keywords

• Diffusion transport problem
• Fourier transform
• Nonlinear inverse problem
• Newton method
• Tikhonov regularization, Spline basis functions
• Optical tomography