Inverse solutions for the diffusion equation

In this paper, we first introduce the concept of computational tomography (CT) for the photon diffusion equation. The forward and the inverse problems are concerned with the diffusion equation, while the solution to the later one is the goal of research in optical CT. The inverse problem can be stated as follows: given the photon density measured from the detectors outside the tissue, we need to find the anomalies (benign or malignant) inside the tissue. We model the forward and the inverse problem using state-space equations and formulate the inverse problem as a system identification problem. Different approaches, namely the nonlinear optimization approach and the nonlinear filtering approach are proposed to solve the inverse problem. Simulation results of different inverse solvers are presented and compared using a real problem of medical image reconstruction