The photon diffusion equation: forward and inverse problems

We compute the solution to the photon diffusion equation using a state-space system. The state-space system under consideration is obtained by applying a FEM technique to the photon diffusion equation. Computing the photon density in the medium is commonly called in computational tomography, the forward problem. We investigate and present some preliminary results on the inverse problem. The inverse problem entails the identification of anomalies (benign or malignant) inside the medium. We investigate the inverse problem using an optimization setup. Specifically, we use a baseline homogeneous experiment that represents the healthy tissue to compare with the actual data. We then form an error signal between the heterogeneous and homogeneous case, which we try to minimize over the absorption coefficient of the true medium. We minimize it using the two-norm of the error signal.