In this paper we address an inverse problem for the time-dependent linear
transport equation (or radiative transfer equation) in 2D having in mind applications in Diffuse Optical Tomography (DOT). We propose two new reconstruction algorithms which so far have not been applied to such a situation and compare their performances in certain practically relevant situations. The first of these reconstruction algorithms uses a sparsity promoting regularization scheme, whereas the second one uses a simultaneous level set reconstruction scheme for two parameters of the linear transport equation. We will also compare the results of both schemes with a third scheme which is a more traditional L2-based Landweber-Kaczmarz scheme. We focus our attention
on the DOT application of imaging the human head of a neonate where the simpler diffusion approximation is not well-suited for the inversion due to the presence of a clear layer beneath the skull which is filled with ‘low-scattering’ cerebrospinal fluid. This layer, even if its location and characteristics are known a-priori, poses significant difficulties for most reconstruction schemes due to its ‘wave-guiding’ property which reduces sensitivity of the data to the interior regions. A further complication arises due to the necessity to reconstruct simultaneously two different parameters of the linear transport equation, the scattering and the absorption cross-section, from the same data set. A significant ‘cross-talk’ between these two parameters is usually expected.
Our numerical experiments indicate that each of the three considered reconstruction schemes do have their merits and perform differently but reasonably well when the clear layer is a-priori known. We also demonstrate the behaviour of the three algorithms in the particular situation where the clear layer is unknown during the reconstruction.