Optical tomography for small volume absorbing inclusions

We present the asymptotic expansion of the solution to a diffusion equation with a finite number of absorbing inclusions of small volume. We use the first few terms in this expansion measured at the domain boundary to reconstruct the absorption parameters of the inclusions and certain geometrical characteristics. We demonstrate theoretically and numerically that the number of inclusions, their location and their capacity can be reconstructed in a stable way even from moderately noisy data. The reconstruction of the absorption parameter, which is important in optical tomography to discriminate between healthy and unhealthy tissues, requires us however to have far less noisy data. Since the reconstruction of absorption maps from boundary measurements is an extremely ill posed problem, the method of asymptotic expansions of small volume inclusions provides a useful framework to decide which information can be reconstructed from boundary measurements with a given noise level.