Imaging by inverse wave scattering is a multidisciplinary area implying concepts from both physics, mathematics and signal processing. Current approaches to the problem fall into two categories : inverse obstacle problem and inverse medium problem. In the former, the scattering object is a homogeneous obstacle one wants to determine from the field on the boundary. In the latter, the scattering object is an inhomogeneous medium with respect to some physical parameters. The inverse problem then consists in estimating those parameters from the field on some boundary. The talk will be focusing on the second problem. Such problem is usually not only non-linear but can also be severely ill-posed. Moreover, obtaining accurate solutions usually requires huge datasets. In the light of recent developments in the compressive sensing (CS) theory together with the increased computing capabilities, we try to obtain high accuracy images of an object with respect to variations in the physical parameters. This is done using somehow advanced understanding of the wave propagation physics while at the same time looking for a decrease in the complexity of current methods.