Clifford Nolan: Applications of Microlocal Analysis to Imaging

We will look at the scattering operator that maps high-frequency perturbations of a coefficient in the wave equation to the corresponding high-frequency perturbed (scattered) wave field. We show in the context of geophysics and radar, that the scattering operator is a Fourier integral operator (FIO). Then we examine under what circumstances the scattering operator may be inverted (an inverse problem). We find that it is rarely invertible and that the best we can do is recover a reasonable estimate of the perturbed coefficient along with artifacts. We will describe these artifacts from both a qualitative and quantitative point of view.