  In this paper, we consider the inverse shape problem of recovering small and
extended isotropic scatterers with a conductive boundary condition. Here, we
assume that the measured far-field data is known at a fixed wave number. We
will provide qualitative reconstruction methods for recovering either small or
extended scatterers. For the case of small scatterers, we model this by a
region(possibly with multiple components) with small volume. We derive an
asymptotic expansion for the far-field pattern which will allow us to study the
MUSIC algorithm for solving the problem. In the case of an extended scatterer,
we derive a new factorization of the far-field operator. This is then used to
provide the resolution analysis for a direct sampling method. The theoretical
results are verified with some numerical experiments in 2--dimensions.

Analysis of PDEs