  In this paper we demonstrate a computational method to solve the inverse
scattering problem for a star-shaped, smooth, penetrable obstacle in 2D. Our
method is based on classical ideas from computational geometry. First, we
approximate the support of a scatterer by a point cloud. Secondly, we use the
Bayesian paradigm to model the joint conditional probability distribution of
the non-convex hull of the point cloud and the constant refractive index of the
scatterer given near field data. Of note, we use the non-convex hull of the
point cloud as spline control points to evaluate, on a finer mesh, the volume
potential arising in the integral equation formulation of the direct problem.
Finally, in order to sample the arising posterior distribution, we propose a
probability transition kernel that commutes with affine transformations of
space. Our findings indicate that our method is reliable to retrieve the
support and constant refractive index of the scatterer simultaneously. Indeed,
our sampling method is robust to estimate a quantity of interest such as the
area of the scatterer. We conclude pointing out a series of generalizations of
our method.
