  In this paper, we are concerned with the recovery of the geometric shapes of
inhomogeneous inclusions from the associated far field data in electrostatics
and acoustic scattering. We present a local resolution analysis and show that
the local shape around a boundary point with a high magnitude of mean curvature
can be reconstructed more easily and stably. In proving this, we develop a
novel mathematical scheme by analyzing the generalized polarisation tensors
(GPTs) and the scattering coefficients (SCs) coming from the associated
scattered fields, which in turn boils down to the analysis of the layer
potential operators that sit inside the GPTs and SCs via the microlocal
analysis. In a delicate and subtle manner, we decompose the reconstruction
process into several steps, where all but one steps depend on the global
geometry, and one particular step depends on the mean curvature at a given
boundary point. Then by a sensitivity analysis with respect to local
perturbations of the curvature of the boundary surface, we establish the local
resolution effects. Our study opens up a new field of mathematical analysis on
wave super-resolution imaging.
