  We are concerned with the inverse scattering problem of extracting the
geometric structures of an unknown/inaccessible inhomogeneous medium by using
the corresponding acoustic far-field measurement. Using the intrinsic geometric
properties of the so-called interior transmission eigenfunctions, we develop a
novel inverse scattering scheme. The proposed method can efficiently capture
the cusp singularities of the support of the inhomogeneous medium. If further a
priori information is available on the support of the medium, say, it is a
convex polyhedron, then one can actually recover its shape. Both theoretical
analysis and numerical experiments are provided. Our reconstruction method is
new to the literature and opens up a new direction in the study of inverse
scattering problems.
