Reconstruction via the intrinsic geometric structures of interior transmission eigenfunctions
This work introduces a new geometric approach for inverse scattering problems, offering a method to recover shape singularities and polyhedral supports, which is a novel direction in the field.
The paper develops a novel inverse scattering scheme using intrinsic geometric properties of interior transmission eigenfunctions to extract geometric structures of an unknown inhomogeneous medium from acoustic far-field measurements. The method efficiently captures cusp singularities and can recover the shape of convex polyhedral supports, with theoretical analysis and numerical validation.
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.