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

Reconstruction via the intrinsic geometric structures of interior transmission eigenfunctions