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We develop a novel wave imaging scheme for reconstructing the shape of aninhomogeneous scatterer and we consider the inverse acoustic obstaclescattering problem as a prototype model for our study. There exists a wealth ofreconstruction methods for the inverse obstacle scattering problem and many ofthem intentionally avoid the interior resonant modes. Indeed, the occurrence ofthe interior resonance may cause the failure of the correspondingreconstruction. However, based on the observation that the interior resonantmodes actually carry the geometrical information of the underlying obstacle, wepropose an inverse scattering scheme of using those resonant modes for thereconstruction. To that end, we first develop a numerical procedure indetermining the interior eigenvalues associated with an unknown obstacle fromits far-field data based on the validity of the factorization method. Then wepropose two efficient optimization methods in further determining thecorresponding eigenfunctions. Using the afore-determined interior resonantmodes, we show that the shape of the underlying obstacle can be effectivelyrecovered. Moreover, the reconstruction yields enhanced imaging resolution,especially for the concave part of the obstacle. We provide rigoroustheoretical justifications for the proposed method. Numerical examples in 2Dand 3D verify the theoretically predicted effectiveness and efficiency of themethod.

On a novel inverse scattering scheme using resonant modes with enhanced imaging resolution