Inverse scattering via low-frequency moments

Abstract

An acoustically soft scatterer defined by a closed and star shape polynomial surface of any degree disturbs the propagation of a time harmonic plane incident wave. It is demonstrated that, under the hypotheses of Schiffer’s uniqueness theorem, all the generalized low-frequency moments corresponding to the capacity potential can be obtained from the scattering amplitude. An analytic algorithm is proposed that recovers the geometry of the body whenever a finite number of generalized moments generated by the leading low-frequency approximation are given. What is striking here is the fact that a surface measure generated by a potential problem is enough to recover the geometry of the scatterer. The idea here is to relate the given moments to a set of particular combined spherical moments that appear as coefficients of an algebraic linear system, whose solution provides the coefficients of the scattering surface in spherical harmonics. This is done with the help of an inner product defined over the surface of the unit sphere with respect to an unknown positive surface measure. In contrast to other existing techniques of shape reconstruction, the one proposed here does not involve the solution of any optimization problem. Instead, only some finite expansions in spherical harmonics and the solution of a linear algebraic system is involved. Tikhonov regularization is used to treat the case of inexact data. The proposed method is illustrated in the case of second degree surfaces where exact analytical data are available.