An analytic algorithm for shape reconstruction from low-frequency moments

In the present work, a novel method, concerning the solution of the inverse scattering problem, is developed and implemented, in the realm of low-frequency acoustics. The method is based on the suitable exploitation of the low-frequency moments, which are the structural pieces of the far-field pattern. The stimulus for the present method has been offered by a recent accomplishment permitting the extraction of the moments from the far-field pattern via a systematic, direct, and stable manner. The aim of the method is to reconstruct polynomial scatterers and to approximate general scatterers by polynomial surfaces. This is accomplished via the formulation of suitable objective functionals involving the unknown coefficients of the Cartesian representation of the sought polynomial surface along with the low-frequency moments. These functionals are constructed by forcing the target polynomial surface to comply with the moments extracted from real data. The minimization of these functionals provides the optimized coefficients of the polynomial manifold, while stability is inherent in the nature of the minimization process. The method has been implemented to the reconstruction of second and fourth order polynomial scatterers as well as to fitting of general scatterers by polynomial surfaces. (C) 2011 American Institute of Physics. [doi:10.1063/1.3638140]