Fourier--Hankel Moment Methods for Topological Counting and Phase-Center Recovery in Acoustic Inverse Scattering

Abstract

We develop a Fourier--Hankel moment framework for extracting topological counting information from full-aperture acoustic far-field data. The method is based on the observation that separated localized components generate distinct phase centers in angular Fourier data. Under the Born approximation, a Bessel--Fourier moment identity shows that suitably scaled row Fourier coefficients form, to leading order, a finite exponential moment sequence. The associated Hankel matrix has rank equal to the number of separated connected components, and the corresponding Hankel pencil recovers their phase-center locations. We prove the exact Hankel rank formula in the phase-center model and establish a perturbation theorem showing stable component counting under a singular-gap condition. We further extend the framework to detectable cavities by introducing a signed phase-center model. In this model, material components and cavities contribute with opposite signs to the moment sequence. The signed Hankel rank counts distinct signed phase centers, and the detectable cavity count is obtained from the excess rank beyond the positive component count. This formulation also identifies an intrinsic degeneracy: cavities whose phase centers coincide with material phase centers, such as perfectly concentric annuli, do not increase the leading signed rank and therefore cannot be detected by the leading phase-center mechanism alone. Numerical experiments validate the proposed theory at several levels: ideal moment sequences, Born far-field data with finite-size components, phase-center location recovery, signed cavity counting, and exact Helmholtz far-field data. The results show that the Fourier--Hankel rank mechanism provides a data-level algebraic approach to component counting and detectable cavity counting, while also making explicit its stability conditions and failure modes.