Spherical ESPRIT by Paired Small Circles and Infinitesimal Rotations

Abstract

We study the recovery of a finite three-dimensional point cloud from its spherical Fourier signal subspace at a fixed wavenumber. Two ESPRIT constructions are analyzed. The first uses finite shifts between paired small circles; the second uses angular derivatives on the whole sphere. Both recover the same commuting coordinate matrices, either through their finite exponentials or directly from a coupled linear system. We prove that the continuous generator system is injective for every distinct point cloud and that one guard harmonic degree preserves the retained equations exactly. For $s$ targets, the cutoff $K\ge s$ gives generic finite identifiability, but not uniform conditioning. We also derive a finite-cutoff perturbation bound. The accuracy of the coordinate estimates is a separate issue. A one-target perturbation calculation shows that the original generator estimate can retain a nonzero bias as the wavenumber increases. To address this bias, we use the generator eigenbasis to separate approximate atoms and recover their locations from amplitude-normalized phases. Under pointwise atom dominance, the resulting error is $O(κ^{-1})$; for a dominant two-atom mixture at fixed separation, it is $O(κ^{-2})$. Experiments on structured point clouds with smooth perturbations compare the two constructions and their use as initializers for a common MUSIC refinement.