Off-Grid Point-Scatterer Localization from Sparse Limited-Aperture Data via Hankel Completion

Abstract

We study the localization of $s$ distinct planar point scatterers from sparsely sampled single-incidence, single-frequency acoustic far-field data over a limited receiver aperture. Under the first Born approximation, the coherent data form a two-dimensional off-grid exponential sum and admit a low-rank Hankel representation. A limited aperture generally admits multiple aperture-admissible Hankel pencils with different lifted dimensions, sampling multiplicities, and Vandermonde conditioning. Hence, the choice of the Hankel pencil affects both data completion and subsequent localization. In this work, we introduce a unified framework for structured completion and off-grid localization based on general aperture admissible Hankel pencils. We establish exact recovery in the noiseless case and stable recovery under bounded noise from a number of random aperture samples that is linear in $s$, up to factors determined by the pencil geometry, source conditioning, and logarithmic terms, together with an explicit localization bound in terms of the completion error. The analysis separates the contribution of the pencil geometry from the conditioning induced by the source configuration and provides verifiable sufficient conditions for stable completion and MUSIC resolution. These results yield quantitative criteria for selecting sample efficient Hankel pencils under a prescribed physical aperture. Numerical experiments on connected and disconnected apertures demonstrate the effects of pencil geometry, sampling, and noise.