Data recovery: from limited-aperture to full-aperture

TL;DR

This paper proposes methods to recover full-aperture far field data from limited measurements in inverse scattering, enhancing scatterer reconstruction accuracy by exploiting symmetry and potential theory.

Contribution

It introduces two schemes leveraging reciprocity and Green's formula to recover missing data in limited-aperture inverse scattering problems.

Findings

Partial data recovery is feasible with the proposed methods.

Recovered data improves scatterer reconstruction accuracy.

Numerical tests validate the effectiveness of the schemes.

Abstract

The inverse scattering problems have been popular for the past thirty years. While very successful in many cases, progress has lagged when only {\em limited-aperture} measurement is available. In this paper, we perform some elementary study to recover data that can not be measured directly. To be precise, we aim at recovering the {\em full-aperture} far field data from {\em limited-aperture} measurement. Due to the reciprocity relation, the multi-static response matrix (MSR) has a symmetric structure. Using the Green's formula and single layer potential, we propose two schemes to recover {\em full-aperture} MSR. The recovered data is tested by a recently proposed direct sampling method and the factorization method. The numerical results show the possibility to recover, at least partially, the missing data and consequently improve the reconstruction of the scatterer.