Inverse electromagnetic source scattering problems with multi-frequency sparse phased and phaseless far field data

TL;DR

This paper develops methods for unique shape reconstruction and phase retrieval in inverse electromagnetic source problems using multi-frequency sparse far field data, addressing challenges with phaseless data and demonstrating robustness with noisy 3D examples.

Contribution

It introduces novel phase retrieval and direct sampling techniques for electromagnetic source reconstruction from sparse multi-frequency data, including phaseless measurements.

Findings

Unique source support determination with phased data.

Stable phase retrieval from limited measurements.

Effective reconstruction methods validated with noisy 3D data.

Abstract

This paper is concerned with uniqueness, phase retrieval and shape reconstruction methods for solving inverse electromagnetic source scattering problems with multi-frequency sparse phased or phaseless far field data. With the phased data, we show that the smallest strip containing the source support with the observation direction as the normal can be uniquely determined by the multi-frequency far field pattern at a single observation direction. A phased direct sampling method is also proposed to reconstruct the strip. The phaseless far field data is closely related to the outward energy flux, which can be measured more easily in practice. We show that the phaseless far field data is invariant under the translation of the sources, which implies that the location of the sources can not be uniquely recovered by the data. To solve this problem, we consider simultaneously the scattering of magnetic dipoles with one fixed source point and at most three scattering strengths. With this technique, a fast and stable phase retrieval approach is proposed based on a simple geometrical result which provides a stable reconstruction of a point in the plane from three distances to the given points. A novel phaseless direct sampling method is also proposed to reconstruct the strip. The phase retrieval approach can also be combined with the phased direct sampling method to reconstruct the strip. Finally, to obtain a source support, we just need to superpose the indicators with respect to the sparse observation directions. Extended numerical examples in three dimensions are conducted with noisy data, and the results further verify the effectiveness and robustness of the proposed phase retrieval approach and direct sampling methods.