Multi-frequency far-field data enrichment for electromagnetic source reconstruction

Reconstructing unknown electromagnetic sources from far-field radiation patterns is a fundamental inverse problem with broad applications in biomedical imaging, non-destructive testing, and telecommunications. In practical settings, however, collecting dense multi-frequency far-field measurements at the Nyquist sampling rate is often infeasible. Under-sampled or sparse data introduce non-radiating source components that sever the uniqueness of the solution, creating severe artifacts when standard inversion techniques are applied. To overcome this limitation, we present a two-stage reconstruction strategy exploiting the physical property that compactly supported, geometrically sparse sources exhibit a finite rate of innovations (FRI). In the first stage, we construct an associated wrap-around structured Hankel matrix. By leveraging the low-rank property of the matrix due to FRI of the unknown sources, we enrich the sub-sampled data. To that end, we convert missing multi-frequency far-field data recovery into a constrained matrix completion task solved via Annihilating Filter-based Low-rank Hankel Matrix Completion Approach (ALOHA). In the second stage, a Fourier inversion scheme reconstructs the current source density from the enriched dataset. Extensive numerical evaluations on electromagnetic source models show that our enrichment framework effectively eliminates under-sampling artifacts and resolves non-uniqueness challenges. The method delivers accurate and stable reconstructions under high sub-sampling rates (e.g., with $30$\% to $50$\% available samples) and strong noise conditions ($10$ dB SNR), outperforming standard $\ell_1$-compressed sensing baselines.