Model-Consistent Structured-Hankel Matrix Completion for 3D Sparse Multi-frequency Electromagnetic Source Reconstruction

Abstract

Reconstruction of 3D electromagnetic currents from sparse, multi-frequency, far-field radiation data is severely ill-posed because sparse sampling masks specific current Fourier modes and the missing spectrum enlarges the null space of the forward operator. Although structured-Hankel completion can recover missing spectrum by exploiting the finite rate of innovations structure of a compactly supported geometrically sparse source, it treats current components independently without guaranteeing compatibility with the model. This often renders non-physical solutions that violate model consistency. To resolve this, we propose a training-free Maxwell-model-consistent 3D structured-Hankel framework (MC-Hankel) that alternates joint low-rank Hankel completion with an exact, closed-form projection onto the true Maxwell synthesis subspace while enforcing conjugate symmetry and noise-aware data fidelity. Across tests on two source models, three sparse sampling rates (30\%, 40\%, and 50\% of a Nyquist-sampled grid), and clean versus noisy (10 dB SNR additive white Gaussian noise) conditions, MC-Hankel consistently outperforms sub-sampled Fourier inversion, scale-normalized $\ell_1$-compressed sensing baseline, and the standard joint 3D annihilating filter-based low-rank Hankel matrix completion approach (ALOHA). Across all considered source-noise-sampling configurations, MC-Hankel improves mean peak SNR (PSNR) by 0.65-4.22 dB and tri-planer structural similarity (3D SSIM) by $0.0266$-$0.0703$, reduces relative full-volume $\ell_2$ reconstruction error by $7.4\%$-$38.9\%$, and brings non-physical model residuals down to machine precision over standard joint 3D ALOHA with merely 17\% computational overhead. Exact signed-rank tests are underpowered for five paired trials, so statistical conclusions are reported together with paired bootstrap intervals, effect sizes, and small-sample limitations.