The reconstruction of multipolar acoustic or electromagnetic sources from their far-field signature plays a crucial role in numerous applications. Most of the existing techniques require dense multi-frequency data at the Nyquist sampling rate. The availability of a sub-sampled grid contributes to the null space of the inverse source-to-data operator, which causes significant imaging artifacts. For this purpose, additional knowledge about the source or regularization is required. In this letter, we propose a novel two-stage strategy for multipolar source reconstruction from sub-sampled sparse data that takes advantage of the sparsity of the sources in the physical domain. The data at the Nyquist sampling rate is recovered from sub-sampled data and then a conventional inversion algorithm is used to reconstruct sources. The data recovery problem is linked to a spectrum recovery problem for the signal with the \textit{finite rate of innovations} (FIR) that is solved using an annihilating filter-based structured Hankel matrix completion approach (ALOHA). For an accurate reconstruction, a Fourier inversion algorithm is used. The suitability of the approach is supported by experiments.

翻译：多极声源或电磁源从其远场特征进行重建在众多应用中发挥着关键作用。现有大多数技术需要以奈奎斯特采样率获取密集的多频数据。欠采样网格的存在会导致源到数据逆算子的零空间，从而产生显著的成像伪影。为此，需要额外的源信息或正则化处理。本文提出一种新颖的两阶段策略，利用物理域中源的稀疏性，从欠采样稀疏数据中重建多极源。首先从欠采样数据中恢复奈奎斯特采样率下的数据，随后采用传统反演算法重建源。该数据恢复问题可转化为具有有限创新率(FIR)信号的频谱恢复问题，通过基于湮灭滤波器的结构化汉克尔矩阵补全方法(ALOHA)求解。为实现精确重建，采用傅里叶反演算法。实验结果验证了该方法的适用性。