AI 中文总结
本文针对三维稀疏多频电磁源重建的不适定问题，提出无需训练的MC-Hankel框架，其性能优于多种基线方法，计算开销低且模型一致性好。
AI 中文摘要
从稀疏、多频、远场辐射数据重建三维电磁电流是严重不适定问题，因为稀疏采样会掩盖特定电流傅里叶模式，缺失的频谱会增大正演算子的零空间。尽管结构化汉克尔补全可利用紧支撑几何稀疏源的有限创新率结构恢复缺失频谱，但它独立处理电流分量，无法保证与模型的兼容性，常产生不符合模型一致性的非物理解。为解决该问题，我们提出了一种无需训练的麦克斯韦模型一致三维结构化汉克尔框架（MC-Hankel），该框架交替进行联合低秩汉克尔补全，同时在精确的闭式麦克斯韦综合子空间上投影，且强制共轭对称性和感知噪声的数据保真度。在两种源模型、三种稀疏采样率（奈奎斯特采样网格的30%、40%和50%）以及干净与含噪（10dB SNR加性高斯白噪声）条件下的测试中，MC-Hankel始终优于子采样傅里叶反演、尺度归一化的ℓ₁压缩感知基线，以及标准的基于联合三维湮灭滤波器的低秩汉克尔矩阵补全方法（ALOHA）。在所有考虑的源-噪声-采样配置中，与标准联合三维ALOHA相比，MC-Hankel将平均峰值信噪比（PSNR）提高了0.65-4.22dB，将平面结构相似度（3D SSIM）提高了0.0266-0.0703，将相对全体积ℓ₂重建误差降低了7.4%-38.9%，且仅需17%的计算开销即可将非物理模型残差降至机器精度。精确符号秩检验在五次配对试验中功效不足，因此统计结论与配对自助区间、效应量及小样本限制一同报告。
英文摘要
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.
Comments16 pages, 8 figures, 6 tables
