球面ESPRIT：成对小圆与无穷小旋转
Spherical ESPRIT by Paired Small Circles and Infinitesimal Rotations
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中文总结 AI 辅助
本文研究从球面傅里叶信号子空间恢复三维点云，提出两种ESPRIT构造方法，证明其可识别性并推导误差界，通过特征基分离原子以消除偏差，实验验证了方法的有效性。
中文摘要 AI 辅助
我们研究了在固定波数下，从球面傅里叶信号子空间中恢复有限三维点云的问题。分析了两种ESPRIT构造。第一种利用成对小圆之间的有限位移；第二种利用整个球面上的角导数。两种方法都恢复相同的交换坐标矩阵，要么通过其有限指数，要么直接从耦合线性系统获得。我们证明了连续生成器系统对每个不同的点云是单射的，并且一个保护谐波次数精确地保留所保留的方程。对于$s$个目标，截止值$K\ge s$给出一般的有限可识别性，但不保证均匀条件数。我们还推导了有限截止扰动界。坐标估计的精度是另一个问题。一个单目标的扰动计算表明，原始生成器估计在波数增加时可能保留非零偏差。为了解决这一偏差，我们使用生成器特征基来分离近似原子，并从幅度归一化相位中恢复其位置。在逐点原子主导下，所得误差为$O(\kappa^{-1})$；对于固定分离下的主导双原子混合物，误差为$O(\kappa^{-2})$。在具有平滑扰动的结构化点云上的实验比较了两种构造及其作为常见MUSIC细化初始化的用途。
英文摘要
We study the recovery of a finite three-dimensional point cloud from its spherical Fourier signal subspace at a fixed wavenumber. Two ESPRIT constructions are analyzed. The first uses finite shifts between paired small circles; the second uses angular derivatives on the whole sphere. Both recover the same commuting coordinate matrices, either through their finite exponentials or directly from a coupled linear system. We prove that the continuous generator system is injective for every distinct point cloud and that one guard harmonic degree preserves the retained equations exactly. For $s$ targets, the cutoff $K\ge s$ gives generic finite identifiability, but not uniform conditioning. We also derive a finite-cutoff perturbation bound. The accuracy of the coordinate estimates is a separate issue. A one-target perturbation calculation shows that the original generator estimate can retain a nonzero bias as the wavenumber increases. To address this bias, we use the generator eigenbasis to separate approximate atoms and recover their locations from amplitude-normalized phases. Under pointwise atom dominance, the resulting error is $O(κ^{-1})$; for a dominant two-atom mixture at fixed separation, it is $O(κ^{-2})$. Experiments on structured point clouds with smooth perturbations compare the two constructions and their use as initializers for a common MUSIC refinement.
