无需对应关系，快速精准对齐旋转数据，抗噪声且支持轴标签和符号混淆。
Fast, Robust, Permutation-and-Sign Invariant SO(3) Pattern Alignment
- 将旋转分解为球面上的基向量，用快速鲁棒匹配对齐各轴方向。
- 在90%极端异常值下仍保持精度，速度比传统方法快6-60倍。
- 适合机器人标定、视觉里程计等存在轴约定不确定的场景。
本文解决SO(3)上两组旋转集的无对应对齐问题，该任务在标定与配准中至关重要，常受时间不同步、异常值及轴约定未知影响。核心思想是将每个旋转分解为三个单位向量（即变换基向量，TBVs），分布在S²球面上，再通过快速鲁棒匹配器（SPMC、FRS及混合方法）对各轴方向进行球面点集对齐。为应对轴重标与符号翻转问题，引入置换与符号不变（PASI）封装器，枚举24种有效带符号排列，通过相关性求和评分，并以投影/卡彻均值融合各轴估计得到最终旋转。整体复杂度保持线性（O(n)），远低于传统球面或SO(3)相关计算的O(N_r³log N_r)。在EuRoC Machine Hall仿真（轴一致）和ETH Hand-Eye基准（robot_arm_real，轴模糊）上的实验表明，本方法准确、速度快6–60倍，且在高达90%异常值下依然鲁棒，全程无需对应搜索。
原文摘要 · Abstract (English)
We address the correspondence-free alignment of two rotation sets on \(SO(3)\), a core task in calibration and registration that is often impeded by missing time alignment, outliers, and unknown axis conventions. Our key idea is to decompose each rotation into its \emph{Transformed Basis Vectors} (TBVs)-three unit vectors on \(S^2\)-and align the resulting spherical point sets per axis using fast, robust matchers (SPMC, FRS, and a hybrid). To handle axis relabels and sign flips, we introduce a \emph{Permutation-and-Sign Invariant} (PASI) wrapper that enumerates the 24 proper signed permutations, scores them via summed correlations, and fuses the per-axis estimates into a single rotation by projection/Karcher mean. The overall complexity remains linear in the number of rotations (\(\mathcal{O}(n)\)), contrasting with \(\mathcal{O}(N_r^3\log N_r)\) for spherical/\(SO(3)\) correlation. Experiments on EuRoC Machine Hall simulations (axis-consistent) and the ETH Hand-Eye benchmark (\texttt{robot\_arm\_real}) (axis-ambiguous) show that our methods are accurate, 6-60x faster than traditional methods, and robust under extreme outlier ratios (up to 90\%), all without correspondence search.
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