Fast, Robust, Permutation-and-Sign Invariant SO(3) Pattern Alignment

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.