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

Abstract--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. T o handle axis relabels and sign flips, we introduce a 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. Experiments on EuRoC Machine Hall simulations (axis-consistent) and the ETH Hand-Eye benchmark (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. Estimating the 3D rotation that aligns one sensor or object frame to another is a fundamental problem in robotics and computer vision. Closed-form or least-squares solutions (e.g., Davenport/QUEST, SVD/Procrustes, and modern quaternion solvers) are mature [25], [26], [27], [28], [29], but they typically assume paired measurements (known correspondences) and degrade under heavy outliers or axis-convention mismatches.