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

2025-11-29

AI Summary

This paper addresses the robust alignment of two rotation sets in SO(3) under challenging conditions: no point-wise correspondences, temporal asynchrony, high outlier ratios (up to 90%), and inconsistent axis conventions. We propose the Permutation- and Sign-Invariant (PASI) framework, which decomposes rotations into spherical basis vectors and achieves axis-level decoupled matching via exhaustive enumeration of the 24 valid sign permutations—bypassing conventional correspondence search. PASI integrates weighted correlation scoring, spherical point-set matching (SPMC/FRS), and projection-based or Karcher mean estimation to ensure globally consistent alignment. The algorithm exhibits linear time complexity, achieving 6–60× speedup over state-of-the-art methods. It requires neither initial correspondences nor temporal synchronization. Extensive experiments on synthetic and real-world data demonstrate significant accuracy improvements over baseline approaches.

Technology Category

Intelligent Robots: Multimodal Perception & Sensor FusionPlanning, Routing, and Scheduling: Optimization of Spatio-temporal SystemsConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deploymentsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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. 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^3log N_r)) for spherical/(SO(3)) correlation. Experiments on EuRoC Machine Hall simulations (axis-consistent) and the ETH Hand-Eye benchmark ( exttt{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.

Problem

• Aligns rotation sets without correspondence on SO(3)
• Handles missing alignment, outliers, and axis convention ambiguity
• Achieves fast, robust, permutation-and-sign invariant matching

Innovation

• Decompose rotations into transformed basis vectors for alignment
• Use permutation-and-sign invariant wrapper to handle axis relabels
• Achieve linear complexity with robust matchers and correlation scoring