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

arXiv: https://arxiv.org/abs/2512.00659 · HTML: https://mlsanity.com/paper/2512.00659/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 57 references that are in the map.
- First posted on arXiv 29 Nov 2025.

## Builds on (most cited first, 12 of 12)

- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [VINS-Mono: A Robust and Versatile Monocular Visual-Inertial State Estimator](https://mlsanity.com/paper/1708.03852/) (2017)
- [Spherical Regression: Learning Viewpoints, Surface Normals and 3D Rotations on n-Spheres](https://mlsanity.com/paper/1904.05404/) (2019)
- [A Quaternion-based Certifiably Optimal Solution to the Wahba Problem with Outliers](https://mlsanity.com/paper/1905.12536/) (2019)
- [Scaling Spherical CNNs](https://mlsanity.com/paper/2306.05420/) (2023)
- [Solving the Robot-World Hand-Eye(s) Calibration Problem with Iterative Methods](https://mlsanity.com/paper/1907.12425/) (2019)
- [Semidefinite descriptions of the convex hull of rotation matrices](https://mlsanity.com/paper/1403.4914/) (2014)
- [Accurate calibration of multi-perspective cameras from a generalization of the hand-eye constraint](https://mlsanity.com/paper/2202.00886/) (2022)
- [A Graph-based Optimization Framework for Hand-Eye Calibration for Multi-Camera Setups](https://mlsanity.com/paper/2303.04747/) (2023)
- [A Certifably Correct Algorithm for Generalized Robot-World and Hand-Eye Calibration](https://mlsanity.com/paper/2507.23045/) (2025)
- [Extrinsic Infrastructure Calibration Using the Hand-Eye Robot-World Formulation](https://mlsanity.com/paper/2305.01407/) (2023)
- [Correspondence-Free Fast and Robust Spherical Point Pattern Registration](https://mlsanity.com/paper/2508.02339/) (2025)

## Abstract (arXiv)

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 Transformed Basis Vectors (TBVs)-three unit vectors on (S2)-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 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 ((O(n))), contrasting with (O(Nr3log Nr)) for spherical/(SO(3)) correlation. Experiments on EuRoC Machine Hall simulations (axis-consistent) and the ETH Hand-Eye benchmark (textttrobotarmreal) (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.