Rubix: Global Correspondence-Free Point Set Alignment through Assignment Geometry

2026-10-07

AI Summary

This study addresses the susceptibility of alternating minimization to local optima and its computational inefficiency in correspondence-free point set alignment. We propose a global optimization method based on support vectors derived from the convex hull vertices of permuted polygons. By proving a tight bound of $n(n-1)$ vertices, we resolve an open problem posed by Rote. Integrating the Procrustes-Wasserstein framework with a branch-and-bound algorithm, our approach achieves exact solutions in 2D and extends naturally to 3D. Evaluated on the MPEG-7 benchmark, the method requires only 12ms on average, achieving a 50-fold speedup over grid search while delivering superior accuracy. These improvements substantially enhance shape retrieval performance, demonstrating both theoretical rigor and practical efficiency for robust point set registration.

Abstract

Procrustes-Wasserstein alignment jointly estimates a matching and rotation without supplied correspondences, but alternating minimization can stop at suboptimal solutions. Rubix solves the equally weighted planar problem globally under squared Euclidean loss. Each matching $σ$ of two centered $n$-point sets defines a complex correlation $z_σ=\sum_i\bar x_i y_{σ(i)}$. Their convex hull is the permutation polygon: supporting vertices give optimal matchings at fixed rotations, and the farthest vertex gives the global alignment. We prove the sharp bound of $n(n-1)$ vertices for $n\ge2$, answering Rote's rotation-assignment open problem. In exact arithmetic, assignment queries recover the polygon in $\mathcal O(n^5)$ operations. Assignment-based bounds extend the approach to three-dimensional rotations and partial matching at a supplied translation through branch-and-bound. On timed MPEG-7 shape pairs, Rubix attains every numerical reference value in 12 ms on average, 50 times faster than a rotation grid at the same accuracy. Its distances improve gravity-aligned matching of real 3D scans, shape retrieval and noisy crystal classification over alternating minimization.

Problem

• point set alignment
• Procrustes-Wasserstein
• correspondence-free
• global optimization
• rotation-assignment

Innovation

• Point Set Alignment
• Procrustes-Wasserstein
• Permutation Polygon
• Global Optimization
• Branch-and-Bound