Rubix: Global Correspondence-Free Point Set Alignment through Assignment Geometry
TL;DR
The sharp bound of $n(n-1)$ vertices for $n\ge2$ is proved, answering Rote's rotation-assignment open problem, and assignment-based bounds extend the approach to three-dimensional rotations and partial matching at a supplied translation through branch-and-bound.
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 $\sigma$ of two centered $n$-point sets defines a complex correlation $z_\sigma=\sum_i\bar x_i y_{\sigma(i)}$. 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.
