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arXiv · 2610.10408

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

Subhransu S. Bhattacharjee·Dylan Campbell·Rahul Shome
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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.

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Subhransu S. Bhattacharjee, Dylan Campbell, Rahul Shome. 2026-10-07. Rubix: Global Correspondence-Free Point Set Alignment through Assignment Geometry. https://arxiv.org/abs/2610.10408

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