Rubix: Global Correspondence-Free Point Set Alignment through Assignment Geometry
Advances mathematical foundations for geometric matching with implications for computer vision and robotics applications.
AI Summary
Rubix presents a global optimization method for correspondence-free point set using assignment geometry, solving a rotation-assignment problem with polynomial complexity and 50x speedup over existing methods.
Excerpt
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 vert