[
    {
        "id": "osp-16559",
        "type": "article-journal",
        "title": "Rubix: Global Correspondence-Free Point Set Alignment through Assignment Geometry",
        "author": [
            {
                "family": "Bhattacharjee",
                "given": "Subhransu S."
            },
            {
                "family": "Campbell",
                "given": "Dylan"
            },
            {
                "family": "Shome",
                "given": "Rahul"
            }
        ],
        "URL": "https://omanscience.com/en/articles/rubix-global-correspondence-free-point-set-alignment-through-assignment-geometry",
        "language": "en",
        "issued": {
            "date-parts": [
                [
                    2026
                ]
            ]
        },
        "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."
    }
]