发表机构
Australian National University（澳大利亚国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
Rubix通过分配几何全局求解无对应点集对齐，证明置换多边形顶点界，实现快速精确的三维旋转与部分匹配，显著提升多项任务性能。
AI 中文摘要
Procrustes-Wasserstein对齐方法在无给定对应关系的情况下联合估计匹配和旋转，但交替最小化可能停滞在次优解。Rubix在平方欧氏损失下全局求解等权平面问题。每个两个中心化n点集的匹配σ定义了一个复相关z_σ=∑_i\bar x_i y_{σ(i)}。它们的凸包是置换多边形：支撑顶点在固定旋转下给出最优匹配，最远顶点给出全局对齐。我们证明了n≥2时顶点数的尖锐界为n(n-1)，回答了Rote的旋转-分配开放问题。在精确算术中，分配查询以O(n^5)次操作恢复多边形。基于分配的下界将方法扩展到三维旋转和给定平移下的部分匹配，通过分支定界实现。在计时的MPEG-7形状对上，Rubix平均在12毫秒内达到所有数值参考值，比相同精度的旋转网格快50倍。其距离改善了真实3D扫描的重力对齐匹配、形状检索和噪声晶体分类，优于交替最小化。
英文摘要
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.
Comments67 pages, 20 figures. Includes full proofs and experimental appendices
