作者：Vitaliy Kurlin
本文解决了欧几里得等距条件下有限个无标记点云的连续分类问题。在实际物体测量中，在点的扰动下，适当度量中所需不变量的Lipschitz连续性受到不可避免的噪声的影响。这种等距分类的最佳解决方案被称为学校几何中的SSS定理，该定理称任何达到同余（平面中的等距）的三角形都具有三个边长的连续完全不变量。然而，SSS定理即使扩展到平面中的四个点也不容易，部分原因是四点云的四参数族具有相同的六个成对距离。在等距下不变的大多数过去度量的计算时间是输入大小的指数。最后的障碍是奇点配置中先前不变量的不连续性，例如，当三角形退化为直线时。以上所有挑战现在都已存在
This paper solves the continuous classification problem for finite clouds ofunlabelled points under Euclidean isometry. The Lipschitz continuity ofrequired invariants in a suitable metric under perturbations of points ismotivated by the inevitable noise in measurements of real objects. The best solved case of this isometry classification is known as the SSStheorem in school geometry saying that any triangle up to congruence (isometryin the plane) has a continuous complete invariant of three side lengths. However, there is no easy extension of the SSS theorem even to four points inthe plane partially due to a 4-parameter family of 4-point clouds that have thesame six pairwise distances. The computational time of most past metrics thatare invariant under isometry was exponential in the size of the input. Thefinal obstacle was the discontinuity of previous invariants at singularconfigurations, for example, when a triangle degenerates to a straight line. All the challenges above are now resolved by the Simplexwise CentredDistributions that combine inter-point distances of a given cloud with the newstrength of a simplex that finally guarantees the Lipschitz continuity. Thecomputational times of new invariants and metrics are polynomial in the numberof points for a fixed Euclidean dimension.
论文链接：http://arxiv.org/pdf/2303.13486v1
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