论文标题
覆盖范围的可计算范围和$ \ mathbb {r}^d $的子集的$ r $ - 概念
Computable bounds for the reach and $r$-convexity of subsets of $\mathbb{R}^d$
论文作者
论文摘要
一套的凸度可以推广到两个较弱的触及范围和$ r $ convexity的概念；两者都描述了布景边界的规律性。对于$ \ mathbb {r}^d $的任何紧凑子集，我们提供了从点云数据计算这些数量上限的方法。随着点云在集合中的密集，边界会收敛到相应的数量，并且在较弱的规则性条件下给出了界限的收敛速率。我们还介绍了$β$ -REACH，即覆盖范围的概括，该覆盖范围排除了大小的小尺寸特征小于[0，\ infty）$中的参数$β\。数值研究表明，如何将$β$ -REACH用于高维度来推断光滑亚曼叶的覆盖范围和其他几何特性。
The convexity of a set can be generalized to the two weaker notions of reach and $r$-convexity; both describe the regularity of a set's boundary. For any compact subset of $\mathbb{R}^d$, we provide methods for computing upper bounds on these quantities from point cloud data. The bounds converge to the respective quantities as the point cloud becomes dense in the set, and the rate of convergence for the bound on the reach is given under a weak regularity condition. We also introduce the $β$-reach, a generalization of the reach that excludes small-scale features of size less than a parameter $β\in[0,\infty)$. Numerical studies suggest how the $β$-reach can be used in high-dimension to infer the reach and other geometric properties of smooth submanifolds.
