中轴的Hausdorff和Gromov-Hausdorff稳定子集
Hausdorff and Gromov-Hausdorff stable subsets of the medial axis
摘要：
在本文中，我们介绍了修剪中轴称为$(λ,α)$ -中轴（$\textrm{ax}_λ^α$）。证明了在弱假设下，集合$K$的$(λ,α)$-中轴在Gromov-Hausdorff意义下是稳定的.更正式地，我们证明了，如果$K$和$K'$在Hausdorff（$d_H$）意义下是接近的，那么$K$和$K'$的$(λ,α)$ -中轴作为度量空间是接近的，即两者之间的Gromov-Hausdorff距离（$d_{GH}$）在$d_{GH} (\textrm{ax}_λ^α(K),\textrm{ax}_λ^α(K')) \lesssim d_H(K,K')^{1/4}$意义下是$\frac{1}{4}$ {<$}lder。两个中轴之间的Hausdorff距离也以$d_{H} (\textrm{ax}_λ^α(K),\textrm{ax}_λ^α(K')) \lesssim d_H(K,K')^{1/2}$为界。这些量化的稳定性结果为近似中轴的实际计算提供了保证。此外，他们提供了关键成分的中轴线的可计算性研究的背景下，可计算分析。
英文摘要：
In this paper we introduce a pruning of the medial axis called the $(λ,α)$-medial axis ($\textrm{ax}_λ^α$). We prove that the $(λ,α)$-medial axis of a set $K$ is stable in a Gromov-Hausdorff sense under weak assumptions. More formally we prove that if $K$ and $K'$ are close in the Hausdorff ($d_H$) sense then the $(λ,α)$-medial axes of $K$ and $K'$ are close as metric spaces, that is the Gromov-Hausdorff distance ($d_{GH}$) between the two is $\frac{1}{4}$-H{ö}lder in the sense that $d_{GH} (\textrm{ax}_λ^α(K),\textrm{ax}_λ^α(K')) \lesssim d_H(K,K')^{1/4}$. The Hausdorff distance between the two medial axes is also bounded, by $d_{H} (\textrm{ax}_λ^α(K),\textrm{ax}_λ^α(K')) \lesssim d_H(K,K')^{1/2}$. These quantified stability results provide guarantees for practical computations of medial axes from approximations. Moreover, they provide key ingredients for studying the computability of the medial axis in the context of computable analysis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
