AI 中文总结
该研究针对欧几里得形状的径向距离滤过，建立了从带边界流形边界的径向持续同调恢复其径向扩展持续同调的方法，还给出了二维像素集的相关计算算法，有望应用于生物医学图像分析领域。
AI 中文摘要
我们研究欧几里得空间中形状$M/subseteq \real^n$的径向滤过的扩展持续同调。径向滤过通过选择一个中心点$c$，取$M$中与$c$距离不超过$r$的点构成。我们证明，在温和假设下，可从带边界流形的边界的径向持续同调恢复该流形的径向扩展持续同调。我们还建立了当$M$为二维数字网格中的像素集时，计算径向扩展持续同调的算法，这些方法与计算嵌入欧几里得空间的形状的高度滤过的扩展持续同调所用方法类似。我们预计这些结果将在生物医学图像分析场景中有用，这类场景中自然会考虑相对于固定中心的径向滤过，例如在神经元结构的研究中。
英文摘要
We study the extended persistent homology of the radial filtration of a shape $M\subseteq \mathbb{R}^n$. A radial filtration is formed by choosing a center point $c$ and taking points of $M$ within distance $r$ of $c$. We show that, under mild assumptions, we can recover the radial extended persistence of a manifold with boundary from the radial persistence of its boundary. We also establish algorithms to compute radial extended persistence when $M$ is a set of pixels in a 2D digital grid. The methods are similar to those used to compute the extended persistent homology of a height filtration of a shape embedded in Euclidean space. We envisage these results will be useful in biomedical image analysis settings where it is natural to consider a radial filtration with respect to a fixed center, for example in the study of neuronal structures.
