论文标题
具有边界的歧管的扩展持续同源转换
The Extended Persistent Homology Transform of manifolds with boundary
论文作者
论文摘要
扩展的持久性同源变换（XPHT）是一种拓扑变换，它以嵌入在欧几里得空间中的形状为输入，并将每个单元向量分配给该形状相对于该方向的高度函数的扩展持久模块。我们可以通过在球体上集成各自的延伸持续模块之间的距离来定义两个形状之间的距离。通过使用扩展的持久性，即使它们具有不同的贝蒂数字，我们也会在形状之间获得有限的距离。我们使用Morse理论表明，可以从限制到边界的高度函数的扩展持久性中推导出高度函数的扩展持久性，而临界点上的标签则是正或负临界点。我们研究了XPHT对二进制图像的应用；概述算法以有效地计算XPHT，从而利用边界曲线的PHT之间的关系到前景的扩展持久性。
The Extended Persistent Homology Transform (XPHT) is a topological transform which takes as input a shape embedded in Euclidean space, and to each unit vector assigns the extended persistence module of the height function over that shape with respect to that direction. We can define a distance between two shapes by integrating over the sphere the distance between their respective extended persistence modules. By using extended persistence we get finite distances between shapes even when they have different Betti numbers. We use Morse theory to show that the extended persistence of a height function over a manifold with boundary can be deduced from the extended persistence for that height function restricted to the boundary, alongside labels on the critical points as positive or negative critical. We study the application of the XPHT to binary images; outlining an algorithm for efficient calculation of the XPHT exploiting relationships between the PHT of the boundary curves to the extended persistence of the foreground.
