One common use of persistent homology is to explore the shape of point clouds, where points are assumed to be sampled from a geometric object. We propose a novel filtration, called ellipsoidal filtration, which assumes that point clouds are sampled from a dynamic smooth flow. Instead of creating topologies from point clouds at increasing scales using isotropic balls (for example, Vietoris-Rips filtration), ellipsoidal filtration creates ellipsoids around points based on local flow variances, approximating the flow's manifold as the scale increases. We show that constructing ellipsoidal neighbourhoods improves the denoising of recurrent signals and the estimation of recurrence times, especially when the data contain bottlenecks. Choosing ellipsoids according to the maximum persistence of the H1 class provides a data-driven threshold for both denoising and recurrence-time estimation.

翻译：持久同调的一种常见用途是探索点云的形状，其中假设点是从几何对象中采样的。我们提出了一种称为椭球过滤的新型过滤方法，该方法假设点云是从动态平滑流中采样的。与使用各向同性球体（例如Vietoris-Rips过滤）在递增尺度上从点云构建拓扑不同，椭球过滤基于局部流方差在点周围创建椭球体，随着尺度增加来逼近流的流形。我们证明，构建椭球邻域能够改善循环信号的去噪和循环时间的估计，尤其是在数据包含瓶颈结构时。根据H1类的最大持久性选择椭球体，为去噪和循环时间估计提供了数据驱动的阈值。