The persistent homology transform (PHT) represents a shape with a multiset of persistence diagrams parameterized by the sphere of directions in the ambient space. In this work, we describe a finite set of diagrams that discretize the PHT such that it faithfully represents the underlying shape. We provide a discretization that is exponential in the dimension of the shape. Moreover, we show that this discretization is stable with respect to various perturbations. Furthermore, we provide an algorithm for computing the discretization. Our approach relies only on knowing the heights and dimensions of topological events, which means that it can be adapted to provide discretizations of other dimension-returning topological transforms, including the Betti curve transform. With mild alterations, we also adapt our methods to faithfully discretize the Euler Characteristic curve transform.

翻译：持续同调变换（PHT）通过环境空间中方向球面参数化的持续图多重集来表示形状。本文描述了一组有限图对PHT进行离散化，使其能够忠实地表示底层形状。我们提出了一种离散化方法，其规模随形状维度呈指数增长，并证明了该离散化对多种扰动具有稳定性。此外，我们给出了计算该离散化的算法。该方法仅依赖于拓扑事件的高度与维度信息，因此可推广至其他返回维度的拓扑变换（包括贝蒂曲线变换）的离散化。通过轻微调整，该方法同样适用于欧拉示性数曲线变换的忠实离散化。