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 and 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 function transform. With mild alterations, we also adapt our methods to faithfully discretize the Euler characteristic function transform.

翻译：持久同调变换（PHT）通过由环境空间中的方向球面参数化的持久图多重集来表示形状。本文描述了一组有限图，用于离散化PHT，使其能忠实地表征底层形状。我们提出了一种随形状维度呈指数增长的离散化方法，并证明该离散化对多种扰动具有稳定性，同时给出了计算该离散化的算法。我们的方法仅依赖于已知拓扑事件的高度和维度，因此可推广至其他返回维度的拓扑变换的离散化，包括贝蒂函数变换。经过适当调整，我们还将该方法应用于忠实离散化欧拉示性数函数变换。