Recent developments in shape reconstruction and comparison call for the use of many different (topological) descriptor types, such as persistence diagrams and Euler characteristic functions. We establish a framework to quantitatively compare the strength of different descriptor types, setting up a theory that allows for future comparisons and analysis of descriptor types and that can inform choices made in applications. We use this framework to partially order a set of six common descriptor types. We then give lower bounds on the size of sets of descriptors that uniquely correspond to simplicial complexes, giving insight into the advantages of using verbose rather than concise topological descriptors.

翻译：形状重建与比较领域的最新进展需要运用多种不同的（拓扑）描述子类型，例如持久图与欧拉示性函数。本文建立了一个量化比较不同描述子类型效力的框架，构建了一套理论体系，以支持未来对描述子类型的比较分析，并为应用中的选择提供依据。利用该框架，我们对六种常见描述子类型进行了偏序划分。随后，我们给出了唯一对应于单纯复形的描述子集合大小的下界，从而揭示了使用详尽而非简明的拓扑描述子所具有的优势。