AI 中文总结
该研究从凸几何视角拓展欧拉特征变换，提出新伪距离定义、施泰纳条形码等，用于形状分类任务提供轻量级特征。
AI 中文摘要
通过研究凸几何的支撑函数与拓扑数据分析中的欧拉特征变换（ECT）之间的关系，我们开发了新工具并提出了一些常见ECT流程的变体。具体而言，我们提出了ECT诱导伪距离的新定义，该定义具有在常见欧几里得等距变换下不变的优势，且无需截断参数即可比较具有不同欧拉特征的形状。这些定义依赖于凸几何中施泰纳点概念的推广，我们将其一般定义为由ECT给出的一个特殊点。我们还展示了凸几何如何提供从平面形状的ECT中恢复其有趣几何信息（即周长）的途径，为此我们给出了显式公式。基于这些概念并利用持续同调，我们将施泰纳条形码定义为形状的等距不变特征，以及支撑函数和施泰纳点的同调变体。最后，我们在形状分类任务中测试了这些构造，为对齐和未对齐数据集提供了轻量级特征。
英文摘要
By examining the relationship between the support function of convex geometry and the Euler Characteristic Transform (ECT) of topological data analysis, we develop new tools and suggest variations on some common ECT pipelines. Specifically, we put forward new definitions of ECT-induced pseudodistances, which have the advantage of being invariant under common euclidean isometries and require no cutoff parameter to compare shapes with distinct Euler characteristic. These definitions rely on a generalization of the convex geometric concept of the Steiner point, which we define in general as a distinguished point given by the ECT. We also show how convex geometry provides a path to recover interesting geometric information of a flat shape from its ECT, namely, its perimeter, for which we give an explicit formula. By building on these concepts and leveraging persistent homology, we define Steiner barcodes as an isometry invariant feature of shapes, as well as homological variants of the support function and Steiner point. Finally, we put these constructions to the test in shape classification tasks, providing lightweight features for aligned and misaligned datasets.
Comments41 pages, 6 figures
