In this article, we establish the mathematical foundations for modeling the randomness of shapes and conducting statistical inference on shapes using the smooth Euler characteristic transform. Based on these foundations, we propose two parametric algorithms for testing hypotheses on random shapes. Simulation studies are presented to validate our mathematical derivations and to compare our algorithms with state-of-the-art methods to demonstrate the utility of our proposed framework. As real applications, we analyze a data set of mandibular molars from four genera of primates and show that our algorithms have the power to detect significant shape differences that recapitulate known morphological variation across suborders. Altogether, our discussions bridge the following fields: algebraic and computational topology, probability theory and stochastic processes, Sobolev spaces and functional analysis, statistical inference, and geometric morphometrics.

翻译：本文建立了一种数学基础，用于通过光滑欧拉特征变换对形状随机性进行建模并对形状进行统计推断。基于这些基础，我们提出了两种参数化算法来检验随机形状的假设。通过仿真研究验证了我们的数学推导，并将我们的算法与最新方法进行了比较，以证明所提出框架的实用性。在实际应用中，我们分析了来自四个灵长类属的下颌磨牙数据集，结果表明我们的算法能够检测出显著的形状差异，这些差异重现了已知的亚目间形态变异。总之，我们的讨论连接了以下领域：代数拓扑与计算拓扑、概率论与随机过程、索伯列夫空间与泛函分析、统计推断以及几何形态测量学。