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 chi-squared statistic-based 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, analysis of variance for functional data, and geometric morphometrics.

翻译：本文建立了利用光滑欧拉特征变换对形状随机性进行建模并实施形状统计推断的数学基础。基于此基础，我们提出了两种基于卡方统计量的算法，用于检验关于随机形状的假设。通过模拟研究验证了我们的数学推导，并将所提算法与前沿方法进行比较，以证明本框架的实用性。在实际应用方面，我们分析了一组来自四个灵长类属的下颌臼齿数据，结果表明我们的算法能够检测到显著的形状差异，这些差异重现了已知的亚目间形态变异。总体而言，我们的讨论连接了以下领域：代数与计算拓扑学、概率论与随机过程、Sobolev空间与泛函分析、函数型数据的方差分析以及几何形态测量学。