Randomness and Statistical Inference of Shapes via the Smooth Euler Characteristic Transform

In this paper, we provide the foundations for deriving the distributional properties of the smooth Euler characteristic transform. Motivated by functional data analysis, we propose two algorithms for testing hypotheses on random shapes based on these foundations. Simulation studies are provided to support our mathematical derivations and show the performance of our hypothesis testing framework. We apply our proposed algorithms to analyze a data set of mandibular molars from four genera of primates to test for shape differences and interpret the corresponding results from the morphology viewpoint. Our discussions connect the following fields: algebraic and computational topology, probability theory and stochastic processes, Sobolev spaces and functional analysis, statistical inference, morphology, and medical imaging.