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

Abstract

In this paper, we provide the mathematical foundations for the randomness of shapes and the distributions of smooth Euler characteristic transform. Based on these foundations, we propose an approach for testing hypotheses on random shapes. Simulation studies are provided to support our mathematical derivations and show the performance of our proposed hypothesis testing framework. Our discussions connect the following fields: algebraic and computational topology, probability theory and stochastic processes, Sobolev spaces and functional analysis, statistical inference, and medical imaging.

Stability and Inference of the Euler Characteristic Transform

“…We propose a new metric on the embeddings of a finite one-dimensional CW complex that is sensitive to changes in arc-length. Next, we introduce a norm on Euler characteristic transforms, in a similar vein to the norm introduced in Meng et al [20,Equation 3.1], defined by taking first the 1-norm over the R component, and then the ∞-norm over the S d−1 component. We then prove a novel stability result for the ECT, showing that the ECT is continuous in our metric of embedded spaces (Theorem 4).…”

“…where γ > 0 is a constant. Define V to be the closed subspace of sequences {w ab } ∈ H such that (a,b)∈N 2 w ab cos a (t) sin b (t) = 0 (20) for all t ∈ [0, 2π). Then the Hilbert space H given by the functions…”