The Euler characteristic transform (ECT) is a signature from topological data analysis (TDA) which summarises shapes embedded in Euclidean space. Compared with other TDA methods, the ECT is fast to compute and it is a sufficient statistic for a broad class of shapes. However, small perturbations of a shape can lead to large distortions in its ECT. In this paper, we propose a new metric on compact one-dimensional shapes and prove that the ECT is stable with respect to this metric. Crucially, our result uses curvature, rather than the size of a triangulation of an underlying shape, to control stability. We further construct a computationally tractable statistical estimator of the ECT based on the theory of Gaussian processes. We use our stability result to prove that our estimator is consistent on shapes perturbed by independent ambient noise; i.e., the estimator converges to the true ECT as the sample size increases.

翻译：欧拉特征变换(ECT)是拓扑数据分析(TDA)中的一种签名，用于总结嵌入欧几里得空间的形状。与其他TDA方法相比，ECT计算速度快，且对一大类形状而言是充分统计量。然而，形状的微小扰动可能导致其ECT发生较大扭曲。本文针对紧致一维形状提出一种新度量，并证明ECT在该度量下具有稳定性。关键之处在于，我们的结果利用曲率而非底层形状三角剖分的大小来控制稳定性。我们进一步基于高斯过程理论构建了一个计算上易于处理的ECT统计估计量。利用稳定性结果，我们证明了该估计量在环境噪声独立扰动下的形状上具有一致性，即随着样本量增加，估计量收敛至真实ECT。