On the Stability of the Euler Characteristic Transform for a Perturbed Embedding

TL;DR

This paper analyzes the stability of the Euler Characteristic Transform (ECT) and its extension, SELECT, providing bounds on their sensitivity to perturbations in shape embeddings and scalar fields, which supports their robustness in shape analysis.

Contribution

It introduces a formal distance measure between ECTs of different embeddings and establishes stability bounds, extending the analysis to the Super Lifted Euler Characteristic Transform (SELECT).

Findings

Provides an upper bound for the distance between ECTs of perturbed embeddings.

Extends stability analysis to the SELECT for scalar fields on shapes.

Supports robustness of ECT and SELECT in shape classification and reconstruction.

Abstract

The Euler Characteristic Transform (ECT) is a robust method for shape classification. It takes an embedded shape and, for each direction, computes a piecewise constant function representing the Euler Characteristic of the shape's sublevel sets, which are defined by the height function in that direction. It has applications in TDA inverse problems, such as shape reconstruction, and is also employed with machine learning methodologies. In this paper, we define a distance between the ECTs of two distinct geometric embeddings of the same abstract simplicial complex and provide an upper bound for this distance. The Super Lifted Euler Characteristic Transform (SELECT), a related construction, extends the ECT to scalar fields defined on shapes. We establish a similar distance bound for SELECT, specifically when applied to fields defined on embedded simplicial complexes.