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Stability and Inference of the Euler Characteristic Transform.

Lewis Marsh1,2, David Beers1

  • 1Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG UK.

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|April 7, 2026
PubMed
Summary
This summary is machine-generated.

We introduce a new metric for shape analysis, proving the Euler characteristic transform (ECT) is stable. This stability ensures reliable shape summarization even with noisy data, enhancing topological data analysis (TDA) methods.

Keywords:
Bayesian statisticsComputational geometryGaussian processesTopological data analysis

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Area of Science:

  • Topological Data Analysis (TDA)
  • Differential Geometry
  • Computational Geometry

Background:

  • The Euler characteristic transform (ECT) is a signature in topological data analysis for shape summarization.
  • While fast and injective for many shapes, the ECT is sensitive to small perturbations.
  • Existing methods often rely on triangulation size, which can be unstable.

Purpose of the Study:

  • To develop a new metric for assessing the stability of the ECT on one-dimensional shapes.
  • To demonstrate that the ECT is stable with respect to curvature, not just triangulation size.
  • To create a statistically sound and computationally efficient estimator for the ECT.

Main Methods:

  • Introduction of a novel metric for compact one-dimensional shapes.
  • Proof of ECT stability using this new metric, focusing on intrinsic shape properties (curvature).
  • Development of a Gaussian process-based statistical estimator for the ECT.

Main Results:

  • The proposed metric establishes the stability of the ECT for one-dimensional shapes.
  • Curvature is identified as a key factor in controlling ECT stability, offering an advantage over mesh-dependent measures.
  • A consistent statistical estimator for the ECT is developed, converging to the true ECT with increasing sample size.

Conclusions:

  • The ECT can be made robust to perturbations through a curvature-based stability analysis.
  • The developed estimator provides a reliable tool for analyzing noisy shape data in TDA.
  • This work advances the practical application of TDA by improving the reliability of shape signatures.