Euler Characteristic Surfaces

TL;DR

This paper extends the use of Euler characteristic to multiparameter topological data analysis, demonstrating its theoretical and computational advantages in higher-dimensional spaces through synthetic and real-world examples.

Contribution

It introduces a framework for applying Euler characteristic in multi-parameter TDA, providing a manageable alternative to more complex invariants like homology.

Findings

Effective in higher-dimensional parameter spaces

Applicable to real-world medical image analysis

Simplifies computational complexity in TDA

Abstract

We study the use of the Euler characteristic for multiparameter topological data analysis. Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, including in the context of random fields. The goal of this paper is to present the extension of using the Euler characteristic in higher-dimensional parameter spaces. While topological data analysis of higher-dimensional parameter spaces using stronger invariants such as homology continues to be the subject of intense research, Euler characteristic is more manageable theoretically and computationally, and this analysis can be seen as an important intermediary step in multi-parameter topological data analysis. We show the usefulness of the techniques using artificially generated examples, and a real-world application of detecting diabetic retinopathy in retinal images.