Euler Characteristic Surfaces

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 multiparameter 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.

$\ell^p$-Distances on Multiparameter Persistence Modules

“…One very simple approach is to define the map in term of the Hilbert function, i.e., the dimension of the vector space at each index [8,52]. Alternatively, when working with homology in all degrees, one can instead take the Euler characteristic at each index [6]. As these approaches do not depend on the internal linear maps in the persistence modules, they are rather coarse.…”