  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.
