<p style='text-indent:20px;'>In this paper, we investigate the use of the Euler characteristic for the topological data analysis, particularly over higher dimensional parameter spaces. The Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, primarily 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. The topological data analysis of higher dimensional parameter spaces using stronger invariants such as homology, has and continues to be the subject of intense research. However, as important theoretical and computational obstacles remain, the use of the Euler characteristic represents an important intermediary step toward multi-parameter topological data analysis. We show the usefulness of the techniques using generated examples as well as a real world dataset of detecting diabetic retinopathy in retinal images.</p>

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.

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Proximity complexes and filtrations are a central construction in topological data analysis. Built using distance functions or more generally metrics, they are often used to infer connectivity information from point clouds. We investigate proximity complexes and filtrations built over the Chebyshev metric, also known as the maximum metric or 8 metric, rather than the classical Euclidean metric. Somewhat surprisingly, the 8 case has not been investigated thoroughly. Our motivation lies in that this metric has the far simpler numerical tests which can lead to computational speedups for high-dimensional data analysis. In this paper, we examine a number of classical complexes under this metric, including the Čech, Vietoris-Rips, and Alpha complexes. We also introduce two new complexes which we call the Alpha clique and Minibox complexes. We provide results on topological properties of these, as well as computational experiments which show that these can often be used to reduce the number of high-dimensional simplices included in Čech filtrations and so speed up the computation of persistent homology.