Euler characteristic surfaces

Abstract

<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>

Hybrid transforms of constructible functions

“…the sublevel-sets constructible function associated to a continuous subanalytic map g : M → R on Z.Example 7.3. The function ϕ γ f is sometimes called Euler characteristic curve for V = R, and Euler characteristic surfaces for V = R 2 , see for instance[Bel+21].Example 7.4. Considering a subset Z ⊆ R d relatively compact subanalytic and locally closed, the Euler characteristic transform defined in[TMB14] is, for ξ ∈ V * and t ∈ R,ECT(Z)(ξ, t) = χ {x ∈ Z ; ξ; x ≤ t} = ϕ − ξ (t).If f : R d → R is continuous subanalytic and ξ ∈ R d , denote by (ξ, f ) : R d → R 2 the map defined by x → ( ξ; x , f (x)).…”