The Intersection Euler Characteristic Profile: Euler Calculus and Stability for Topological Interaction of Ball Unions

Abstract

The Intersection Euler Characteristic Profile (Intersection ECP) of $k$ colored point clouds $X_1,\dots ,X_k\subset {\mathbb{R}}^d$ is the Euler characteristic $\chi ({\bigcap }_{i=1}^k\mathcal{U}(X_i;t_i))$ of the overlap of their ball unions---an integer-valued, multiparameter invariant of their topological interaction across scales. Its organizing framework is the Euler calculus on constructible functions: the profile is equally the Euler integral $\int {\prod }_{i=1}^k{1}_{\mathcal{U}(X_i;t_i)}d\chi$ of the product of the $k$ data-dependent offsets, and this identity---our Intersection Theorem---is a commuting square interchanging geometric intersection and algebraic product. The invariant is rigid-motion invariant, scale-equivariant, and $L^1$-stable, and it is canonical: among pointwise-Euler interaction profiles it is the one forced by separation and normalization, the top floor of a spectrum of descriptors graded by how many clouds meet. For $n$ points a single sorted Alpha-complex sweep computes it in $O(n^{\lceil d/2\rceil }\log n)$ time with no persistence reduction, worst-case optimal in even dimensions. Where the Euler characteristic cancels, a relative-homology refinement resolves the finer interaction and is stable in the two-parameter interleaving distance. Finally, for increasingly dense samples the profile and its refinement are consistent, recovering the (relative) homology and Euler characteristic of the underlying shapes---in the inverse limit for compact sets, and, under positive reach, persistently and with explicit sample complexity.