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

2026-08-06

AI Summary

This work investigates the topological interaction structure of overlapping regions among multiple point clouds across varying scales. To this end, it introduces the Intersection Euler Characteristic Profile (Intersection ECP), which for the first time yields a pointwise Euler interaction descriptor uniquely determined by separability and normalization, thereby unifying geometric intersections with algebraic products and establishing a commutation theorem between them. The descriptor is multiparametric, rigid-motion invariant, and L¹-stable, and admits optimal computation without requiring persistence reduction. Leveraging tools from Euler calculus, Alpha complexes, and relative homology, the proposed algorithm runs efficiently in O(n^{⌈d/2⌉} log n) time—achieving worst-case optimality in even dimensions—and consistently recovers the underlying shape’s (relative) homology and Euler characteristic under dense sampling.

Abstract

The Intersection Euler Characteristic Profile (Intersection ECP) of $k$ colored point clouds $X_1, \ldots, X_k \subset \mathbb{R}^d$ is the Euler characteristic $χ(\bigcap_{i=1}^k \mathcal{U}(X_i; t_i))$ 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 $\int \prod_{i=1}^k \mathbf{1}_{\mathcal{U}(X_i; t_i)} \, dχ$ 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.

Problem

• Euler characteristic
• topological interaction
• ball unions
• multiparameter invariant
• point clouds

Innovation

• Intersection Euler Characteristic Profile
• Euler calculus
• topological interaction
• Alpha-complex
• relative homology