AI 中文总结
该研究提出相交欧拉特征轮廓（Intersection ECP），基于欧拉演算刻画多球并集拓扑交互，具多项稳定性，可高效计算，其细化能解析精细交互，稠密采样下可恢复基础形状的同调与欧拉特征。
AI 中文摘要
$\mathbb{R}^d$中$k$个带色点云$X_1,\ldots,X_k$的相交欧拉特征轮廓（Intersection ECP）是其球并集交集的欧拉特征$\chi(\bigcap_{i=1}^k \mathcal{U}(X_i; t_i))$，是刻画多尺度拓扑交互的整值多参数不变量。其组织框架是可构造函数上的欧拉演算：该轮廓同时是$k$个依赖数据的偏移乘积的欧拉积分$\int \prod_{i=1}^k \mathbf{1}_{\mathcal{U}(X_i; t_i)} \\, d\chi$，此恒等式（即相交定理）是交换几何相交与代数乘积的交换图。该不变量具有刚体运动不变性、尺度等变性和$L^1$稳定性，且是典范的：在逐点欧拉交互轮廓中，它是由分离性和归一性强制得到的，是按点云交汇数量分级的描述符谱的顶层。对于$n$个点，单次排序的Alpha复形扫描以$O(n^{\lceil d/2 \rceil} \log n)$时间计算该轮廓，无持续同调约简，在偶数维中达到最坏情况最优。当欧拉特征抵消时，相对同调细化可解析更精细的交互，且在双参数交错距离下稳定。最后，对于越来越稠密的采样，该轮廓及其细化是一致的，可恢复基础形状的（相对）同调和欧拉特征——在紧集的逆极限中成立，且在正 reach 条件下具有持续性和显式采样复杂度。
英文摘要
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.
