The Delaunay filtration $\mathcal{D}_{\bullet}(X)$ of a point cloud $X\subset \mathbb{R}^d$ is a central tool of computational topology. Its use is justified by the topological equivalence of $\mathcal{D}_{\bullet}(X)$ and the offset (i.e., union-of-balls) filtration of $X$. Given a function $\gamma: X \to \mathbb{R}$, we introduce a Delaunay bifiltration $\mathcal{DC}_{\bullet}(\gamma)$ that satisfies an analogous topological equivalence, ensuring that $\mathcal{DC}_{\bullet}(\gamma)$ topologically encodes the offset filtrations of all sublevel sets of $\gamma$, as well as the topological relations between them. $\mathcal{DC}_{\bullet}(\gamma)$ is of size $O(|X|^{\lceil\frac{d+1}{2}\rceil})$, which for $d$ odd matches the worst-case size of $\mathcal{D}_{\bullet}(X)$. Adapting the Bowyer-Watson algorithm for computing Delaunay triangulations, we give a simple, practical algorithm to compute $\mathcal{DC}_{\bullet}(\gamma)$ in time $O(|X|^{\lceil \frac{d}{2}\rceil +1})$. Our implementation, based on CGAL, computes $\mathcal{DC}_{\bullet}(\gamma)$ with modest overhead compared to computing $\mathcal{D}_{\bullet}(X)$, and handles tens of thousands of points in $\mathbb{R}^3$ within seconds.

翻译：点云 $X\subset \mathbb{R}^d$ 的Delaunay滤子 $\mathcal{D}_{\bullet}(X)$ 是计算拓扑的核心工具。其应用基于 $\mathcal{D}_{\bullet}(X)$ 与 $X$ 的偏移（即球并）滤子之间的拓扑等价性。给定函数 $\gamma: X \to \mathbb{R}$，我们引入一种满足类似拓扑等价性的Delaunay双滤子 $\mathcal{DC}_{\bullet}(\gamma)$，确保 $\mathcal{DC}_{\bullet}(\gamma)$ 在拓扑上编码 $\gamma$ 所有子水平集的偏移滤子及其拓扑关系。$\mathcal{DC}_{\bullet}(\gamma)$ 的大小为 $O(|X|^{\lceil\frac{d+1}{2}\rceil})$，对于奇数 $d$ 而言，该复杂度与 $\mathcal{D}_{\bullet}(X)$ 的最坏情况大小相匹配。通过改编计算Delaunay三角剖分的Bowyer-Watson算法，我们提出一种简单实用的算法，能在 $O(|X|^{\lceil \frac{d}{2}\rceil +1})$ 时间内计算 $\mathcal{DC}_{\bullet}(\gamma)$。基于CGAL的实现显示，与计算 $\mathcal{D}_{\bullet}(X)$ 相比，计算 $\mathcal{DC}_{\bullet}(\gamma)$ 的开销适度，且能在数秒内处理 $\mathbb{R}^3$ 中数万个点。