A key property of the Delaunay filtration is that it is topologically (i.e., weakly) equivalent to the offset (union-of-balls) filtration. Recently, this filtration has been extended to point clouds equipped with an $\mathbb{R}$-valued function, yielding a computable 2-parameter filtration that satisfies an analogous weak equivalence. Motivated in part by the study of time-varying data, we introduce a 3-parameter extension of the Delaunay filtration for point clouds equipped with an $\mathbb{R}^2$-valued function, also satisfying an analogous weak equivalence. For a point cloud $X \subset \mathbb{R}^d$, our trifiltration has size $O\bigl(|X|^{\lceil(d+1)/2\rceil+1}\bigr)$. We present an algorithm that computes this trifiltration in time $O\bigl(|X|^{\lceil d/2\rceil+2}\bigr)$, together with an implementation. Our experiments demonstrate that implementation can handle thousands of points in $\mathbb{R}^3$, with memory growth that is nearly linear.

翻译：Delaunay过滤的一个关键性质是它在拓扑上（即弱）等价于偏移（球并）过滤。最近，该过滤已被扩展至配备 $\mathbb{R}$-值函数的点云，从而产生一个满足类似弱等价关系的可计算双参数过滤。部分受时变数据研究的启发，我们为配备 $\mathbb{R}^2$-值函数的点云引入了一种三参数扩展的Delaunay过滤，同样满足类似的弱等价关系。对于点云 $X \subset \mathbb{R}^d$，我们的三重过滤规模为 $O\bigl(|X|^{\lceil(d+1)/2\rceil+1}\bigr)$。我们提出了一种算法，可在 $O\bigl(|X|^{\lceil d/2\rceil+2}\bigr)$ 时间内计算该三重过滤，并附有实现。实验表明，该实现能够处理 $\mathbb{R}^3$ 中数千个点，且内存增长近乎线性。