We propose an extension of the classical union-of-balls filtration of persistent homology: fixing a point $q$, we focus our attention to a ball centered at $q$ whose radius is controlled by a second scale parameter. We discuss an absolute variant, where the union is just restricted to the $q$-ball, and a relative variant where the homology of the $q$-ball relative to its boundary is considered. Interestingly, these natural constructions lead to bifiltered simplicial complexes which are not $k$-critical for any finite $k$. Nevertheless, we demonstrate that these bifiltrations can be computed exactly and efficiently, and we provide a prototypical implementation using the CGAL library. We also argue that some of the recent algorithmic advances for $2$-parameter persistence (which usually assume $k$-criticality for some finite $k$) carry over to the $\infty$-critical case.

翻译：我们提出了持久同调中经典球并集过滤的一个推广：固定一点$q$，我们将注意力集中在以$q$为中心、半径由第二个尺度参数控制的球上。我们讨论了一种绝对变体，其中并集仅限制在$q$球内，以及一种相对变体，其中考虑了$q$球相对于其边界的同调。有趣的是，这些自然的构造导致了双过滤的单纯复形，这些复形对于任何有限$k$都不是$k$-临界的。尽管如此，我们证明了这些双过滤可以精确且高效地计算，并提供了使用CGAL库的原型实现。我们还论证了最近关于2参数持久性的某些算法进展（通常假设对于某个有限$k$的$k$-临界性）可以推广到$\infty$-临界情形。