  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.
