  We show how a filtration of Delaunay complexes can be used to approximate the
persistence diagram of the distance to a point set in $R^d$. Whereas the full
Delaunay complex can be used to compute this persistence diagram exactly, it
may have size $O(n^{\lceil d/2 \rceil})$. In contrast, our construction uses
only $O(n)$ simplices. The central idea is to connect Delaunay complexes on
progressively denser subsamples by considering the flips in an incremental
construction as simplices in $d+1$ dimensions. This approach leads to a very
simple and straightforward proof of correctness in geometric terms, because the
final filtration is dual to a $(d+1)$-dimensional Voronoi construction similar
to the standard Delaunay filtration complex. We also, show how this complex can
be efficiently constructed.
