Abstract: We apply ideas from mesh generation to improve the time and space complexities of computing the full persistent homological information associated with a point cloud $P$ in Euclidean space $\R^d$. Classical approaches rely on the \v Cech, Rips, $\alpha$-complex, or witness complex filtrations of $P$, whose complexities scale up very badly with $d$. For instance, the $\alpha$-complex filtration incurs the $n^{\Omega(d)}$ size of the Delaunay triangulation, where $n$ is the size of $P$. The common alternative is to truncate the filtrations when the sizes of the complexes become prohibitive, possibly before discovering the most relevant topological features. In this paper we propose a new collection of filtrations, based on the Delaunay triangulation of a carefully-chosen superset of $P$, whose sizes are reduced to $2^{O(d^2)}n$. A nice property of these filtrations is to be interleaved multiplicatively with the family of offsets of $P$, so that the persistence diagram of $P$ can be approximated in $2^{O(d^2)}n^3$ time in theory, with a near-linear observed running time in practice (ignoring the constant factors depending exponentially on $d$). Thus, our approach remains tractable in medium dimensions, say 4 to 10.