  We construct a filtered simplicial complex $(X_L,f_L)$ associated to a subset
$X\subset \mathbb{R}^d$, a function $f:X\rightarrow \mathbb{R}$ with compactly
supported sublevel sets, and a collection of landmark points $L\subset
\mathbb{R}^d$. The persistence values $f_L(\Delta)$ are defined as the
minimizing values of a family of constrained optimization problems, whose
domains are certain higher order Voronoi cells associated to $L$. We prove that
$H_k^{a,b}(X_L)\cong H^{a,b}_k(X)$ provided that $f$ is the restriction of a
smooth function, the landmarks are sufficiently dense, and $a<b$ are generic,
and we show that the construction produces desirable results in some examples.
