Persistence-sensitive simplification functions on 2-manifolds

We continue the study of topological persistence [5] by investigating the problem of simplifying a function <i>f</i> in a way that removes topological noise as determined by its persistence diagram [2]. To state our results, we call a function <i>g</i> an ε-<i>simplification</i> of another function <i>f</i> if ¦¦<i>f−g</i>¦¦_∞≤ε, and the persistence diagrams of <i>g</i> are the same as those of <i>f</i> except all points within <i>L</i>₁-distance at most ε from the diagonal have been removed. We prove that for functions <i>f</i> on a 2-manifold such ε-simplification exists, and we give an algorithm to construct them in the piecewise linear case.