Optimal Topological Simplification of Discrete Functions on Surfaces

Abstract

Given a function f on a surface and a tolerance δ > 0, we construct a function f δ subject to {norm of matrix}f δ-f{norm of matrix} ∞≤δ such that f δ has a minimum number of critical points. Our construction relies on a connection between discrete Morse theory and persistent homology and completely removes homological noise with persistence ≤2δ from the input function f. The number of critical points of the resulting simplified function f δ achieves the lower bound dictated by the stability theorem of persistent homology. We show that the simplified function can be computed in linear time after persistence pairs have been computed. © 2011 The Author(s).