Optimal topological simplification of discrete functions on surfaces

We present an efficient algorithm for computing a function that minimizes the number of critical points among all functions within a prescribed distance δ from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2δ, constructively proving that the lower bound on the number of critical points given by the stability theorem of persistent homology is tight in dimension two for any input function. 1Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16–18, 37083 Göttingen, Germany. {bauer,wardetzky}@math.uni-goettingen.de, http://ddg.math.uni-goettingen.de/ 2Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany. [email protected], http://geom.mi.fu-berlin.de/