Optimal topological simplification of discrete functions on surfaces

TL;DR

This paper introduces a method to optimally simplify functions on surfaces by minimizing critical points within a specified distance, leveraging discrete Morse theory and persistent homology, and achieving linear-time computation after persistence pairing.

Contribution

It establishes a novel connection between discrete Morse theory and persistent homology to achieve topological simplification with proven optimality and efficiency.

Findings

Successfully minimizes critical points within a given distance

Proves the tightness of a lower bound on critical points

Achieves linear-time computation after persistence pairs are computed

Abstract

We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} 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 2{\delta}, constructively proving the tightness of a lower bound on the number of critical points given by the stability theorem of persistent homology in dimension two for any input function. We also show that an optimal solution can be computed in linear time after persistence pairs have been computed.