Topological Analysis of Rough Surfaces Using Persistent Homology

Abstract

This letter investigates rough surfaces using a topological method. The horizontal cross section of a rough surface consists of “islands”, and we focus on the topological changes in the island shapes (generation and annihilation of islands and lakes) with changes in elevation. We apply persistent homology to track these topological changes. We numerically confirm that the life spans of the islands and lakes follow power-law distributions, whose scaling exponents vary according to the roughness of the surface. We also provide a theoretical explanation for the relation between these scaling exponents and the roughness exponent with a simple scaling argument. The proposed method successfully connects a topological property with the roughness of a surface.