Generalized Morse Theory of Distance Functions to Surfaces for Persistent Homology

Abstract

This paper brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Distance fields are central objects in shape representation, while topological data analysis uses algebraic topology to characterize geometric and topological patterns in shapes. The most well-known and widely applied tool from this approach is persistent homology, which tracks the evolution of topological features in a dynamic manner as a barcode. Morse theory is a framework from differential topology that studies critical points of functions on manifolds; it has been used to characterize the birth and death of persistent homology features. However, a significant limitation to Morse theory is that it cannot be readily applied to distance functions because distance functions lack smoothness, which is required in Morse theory. Our contributions to addressing this issue is two fold. First, we generalize Morse theory to Euclidean distance functions of bounded sets with smooth boundaries. We focus in particular on distance fields for shape representation and we study the persistent homology of shape textures using a sublevel set filtration induced by the signed distance function. We use transversality theory to prove that for generic embeddings of a smooth compact surface in $\mathbb{R}^3$, signed distance functions admit finitely many non-degenerate critical points. This gives rise to our second contribution, which is that shapes and textures can both now be quantified and rigorously characterized in the language of persistent homology: signed distance persistence modules of generic shapes admit a finite barcode decomposition whose birth and death points can be classified and described geometrically. We use this approach to quantify shape textures on both simulated data and real vascular data from biology.