Persistent Homology of the Signed Distance and Morse Theory for Shape Texture Analysis

This talk brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Signed distance fields are fundamental to shape representation and analysis. Topological data analysis uses algebraic topology to characterize geometric and topological patterns in shapes. The most well-known tool from this approach is persistent homology that tracks the evolution of topological features in a dynamic manner as a barcode; here we study the persistent homology of shape textures using sublevel set filtration induced by the signed distance. Morse theory is a framework from differential topology that studies critical points of functions on manifolds and has been used to characterize the lifetimes of persistent homology features. However, a significant limitation here is that it cannot be readily applied to distance functions because of their lack of smoothness. In this paper, we generalize Morse theory to Euclidean distance functions of bounded sets with smooth boundaries. We use transversality theory to prove that for generic embeddings of a smooth compact surface in R3, signed distance functions admit finitely-many non-degenerate critical points. Thus, signed distance persistence modules of generic shapes admit a finite barcode decomposition whose birth and death points can be classified and described geometrically, providing a rigorous characterization of shape textures. We use this approach to quantify shape textures on both simulated data and real vascular data from biology. This is joint work with Anna Song (Imperial College London, The Francis Crick Institute) and Ka Man (Ambrose) Yim (Oxford).