Generalized Morse Theory of Distance Functions to Surfaces for Persistent Homology

Explore a groundbreaking lecture that merges three distinct theories to quantify complex shape textures. Delve into the intersection of distance fields, persistent homology, and Morse theory as applied to shape representation and analysis. Learn how this innovative approach generalizes Morse theory to Euclidean distance functions of bounded sets with smooth boundaries. Discover how transversality theory is used to prove that generic embeddings of smooth compact surfaces in R3 yield signed distance functions with finitely many non-degenerate critical points. Understand the implications of this research for creating finite barcode decompositions of signed distance persistence modules, allowing for geometric classification of birth and death points. Examine practical applications of this methodology in both simulated data using the "curvatubes" model and real-world vascular data from leukaemic samples, gaining insights into how this approach can provide new biological understanding.