A Variational Loop Shrinking Analogy for Handle and Tunnel Detection and Reeb Graph Construction on Surfaces

Abstract

The humble loop shrinking property played a central role in the inception of modern topology but it has been eclipsed by more abstract algebraic formalisms. This is particularly true in the context of detecting relevant non-contractible loops on surfaces where elaborate homological and/or graph theoretical constructs are favored in algorithmic solutions. In this work, we devise a variational analogy to the loop shrinking property and show that it yields a simple, intuitive, yet powerful solution allowing a streamlined treatment of the problem of handle and tunnel loop detection. Our formalization tracks the evolution of a diffusion front randomly initiated on a single location on the surface. Capitalizing on a diffuse interface representation combined with a set of rules for concurrent front interactions, we develop a dynamic data structure for tracking the evolution on the surface encoded as a sparse matrix which serves for performing both diffusion numerics and loop detection and acts as the workhorse of our fully parallel implementation. The substantiated results suggest our approach outperforms state of the art and robustly copes with highly detailed geometric models. As a byproduct, our approach can be used to construct Reeb graphs by diffusion thus avoiding commonly encountered issues when using Morse functions.