Arrange and Traverse Algorithm for Computation of Reeb Spaces of Piecewise Linear Maps

Abstract

We present the first combinatorial algorithm for efficiently computing the Reeb space in all dimensions. The Reeb space is a higher-dimensional generalization of the Reeb graph, which is standard practice in the analysis of scalar fields, along with other computational topology tools such as persistent homology and the Morse-Smale complex. One significant limitation of topological tools for scalar fields is that data often involves multiple variables, where joint analysis is more insightful. Generalizing topological data structures to multivariate data has proven challenging and the Reeb space is one of the few available options. However, none of the existing algorithms can efficiently compute the Reeb space in arbitrary dimensions and there are no available implementations which are robust with respect to numerical errors. We propose a new algorithm for computing the Reeb space of a generic piecewise linear map over a simplicial mesh of any dimension called arrange and traverse. We implement a robust specialization of our algorithm for tetrahedral meshes and evaluate it on real-life data.