Robust Geometric Predicates for Bivariate Computational Topology

Abstract

We present theory and practice for robust implementations of bivariate Jacobi set and Reeb space algorithms. Robustness is a fundamental topic in computational geometry that deals with the issues of numerical errors and degenerate cases in algorithm implementations. Computational topology already uses some robustness techniques for the development of scalar field algorithms, such as those for computing critical points, merge trees, contour trees, Reeb graphs, Morse-Smale complexes, and persistent homology. In most cases, robustness can be ensured with floating-point arithmetic, and degenerate cases can be resolved with a standard symbolic perturbation technique called Simulation of Simplicity. However, this becomes much more complex for topological data structures of multifields, such as Jacobi sets and Reeb spaces. The geometric predicates used in their computation require exact arithmetic and a more involved treatment of degenerate cases to ensure correctness. Neither of these challenges has been fully addressed in the literature so far. In this paper, we describe how exact arithmetic and symbolic perturbation schemes can be used to enable robust implementations of bivariate Jacobi set and Reeb space algorithms. In the process, we develop a method for automatically evaluating predicates that can be expressed as large symbolic polynomials, which are difficult to factor appropriately by hand, as is typically done in the computational geometry literature. We provide implementations of all proposed approaches and evaluate their efficiency.