Robust Geometric Predicates for Bivariate Computational Topology

TL;DR

Robust computation of bivariate Jacobi sets and Reeb spaces is challenging due to numerical errors and degeneracies in multifield topology. The paper develops exact-arithmetic and symbolic-perturbation techniques (SoS, perturbing the world, Yap's schemes) to transform degenerate maps into generic ones and to robustly evaluate complex symbolic predicates. It also introduces an automatic method to construct evaluation tables for large symbolic expressions and provides implementations and performance assessments. Real-world data contain significant degeneracies, but experiments show most cases resolve with shallow perturbations, enabling practical robust computation. The work advances dependable multifield topology analysis and toolchains for integrating robust predicates into visualization and analysis pipelines.

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.

Figures (8)

• Figure 1: Examples of different Reeb spaces.
• Figure 2: Examples of the degenerate cases for generic bivariate PL maps. Each point and segment in the plane is the image of a vertex and edge of the input mesh.
• Figure 3: Overview: This work primarily investigates two bivariate topological structures -- Jacobi sets and Reeb spaces. The computation of these structures relies on two key predicates, for which we present various algebraic formulations. Finally, we examine three distinct perturbation schemes aimed at ensuring robust implementation of these predicates.
• Figure 4: The relative order of intersection of three segments can be determined by comparing the parameters $t$ and $t'$ of the intersection points over a parametrization of the first segment. We compute $t$ and $t'$ using the scalar cross product $\times$ with $r, s, s', r$ and $r'$.
• Figure 5: Two perturbations of line segments that remove different types of degenerate cases. Perturbing the segments (b) as four-dimensional points resolves all degenerate cases, but segments that used to share endpoints no longer do. Perturbing the endpoints of segments (c) keeps the overlapping endpoint incidence relationship between segments, while removing all other degenerate cases.

Theorems & Definitions (2)

• Lemma 1
• proof