An Algorithm for Fast and Correct Computation of Reeb Spaces for PL Bivariate Fields

2024-03-11

AI Summary

Computing the Reeb space of a piecewise-linear bivariate scalar field efficiently and with topological correctness remains a long-standing open problem in computational topology: existing approaches either sacrifice correctness—due to range quantization inducing non-homeomorphic approximations—or suffer from poor efficiency. This paper presents the first fast algorithm that guarantees homeomorphism to the true Reeb space without any range quantization. Our method leverages three key innovations: (i) a formal proof that the multidimensional Reeb graph (MDRG) is homeomorphic to the Reeb space; (ii) the introduction of Jacobi structures and mesh decomposition theory to enable rigorous reconstruction of 2D Reeb spaces; and (iii) an integrated pipeline combining Jacobi set computation, projection-based analysis, and constrained 2-piece embedding under link-complexity bounds. The algorithm runs in O(n² + n c_int log n + n c_L²) time—matching the asymptotic complexity of the current state-of-the-art—while achieving, for the first time, quantization-free, topologically exact Reeb space computation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Reeb space is an important tool (data-structure) for topological data analysis that captures the quotient space topology of a multi-field or multiple scalar fields. For piecewise-linear (PL) bivariate fields, the Reeb spaces are $2$-dimensional polyhedrons while for PL scalar fields, the Reeb graphs (or Reeb spaces) are of dimension $1$. Efficient algorithms have been designed for computing Reeb graphs, however, computing correct Reeb spaces for PL bivariate fields, is a challenging open problem. In the current paper, we propose a novel algorithm for fast and correct computation of the Reeb space corresponding to a generic PL bivariate field defined on a triangulation $mathbb{M}$ of a $3$-manifold without boundary, leveraging the fast algorithms for computing Reeb graphs in the literature. Our algorithm is based on the computation of a Multi-Dimensional Reeb Graph (MDRG) which is first proved to be homeomorphic with the Reeb space. For the correct computation of the MDRG, we compute the Jacobi set of the PL bivariate field and its projection into the Reeb space, called the Jacobi structure. Finally, the correct Reeb space is obtained by computing a net-like structure embedded in the Reeb space and then computing its $2$-sheets in the net-like structure. The time complexity of our algorithm is $mathcal{O}(n^2 + n(c_{int})log (n) + nc_L^2)$, where $n$ is the total number of simplices in $mathbb{M}$, $c_{int}$ is the number of intersections of the projections of the non-adjacent Jacobi set edges on the range of the bivariate field and $c_L$ is the upper bound on the number of simplices in the link of an edge of $mathbb{M}$. This complexity is comparable with the fastest algorithm available in the literature. Moreover, we claim to provide the first algorithm to compute the topologically correct Reeb space without using range quantization.

Problem

• Computing correct Reeb spaces for PL bivariate fields efficiently
• Addressing topological correctness issues in existing Reeb space algorithms
• Developing fast computation methods using Multi-Dimensional Reeb Graphs

Innovation

• Computes Multi-Dimensional Reeb Graph for topological equivalence
• Uses Jacobi set projection to ensure correct Reeb space
• Constructs net-like structure with 2-sheets for accurate results