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Reeb Graph of Sample Thickenings

Håvard Bakke Bjerkevik, Nello Blaser, Lars M. Salbu

TL;DR

The paper develops a framework to approximate the Reeb graph of an unknown space from samples by thickening point clouds and leveraging a stability theory via interleaving distances. It generalizes Reeb-graph stability to all $\mathbb{R}$-spaces using a Reeb precosheaf, and introduces a Reeb-approximation template based on path deformation retractions to transfer geometric reconstruction results to Reeb graphs. It then specializes the theory to closed Euclidean subsets with positive reach and closed subsets of Riemannian manifolds, providing concrete error bounds under sampling assumptions. Finally, it presents a practical algorithm for computing the Reeb graph of a sample thickening with an analysis showing near-optimal time complexity for the problem class.

Abstract

We consider the Reeb graph of a thickening of points sampled from an unknown space. Our main contribution is a framework to transfer reconstruction results similar to the well-known work of Niyogi, Smale, and Weinberger to the setting of Reeb graphs. To this end, we first generalize and study the interleaving distances for Reeb graphs. We find that many of the results previously established for constructible spaces also hold for general topological spaces. We use this to show that under certain conditions for topological spaces with real-valued Lipschitz maps, the Reeb graph of a sample thickening approximates the Reeb graph of the underlying space. Finally, we provide an algorithm for computing the Reeb graph of a sample thickening.

Reeb Graph of Sample Thickenings

TL;DR

The paper develops a framework to approximate the Reeb graph of an unknown space from samples by thickening point clouds and leveraging a stability theory via interleaving distances. It generalizes Reeb-graph stability to all -spaces using a Reeb precosheaf, and introduces a Reeb-approximation template based on path deformation retractions to transfer geometric reconstruction results to Reeb graphs. It then specializes the theory to closed Euclidean subsets with positive reach and closed subsets of Riemannian manifolds, providing concrete error bounds under sampling assumptions. Finally, it presents a practical algorithm for computing the Reeb graph of a sample thickening with an analysis showing near-optimal time complexity for the problem class.

Abstract

We consider the Reeb graph of a thickening of points sampled from an unknown space. Our main contribution is a framework to transfer reconstruction results similar to the well-known work of Niyogi, Smale, and Weinberger to the setting of Reeb graphs. To this end, we first generalize and study the interleaving distances for Reeb graphs. We find that many of the results previously established for constructible spaces also hold for general topological spaces. We use this to show that under certain conditions for topological spaces with real-valued Lipschitz maps, the Reeb graph of a sample thickening approximates the Reeb graph of the underlying space. Finally, we provide an algorithm for computing the Reeb graph of a sample thickening.

Paper Structure

This paper contains 12 sections, 24 theorems, 40 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Acknowledgments
  4. The Reeb Functor
  5. Reeb Precosheaf and Interleaving
  6. Reeb Approximation Framework
  7. Reeb Approximation from Reconstruction
  8. Closed Euclidean Subsets
  9. Reeb approximation from closed subsets of Riemannian manifolds
  10. Computation
  11. The algorithm
  12. Analysis of the algorithm

Key Result

Lemma 2.2

If $f:X\to \mathbb{R}$ is a continuous map and $U\subseteq \mathbb{R}$ is an open subset, then

Figures (1)

  • Figure 1: From left to right: the discs of radius $1$ around the points in $A$, the images of the discs and pairwise intersections, and the constructed Reeb graphs with partitions before and after handling the last interval $I_{r,s}$. We use the shorthand $\{pq,r\}$ for $\{\{p,q\},\{r\}\}$, etc.

Theorems & Definitions (51)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: Generalizing bauer_2021
  • proof
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • ...and 41 more