skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Stability and Approximations for Decorated Reeb Spaces
Given a map f:X → M from a topological space X to a metric space M, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when M = ℝ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of f. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data.  more » « less
Award ID(s):
2324962 2107808
PAR ID:
10529430
Author(s) / Creator(s):
; ; ; ;
Editor(s):
Mulzer, Wolfgang; Phillips, Jeff M
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Volume:
293
ISSN:
1868-8969
ISBN:
978-3-95977-316-4
Page Range / eLocation ID:
293-293
Subject(s) / Keyword(s):
Reeb spaces Gromov-Hausdorff distance Persistent homology Mathematics of computing → Algebraic topology Theory of computation → Computational geometry
Format(s):
Medium: X Size: 17 pages; 4726540 bytes Other: application/pdf
Size(s):
17 pages 4726540 bytes
Right(s):
Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ; (, 40th International Symposium on Computational Geometry (SoCG 2024), Leibniz International Proceedings in Informatics (LIPIcs))
    Mulzer, Wolfgang; Phillips, Jeff M (Ed.)
    A Reeb graph is a graphical representation of a scalar function on a topological space that encodes the topology of the level sets. A Reeb space is a generalization of the Reeb graph to a multiparameter function. In this paper, we propose novel constructions of Reeb graphs and Reeb spaces that incorporate the use of a measure. Specifically, we introduce measure-theoretic Reeb graphs and Reeb spaces when the domain or the range is modeled as a metric measure space (i.e., a metric space equipped with a measure). Our main goal is to enhance the robustness of the Reeb graph and Reeb space in representing the topological features of a scalar field while accounting for the distribution of the measure. We first introduce a Reeb graph with local smoothing and prove its stability with respect to the interleaving distance. We then prove the stability of a Reeb graph of a metric measure space with respect to the measure, defined using the distance to a measure or the kernel distance to a measure, respectively. 
    more » « less
  2. ; ; (, Computing in Geometry and Topology)
    Reeb graphs are widely used in a range of fields for the purposes of analyzing and comparing complex spaces via a simpler combinatorial object. Further, they are closely related to extended persistence diagrams, which largely but not completely encode the information of the Reeb graph. In this paper, we investigate the effect on the persistence diagram of a particular continuous operation on Reeb graphs; namely the (truncated) smoothing operation. This construction arises in the context of the Reeb graph interleaving distance, but separately from that viewpoint provides a simplification of the Reeb graph which continuously shrinks small loops. We then use this characterization to initiate the study of inverse problems for Reeb graphs using smoothing by showing which paths in persistence diagram space (commonly known as vineyards) can be realized by a path in the space of Reeb graphs via these simple operations. This allows us to solve the inverse problem on a certain family of piecewise linear vineyards when fixing an initial Reeb graph. While this particular application is limited in scope, it suggests future directions to more broadly study the inverse problem on Reeb graphs. 
    more » « less
  3. (, Canadian Mathematical Bulletin)
    Abstract We give necessary and sufficient conditions for certain pushouts of topological spaces in the category of Čech’s closure spaces to agree with their pushout in the category of topological spaces. We prove that in these two categories, the constructions of cell complexes by a finite sequence of closed cell attachments, which attach arbitrarily many cells at a time, agree. Likewise, the constructions of CW complexes relative to a compactly generated weak Hausdorff space that attach only finitely many cells, also agree. On the other hand, we give examples showing that the constructions of finite-dimensional CW complexes, CW complexes of finite type, and relative CW complexes that attach only finitely many cells, need not agree. 
    more » « less
  4. ; ; ; (, Geometry & Topology)
    It is known that every contact form on a closed three-manifold has at least two simple Reeb orbits, and a generic contact form has infinitely many. We show that if there are exactly two simple Reeb orbits, then the contact form is nondegenerate. Combined with a previous result, this implies that the three-manifold is diffeomorphic to the three-sphere or a lens space, and the two simple Reeb orbits are the core circles of a genus-one Heegaard splitting. We also obtain further information about the Reeb dynamics and the contact structure. For example, the Reeb flow has a disk-like global surface of section and so its dynamics are described by a pseudorotation, the contact structure is universally tight, and in the case of the three-sphere the contact volume and the periods and rotation numbers of the simple Reeb orbits satisfy the same relations as for an irrational ellipsoid. 
    more » « less
  5. ; ; (, Forensic toxicology)
    Buchin, Kevin; Colin de Verdi\` (Ed.)
    In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we "chop off" parts near local minima and maxima during the course of smoothing, where the amount cut is controlled by a parameter τ. After formalizing truncation as a functor, we show that when applied after the smoothing functor, this prevents extensive expansion of the range of the function, and yields particularly nice properties (such as maintaining connectivity) when combined with smoothing for 0 ≤ τ ≤ 2ε, where ε is the smoothing parameter. Then, for the restriction of τ ∈ [0,ε], we have additional structure which we can take advantage of to construct a categorical flow for any choice of slope m ∈ [0,1]. Using the infrastructure built for a category with a flow, this then gives an interleaving distance for every m ∈ [0,1], which is a generalization of the original interleaving distance, which is the case m = 0. While the resulting metrics are not stable, we show that any pair of these for m, m' ∈ [0,1) are strongly equivalent metrics, which in turn gives stability of each metric up to a multiplicative constant. We conclude by discussing implications of this metric within the broader family of metrics for Reeb graphs. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email