Discrete & Computational Geometry

, Volume 55, Issue 4, pp 854–906 | Cite as

Categorified Reeb Graphs

Article

Abstract

The Reeb graph is a construction which originated in Morse theory to study a real-valued function defined on a topological space. More recently, it has been used in various applications to study noisy data which creates a desire to define a measure of similarity between these structures. Here, we exploit the fact that the category of Reeb graphs is equivalent to the category of a particular class of cosheaf. Using this equivalency, we can define an ‘interleaving’ distance between Reeb graphs which is stable under the perturbation of a function. Along the way, we obtain a natural construction for smoothing a Reeb graph to reduce its topological complexity. The smoothed Reeb graph can be constructed in polynomial time.

Keywords

Reeb graph Cosheaf Interleaving distance Stability Smoothing 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.ClaremontUSA
  2. 2.AlbanyUSA
  3. 3.PrincetonUSA

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