Stability of Reeb graphs under function perturbations: the case of closed curves

TL;DR

This paper demonstrates that Reeb graphs of closed curves are stable under small perturbations of Morse functions, ensuring their robustness for shape analysis in computer graphics.

Contribution

It introduces an editing distance for Reeb graphs of curves and proves their stability under function perturbations, advancing shape analysis methods.

Findings

Reeb graphs are stable under small Morse function perturbations.

An editing distance between Reeb graphs quantifies their differences.

Stability results support robustness in shape analysis applications.

Abstract

Reeb graphs provide a method for studying the shape of a manifold by encoding the evolution and arrangement of level sets of a simple Morse function defined on the manifold. Since their introduction in computer graphics they have been gaining popularity as an effective tool for shape analysis and matching. In this context one question deserving attention is whether Reeb graphs are robust against function perturbations. Focusing on 1-dimensional manifolds, we define an editing distance between Reeb graphs of curves, in terms of the cost necessary to transform one graph into another. Our main result is that changes in Morse functions induce smaller changes in the editing distance between Reeb graphs of curves, implying stability of Reeb graphs under function perturbations.