Di Fabio, Barbara ; Landi, Claudia
(2011)
Reeb graphs of curves are stable under function perturbations.
[Preprint]
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Abstract
Reeb graphs provide a method to combinatorially describe the shape of a manifold endowed with a Morse function. 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 through editing moves. 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. We also prove that our editing distance is equal to the natural pseudo-distance, and, moreover, that it is lower bounded by the bottleneck distance of persistent homology.
Abstract
Reeb graphs provide a method to combinatorially describe the shape of a manifold endowed with a Morse function. 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 through editing moves. 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. We also prove that our editing distance is equal to the natural pseudo-distance, and, moreover, that it is lower bounded by the bottleneck distance of persistent homology.
Document type
Preprint
Creators
Keywords
shape similarity, editing distance, Morse function, natural stratification, natural pseudo-distance
Subjects
DOI
Deposit date
30 Mar 2011 12:15
Last modified
16 May 2011 12:17
URI
Other metadata
Document type
Preprint
Creators
Keywords
shape similarity, editing distance, Morse function, natural stratification, natural pseudo-distance
Subjects
DOI
Deposit date
30 Mar 2011 12:15
Last modified
16 May 2011 12:17
URI
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