Abstract
Reeb graphs are structural descriptors that capture shape properties of a topological space from the perspective of a chosen function. In this work, we define a combinatorial distance for Reeb graphs of orientable surfaces in terms of the cost necessary to transform one graph into another by edit operations. The main contributions of this paper are the stability property and the optimality of this edit distance. More precisely, the stability result states that changes in the Reeb graphs, measured by the edit distance, are as small as changes in the functions, measured by the maximum norm. The optimality result states that the edit distance discriminates Reeb graphs better than any other distance for Reeb graphs of surfaces satisfying the stability property.
Highlights
In shape comparison, a widely used scheme is to measure the dissimilarity between descriptors associated with each shape rather than to match shapes directly
A shape is modeled as a topological space X endowed with a scalar function f : X → R
Reeb graphs have been used as an effective tool for shape analysis and description tasks since [24, 23]
Summary
A widely used scheme is to measure the dissimilarity between descriptors associated with each shape rather than to match shapes directly. The basic properties we consider important for a metric between Reeb graphs are: the robustness to perturbations of the input functions; the ability to discriminate functions on the same manifold; the deployment of the combinatorial nature of graphs For this reason, we apply to the case of surfaces the same underlying ideas as used in [7] for curves. We deduce that the edit distance between the Reeb graphs of two functions f and g defined on a surface is a lower bound for the natural pseudo-distance between f and g obtained by minimizing the change in the functions due to the application of a self-diffeomorphism of the manifold, with respect to the maximum norm. A third question that would deserve investigation is how to generalize the edit distance to compare functions on non-homeomorphic surfaces as well, and the relationship with the functional distortion distance in that case
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