Let @math be a simplicial complex with a piecewise linear function @math . The Reeb graph @math is the quotient of @math , where we collapse each connected component of @math to a single point. Let the nodes of @math be all homologically critical points where any homology of the corresponding component of the level set @math changes. Then we can label every arc of @math with the Betti numbers @math of the corresponding @math -dimensional component of a level set. The homology labels give more information about the original complex @math than the classical Reeb graph. We describe a canonical embedding of a Reeb graph into a multi-page book (a star cross a line) and give a unique linear code of this book embedding.
Reeb graphs are used in image analysis @cite_9 as simple representations of shapes. The Reeb graph is an abstract graph that describes the linking structure of connected components in level sets of a real-valued function on a complex. Briefly, a complex is a union of vertices, edges, triangles, tetrahedra and so on. Formal Definition of a Reeb graph in section doesn't specify any canonical way to visualize this abstract graph in a low dimensional space.
@cite_9
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Reeb graphs are compact shape descriptors which convey topological information related to the level sets of a function defined on the shape. Their definition dates back to 1946, and finds its root in Morse theory. Reeb graphs as shape descriptors have been proposed to solve different problems arising in Computer Graphics, and nowadays they play a fundamental role in the field of computational topology for shape analysis. This paper provides an overview of the mathematical properties of Reeb graphs and reconstructs its history in the Computer Graphics context, with an eye towards directions of future research. 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 metric 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 functions, measured by the maximum norm, imply not greater changes in the corresponding Reeb graphs, measured by the edit distance. The optimality result states that our edit distance discriminates Reeb graphs better than any other metric for Reeb graphs of surfaces satisfying the stability property. 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. 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.