Reeb graph
Encyclopedia
In Morse theory
, a branch of mathematics, a Reeb graph of a scalar function describes the connectivity
of its level sets.Reeb graphs are named after Georges Reeb
.
If the function is defined over a vector space
rather than over a more general manifold
, the Reeb graph forms a polytree
(a directed graph formed by assigning orientations to the edges of an undirected forest). In this special case, it is also called a contour tree.
Reeb graphs and contour trees have a wide variety of applications including computer aided geometric design, topology-based shape matching
, topological simplification and cleaning, surface segmentation and parametrization, and efficient computation of level sets.
Morse theory
In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
, a branch of mathematics, a Reeb graph of a scalar function describes the connectivity
Connectivity (graph theory)
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements which need to be removed to disconnect the remaining nodes from each other. It is closely related to the theory of network flow problems...
of its level sets.Reeb graphs are named after Georges Reeb
Georges Reeb
Georges Henri Reeb was a French mathematician. He worked in differential topology, differential geometry, differential equations, topological dynamical systems theory and non-standard analysis....
.
If the function is defined over a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
rather than over a more general manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, the Reeb graph forms a polytree
Polytree
In graph theory, a polytree is a directed graph with at most one undirected path between any two vertices. In other words, a polytree is a directed acyclic graph for which there are no undirected cycles either...
(a directed graph formed by assigning orientations to the edges of an undirected forest). In this special case, it is also called a contour tree.
Reeb graphs and contour trees have a wide variety of applications including computer aided geometric design, topology-based shape matching
, topological simplification and cleaning, surface segmentation and parametrization, and efficient computation of level sets.