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Reeb graph

AReeb graph^[1] (named after Georges ReebbyRené Thom) is a mathematical object reflecting the evolution of the level sets of a real-valued function on a manifold.^[2] According to ^[3] a similar concept was introduced by G.M. Adelson-Velskii and A.S. Kronrod and applied to analysis of Hilbert's thirteenth problem.^[4] Proposed by G. Reeb as a tool in Morse theory,^[5] Reeb graphs are the natural tool to study multivalued functional relationships between 2D scalar fields ${\displaystyle \psi }$, ${\displaystyle \lambda }$, and ${\displaystyle \phi }$ arising from the conditions ${\displaystyle \nabla \psi =\lambda \nabla \phi }$ and ${\displaystyle \lambda \neq 0}$, because these relationships are single-valued when restricted to a region associated with an individual edge of the Reeb graph. This general principle was first used to study neutral surfacesinoceanography.^[6]

Reeb graphs have also found a wide variety of applications in computational geometry and computer graphics,^[1][7] including computer aided geometric design, topology-based shape matching,^[8][9][10] topological data analysis,^[11] topological simplification and cleaning, surface segmentation ^[12] and parametrization, efficient computation of level sets, neuroscience,^[13] and geometrical thermodynamics.^[3] In a special case of a function on a flat space (technically a simply connected domain), the Reeb graph forms a polytree and is also called a contour tree.^[14]

Level set graphs help statistical inference related to estimating probability density functions and regression functions, and they can be used in cluster analysis and function optimization, among other things. ^[15]

Formal definition[edit]

Given a topological space X and a continuous function f: X → R, define an equivalence relation ~ on X where p~q whenever p and q belong to the same connected component of a single level set f⁻¹(c) for some real c. The Reeb graph is the quotient space X /~ endowed with the quotient topology.

Generally, this quotient space does not have the structure of a finite graph. Even for a smooth function on a smooth manifold, the Reeb graph can be not one-dimensional and even non-Hausdorff space.^[16]

In fact, the compactness of the manifold is crucial: The Reeb graph of a smooth function on a closed manifold is a one-dimensional Peano continuum that is homotopy equivalent to a finite graph.^[16] In particular, the Reeb graph of a smooth function on a closed manifold with a finite number of critical values---which is the case of Morse functions, Morse-Bott functions or functions with isolated critical points---has the structure of a finite graph.^[17]

Description for Morse functions[edit]

Iff is a Morse function with distinct critical values, the Reeb graph can be described more explicitly. Its nodes, or vertices, correspond to the critical level sets f⁻¹(c). The pattern in which the arcs, or edges, meet at the nodes/vertices reflects the change in topology of the level set f⁻¹(t) as t passes through the critical value c. For example, if c is a minimum or a maximum of f, a component is created or destroyed; consequently, an arc originates or terminates at the corresponding node, which has degree 1. If c is a saddle point of index 1 and two components of f⁻¹(t) merge at t = cast increases, the corresponding vertex of the Reeb graph has degree 3 and looks like the letter "Y"; the same reasoning applies if the index of c is dim X−1 and a component of f⁻¹(c) splits into two.

References[edit]