Layered Reeb graphs for three-dimensional manifolds in boundary representation

Abstract

Reeb graphs are topological graphs originating in Morse theory, which represent the topological structure of a manifold by contracting the level set components of a scalar-valued function defined on it. The generalization to several functions leads to Reeb spaces, which are thus able to capture more features of an object. We introduce the layered Reeb graph as a discrete representation for Reeb spaces of 3D solids (embedded three-dimensional manifolds with boundary) with respect to two scalar-valued functions. After that we present an efficient algorithm for computing the layered Reeb graph, which uses only a boundary representation of the underlying three-dimensional manifold. This leads to substantial computational advantages if the manifold is given in a boundary representation, since no volumetric representation has to be constructed. However, this algorithm is applicable only if the defining functions satisfy certain conditions.