Surface Shape Understanding based on Extended Reeb Graph Representation

In this chapter we introduce and describe the notion of Extended Reeb Graph (ERG). First of all, an overview of the usual definition of critical points and Morse complexes for smooth manifolds is given. Then, we will describe some existing methods which extend these concepts to piecewise linear 2-manifolds, focusing, in particular, on topological structures as surface networks and quasi-Morse complexes, available for analysis and simplification of triangular meshes. To avoid the dependency of such structures on the locality of the critical point definition we will propose to consider critical areas and influence zone instead of usual ones, and we will give their formal definition. We will base the ERG representation on this characterization and we will compare it with the surface network structure. Finally, we will describe the ERG structure and its construction process, also introducing several examples. Discussions and conclusions will appear in the last section.