With the ongoing development of 3D technologies, 3D shapes are becoming an interactive media of major importance. Their commonest representation, the surface mesh, suffers however from high variability towards standard shape-preserving surface transformations.It is necessary thus to design intrinsic shape modeling techniques. In this thesis, we explore topological modeling by studying Reeb graph based structures. In particular, we introduce a novel shape abstraction, called the enhanced topological skeleton, which enables not only the study of the topological evolution of Morse functions' level sets but also that of their geometrical evolution. We show the utility of this intrinsic shape representation in three research problems related to Computer Graphics and Computer Vision. First, we introduce the notion of geometrical calculus on Reeb graphs for the stable and automatic computation of control skeletons for interactive shape handling. Then, by introducing the notions of Reeb chart and Reeb pattern, we propose a new method for partial 3D shape similarity estimation. We show this approach outperforms the competing methods of the international SHape REtrieval Contest 2007 by a gain of 14%. Finally, we present two techniques for the functional decomposition computation of a 3D shape, both from human perception based heuristics and from the analysis of time-varying 3D data. For each of these research problems, concrete applicative examples are presented to assess the utility of our approach.
Reeb graph based 3D shape modeling and applications