Abstract
In the last decade, the technological progress in telecommunication, hardware design and multimedia, allows access to an ever finer three-dimensional (3-D) modeling of the world. While most researchers have focused on the field of 3D objects, now it is necessary to turn to 3D time domain (3D+t). 3D dynamic meshes are becoming a media of increasing importance. This 3D content is subject to various processing operations such as indexation, segmentation or compression. However, surface mesh is an extrinsic shape representation. Therefore, it suffers from important variability under different sampling strategies and canonical shape-non-altering surface transformations, such as affine or isometric transformations. Consequently it needs an intrinsic structural descriptor before being processed by one of the aforementioned processing operations. The research topic of this thesis work is the topological modeling based on Reeb graphs. Specifically, we focus on 3D shapes represented by triangulated surfaces. Our objective is to propose a new approach, of Reeb graph construction, which exploits the temporal information. The main contribution consists in defining a new continuous function based on the heat diffusion properties. The latter is computed from the discrete representation of the shape to obtain a topological structure.The restriction of the heat kernel to temporal domain makes the proposed function intrinsic and stable against transformation. Due to the presence of neighborhood information in the heat kernel, the proposed Reeb Graph construction approach can be extremely useful as local shape descriptor for non-rigid shape retrieval. It can also be introduced into a segmentation-based dynamic compression scheme in order to infer the functional parts of a 3D shape by decomposing it into parts of uniform motion. In this context, we apply the concept of Reeb graph in two widely used applications which are pattern recognition and compression.Reeb graph has been known as an interesting candidate for 3D shape intrinsic structural representation. we propose a 3D non rigid shape recognition approach. The main contribution consists in defining a new scalar function to construct the Reeb graph. This function is computed based on the diffusion distance. For matching purpose, the constructed Reeb graph is segmented into Reeb charts, which are associated with a couple of geometrical signatures. The matching between two Reeb charts is performed based on the distances between their corresponding signatures. As a result, the global similarity is estimated based on the minimum distance between Reeb chart pairs. Skeletonisation and segmentation tasks are closely related. Mesh segmentation can be formulated as graph clustering. First we propose an implicit segmentation method which consists in partitioning mesh sequences, with constant connectivity, based on the Reeb graph construction method. Regions are separated according to the values of the proposed continuous function while adding a refinement step based on curvature and boundary information.Intrinsic mesh surface segmentation has been studied in the field of computer vision, especially for compression and simplification purposes. Therefore we present a segmentation-based compression scheme for animated sequences of meshes with constant connectivity. The proposed method exploits the temporal coherence of the geometry component by using the heat diffusion properties during the segmentation process. The motion of the resulting regions is accurately described by 3D affine transforms. These transforms are computed at the first frame to match the subsequent ones. In order to improve the performance of our coding scheme, the quantization of temporal prediction errors is optimized by using a bit allocation procedure. The objective aimed at is to control the compression rate while minimizing the reconstruction error.