Abstract

The great challenge in 3D object processing is to devise computationally efficient algorithms for recovering 3D models contaminated by noise and preserving their geometrical structure. The first problem addressed in this thesis is object denoising formulated in the discrete variational framework. We introduce a 3D mesh denoising method based on kernel density estimation. The proposed approach is able to reduce the over-smoothing effect and effectively remove undesirable noise while preserving prominent geometric features of a 3D mesh such as sharp features and fine details. The feasibility of the approach is demonstrated through extensive experiments. The rest of the thesis is devoted to a joint exploitation of geometry and topology of 3D objects for as parsimonious as possible representation of models and its subsequent application in object modeling, compression, and hashing problems. We introduce a 3D mesh compression technique using the centroidal mesh neighborhood information. The key idea is to apply eigen-decomposition to the mesh umbrella matrix, and then discard the smallest eigenvalues/eigenvectors in order to reduce the dimensionality of the new spectral basis so that most of the energy is concentrated in the low frequency coefficients. We also present a hashing technique for 3D models using spectral graph theory and entropic spanning trees by partitioning a 3D triangle mesh into an ensemble of submeshes, and then applying eigen-decomposition to the Laplace-Beltrami matrix of each sub-mesh, followed by computing the hash value of each sub-mesh. Moreover, we introduce several statistical distributions to analyze the topological properties of 3D objects. These probabilistic distributions provide useful information about the way 3D mesh models are connected. Illustrating experiments with synthetic and real data are provided to demonstrate the feasibility and the much improved performance of the proposed approaches in 3D object compression, hashing, and modeling.