Abstract

A key issue in developing an accurate 3D shape recognition system is to design an efficient shape
descriptor for which an index can be built, and similarity queries can be answered efficiently. While
the overwhelming majority of prior work on 3D shape analysis has concentrated primarily on rigid
shape retrieval, many real objects such as articulated motions of humans are nonrigid and hence
can exhibit a variety of poses and deformations.
Motivated by the recent surge of interest in content-based analysis of 3D objects in computeraided
design and multimedia computing, we develop in this thesis a unified theoretical and computational
framework for 3D shape denoising and retrieval by incorporating insights gained from
algebraic graph theory and spectral geometry. We first present a regularized kernel diffusion for
3D shape denoising by solving partial differential equations in the weighted graph-theoretic framework.
Then, we introduce a computationally fast approach for surface denoising using the vertexcentered
finite volume method coupled with the mesh covariance fractional anisotropy. Additionally,
we propose a spectral-geometric shape skeleton for 3D object recognition based on the second
eigenfunction of the Laplace-Beltrami operator in a bid to capture the global and local geometry
of 3D shapes. To further enhance the 3D shape retrieval accuracy, we introduce a graph matching
approach by assigning geometric features to each endpoint of the shape skeleton. Extensive experiments
are carried out on two 3D shape benchmarks to assess the performance of the proposed
shape retrieval framework in comparison with state-of-the-art methods. The experimental results
show that the proposed shape descriptor delivers best-in-class shape retrieval performance.