The Taub Faculty of Computer Science Events and Talks

We will address the problem of representing shapes using eigenvalues and eigenvectors of the discrete diffusion operator. A discrete shape, such as a mesh or a point cloud, can be viewed as an undirected weighted graph, hence one can use spectral graph theory to both embed and analyse shapes. We propose two graph diffusion operators that are built based on two widely used graph Laplacians: The combinatorial Laplacian and the normalized Laplacian. We recapitulate the basic spectral properties of these matrices and we thoroughly motivate the choice of the combinatorial Laplacian. The embedded shape is therefore treated as a distribution in the corresponding embedded (or feature) space. We propose several possible normalizations and we characterize some of their statistical properties. We will illustrate the usefulness of this shape representations for the task of sparse shape matching.