In digital geometry processing and shape modelling, the Laplace-Beltrami and the heat diffusion operator, together with the corresponding Laplacian eigenmaps, harmonic and geometry-aware functions, have been used in several applications, which range from surface parameterization, deformation, and compression to segmentation, clustering, and comparison.

In this context, our research has been focused on the following activities.

(1) We have improved the results of previous work on the study of the spectral properties of the Laplacian eigenfunctions through the analysis of the numerical aspects behind the computation, robustness, and applications of mesh Laplacian operators. More precisely, the perturbation theory related to the eigenvalue problem has been used to address the numerical (in)stability in the computation of the Laplacian eigenfunctions and show at which extent it affects the analysis of the input surface. Then, the spectral properties of the Laplacian matrix have been exploited to introduce the Tikhonov regularization, the feature and scale-based spaces associated to triangulated surfaces. Finally, these concepts have been characterized through the analysis of their main properties and applications.

(2) We have derived a novel discretization of the heat kernel, which is linear, stable to an irregular sampling density of the input surface, and scale covariant. With respect to previous work, this last property makes the kernel particularly suitable for shape analysis and comparison; in fact, local and global changes of the surface correspond to a re-scaling of the time parameter without affecting its spectral component.

(3) We have studied the feature spaces and scale spaces that are induced by the proposed Laplace-Beltrami operator and heat kernel on surfaces, respectively. These spaces have been used to provide a multi-scale approximation of scalar functions defined on 3D shapes.

4. (4)We are investigating the selection of the Laplacian eigenvalues and the definition of novel signatures for shape
   

comparison. The shape can be represented by manifold as well as non-manifold triangle meshes; specializations of our approach to point-sampled surfaces are currently under investigation.