Résumé
Diffusion processes capture information about the geometry of an object such as its curvature, symmetries and particular points. The evolution of the diffusion is governed by the Laplace-Beltrami operator which presides to the diffusion on the manifold. In this paper, we define a new discrete adaptive Laplacian for digital objects, gener- alizing the operator defined on meshes. We study its eigenvalues and eigenvectors recovering interesting geometrical informations. We discuss its convergence towards the usual Laplacian operator especially on lat- tice of diamonds. We extend this definition to 3D shapes. Finally we use this Laplacian in classical but adaptive denoising of pictures preserving zones of interest like thin structures.