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, generalizing 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 lattice 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.