Clifford Bundles: a Unifying Framework for Images(videos), Vector Fields and Orthonormal Frame Fields Regularization

The aim of this paper is to present a new framework for regularization by diffusion. The methods we develop in the sequel can be used to smooth nD images, nD videos, vector fields and orthonormal frame fields in any dimension.1 From a mathematical viewpoint, we deal with vector bundles over Riemannian manifolds and socalled generalized Laplacians. Sections are regularized from heat equations associated to generalized Laplacians, the solution being given as convolution by generalized heat kernels. The anisotropy of the diffusion is controlled by the metric of the base manifold and by the connection of the vector bundle. It finds applications to images and videos anisotropic regularization. The main topic of this paper is to show that this approach can be extended to other fields such as vector fields and orthonormal frame fields by considering the context of Clifford algebras. We introduce a Clifford-Hodge operator (and the corresponding Clifford-Hodge flow) as a generalized Laplacian on the Clifford bundle of a Riemannian manifold. Laplace-Beltrami diffusion appears as a particular case of Clifford-Hodge diffusion, namely diffusion for degree 0 sections (functions). Considering base manifolds of dimension 2 and 3, applications to multispectral images, multispectral videos, 2D and 3D vector fields and orthonormal frame fields regularization may be envisaged. Some of them are presented in this paper.