Clifford Bundles: A Common Framework for Image, Vector Field, and Orthonormal Frame Field Regularization

The aim of this paper is to present a new framework for regularization by diffusion. The methods we develop in what follows can be used to smooth multichannel images, multichannel image sequences (videos), vector fields, and orthonormal frame fields in any dimension. From a mathematical viewpoint, we deal with vector bundles over Riemannian manifolds and so-called generalized Laplacians. Sections are regularized from heat equations associated with generalized Laplacians, the solutions being approximated by convolutions with kernels. Then, the behavior of the diffusion is determined by the geometry of the vector bundle, i.e., by the metric of the base manifold and by a connection on the vector bundle. For instance, the heat equation associated with the Laplace-Beltrami operator can be considered from this point of view for applications to images and video regularization. The main topic of this paper is to show that this approach can be extended in several ways to vector fields and orthonormal frame fields by considering the context of Clifford algebras. We introduce Clifford-Beltrami and Clifford-Hodge operators as generalized Laplacians on Clifford bundles over Riemannian manifolds. Laplace-Beltrami diffusion appears as a particular case of diffusion for degree 0 sections (functions). Dealing with base manifolds of dimension 2, applications to multichannel image, two-dimensional vector field, and orientation field regularization are presented.