The Clifford-Hodge Flow: An Extension of the Beltrami Flow

Abstract

In this paper, we make use of the theory of Clifford algebras for anisotropic smoothing of vector-valued data. It provides a common framework to smooth functions, tangent vector fields and mappings taking values in $\mathfrak{so}(m)$, the Lie algebra of SO(m), defined on surfaces and more generally on Riemannian manifolds. Smoothing process arises from a convolution with a kernel associated to a second order differential operator: the Hodge Laplacian. It generalizes the Beltrami flow in the sense that the Laplace-Beltrami operator is the restriction to functions of minus the Hodge operator. We obtain a common framework for anisotropic smoothing of images, vector fields and oriented orthonormal frame fields defined on the charts.

This work was partially supported by the ONR Grant N00014-09-1-0493, and by the ”Communauté d’Agglomération de La Rochelle”.