A Class of generalized Laplacians on vector bundles devoted to multi-channels image processing

Thomas Batard (1) , Nir Sochen (1)

1 TAU - School of Mathematical Sciences [Tel Aviv]

In the context of fiber bundles theory, there exist some differential operators of order 2 , called generalized Laplacians, acting on sections of vector bundles over Riemannian manifolds, and generalizing the Laplace-Beltrami operator. Such operators are determined by the choice of a covariant derivative on the vector bundle. In this paper, we construct a class of generalized Laplacians, devoted to multi-channels image processing, from the construction of particular covariant derivatives. The construction requires to deal with the notion of associated bundle, that relates principal and vector bundles by the choice of a group representation. In particular, covariant derivatives are determined by connection 1-forms on principal bundles. We consider a minimization problem to construct particular connection 1-forms. Then, from the heat equation of the corresponding Laplacian, we obtain a class of diffusions whose behaviours depend of the choice of the group representation. We provide experiments on grey-level and color images.

Mots clés: Multi-Channel Image Processing; Variational Methods; Heat Equation; Generalized Laplacian; Differential Geometry

Domaines: Géométrie différentielle [math.DG]; Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Thomas Batard, Nir Sochen. A Class of generalized Laplacians on vector bundles devoted to multi-channels image processing. 2012. ⟨hal-00683953v3⟩