Heat Equations on Vector Bundles—Application to Color Image Regularization

We use the framework of heat equations associated to generalized Laplacians on vector bundles over Riemannian manifolds in order to regularize color images. We show that most methods devoted to image regularization may be considered in this framework. From a geometric viewpoint, they differ by the metric of the base manifold and the connection of the vector bundle involved. By the regularization operator we propose in this paper, the diffusion process is completely determined by the geometry of the vector bundle. More precisely, the metric of the base manifold determines the anisotropy of the diffusion through the computation of geodesic distances whereas the connection determines the data regularized by the diffusion process through the computation of the parallel transport maps. This regularization operator generalizes the ones based on short-time Beltrami kernel and oriented Gaussian kernels. Then we construct particular connections and metrics involving color information such as luminance and chrominance in order to perform new kinds of regularization.