Generalized Gradient on Vector Bundle - Application to Image Denoising

Thomas Batard (1) , Marcelo Bertalmío (1)

1 ETIC / UPF - Departament de Tecnologies de la Informació i les Comunicacions [Barcelona]

We introduce a gradient operator that generalizes the Euclidean and Riemannian gradients. This operator acts on sections of vector bundles and is determined by three geometric data: a Riemannian metric on the base manifold, a Riemannian metric and a covariant derivative on the vector bundle. Under the assumption that the covariant derivative is compatible with the metric of the vector bundle, we consider the problems of minimizing the L2 and L1 norms of the gradient. In the L2 case, the gradient descent for reaching the solutions is a heat equation of a differential operator of order two called connection Laplacian. We present an application to color image denoising by replacing the regularizing term in the Rudin-Osher-Fatemi (ROF) denoising model by the L1 norm of a generalized gradient associated with a well-chosen covariant derivative. Experiments are validated by computations of the PSNR and Q-index.

Mots clés: Generalized gradient; Riemannian manifold; Vector bundle; Total variation; Color image denoising; Rudin-Osher-Fatemi model

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, Marcelo Bertalmío. Generalized Gradient on Vector Bundle - Application to Image Denoising. 2013. ⟨hal-00782496⟩