Optimal second order covariant derivatives on associated bundles — Application to color image restoration

This paper deals with the application of a geometric setting widely employed in theoretical physics, namely the fiber bundles, to color image restoration. The key idea of this approach is to model an image as a function on a principal bundle satisfying an equivariance property with respect to the action of a Lie group acting on its pixel values, and which can model the lighting changes in a scene. In this context, a natural tool for the differentiation of an image is by means of a covariant derivative. In previous works, optimal covariant derivatives have been constructed as solutions of a variational model consisting of the minimization of the L2 norm of the covariant derivative of the image, and applied to various tasks in color image restoration through the extension of the Total Variation regularizer to vector bundles. The aim of this paper is to extend these works by constructing optimal second order covariant derivatives as solutions of the minimization of the norm of the second order covariant derivative of the image. Experiments on deblurring and super-resolution corroborate the relevance of the proposed model for color image restoration. More generally, this paper validates the use of the geometric setting of fiber bundles in imaging sciences.