Abstract. We analyze a variational problem for the recovery of vector valued functions and we compute its numerical solution. The data of the problem are a small set of complete samples of the vector valued function and a significant incomplete information where the former are missing. The incomplete information is assumed as the result of a distortion, with values in a lower dimensional manifold. For the recovery of the function we minimize a functional which is formed by the discrepancy with respect to the data and total variation regularization constraints. We show existence of minimizers in the space of vector valued BV functions. For the computation of minimizers we provide a stable and efficient method. First we approximate the functional by coercive functionals on W 1,2 in terms of Γ-convergence. Then we realize approximations of minimizers of the latter functionals by an iterative procedure to solve the PDE system of the corresponding Euler-Lagrange equations. The numerical implementation comes naturally by finite element discretization. We apply the algorithm to the restoration of color images from a limited color information and gray levels where the colors are missing. The numerical experiments show that this scheme is very fast and robust. The reconstruction capabilities of the model are shown, also from very limited (randomly distributed) color data. Several examples are included from the real restoration problem of the A. Mantegna’s art frescoes in Italy.

Abstract. In this paper the reconstruction of damaged piecewice constant color images is studied using a RGB total variation based model for colorization/inpainting. In particular, it is shown that when color is known in a uniformly distributed region, then reconstruction is possible with maximal fidelity. Contents

Abstract Can one of the most important Italian Renaissance frescoes reduced in hundreds of thousand fragments by a bombing during the Second World War be re-composed after more than 60 years from its damage? Can we reconstruct the missing parts and can we say something about their original color? In this short paper we want to exemplify, hopefully effectively by taking advantage of the seduction of art, how mathematics today can be applied in real-life problems which were considered unsolvable only few years ago. 1