Restoration of color images by vector valued BV functions and variational calculus

We analyze a variational problem for the recovery of vector valued functions and compute its numerical solution. The data of the problem are a small set of complete samples of the vector valued function and some 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 the existence of minimizers in the space of vector valued bounded variation functions. For the computation of minimizers we provide a stable and efficient method. First, we approximate the functional by coercive functionals on W-1,W-2 in terms of Gamma- 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 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 A. Mantegna's art frescoes in Italy.