Calculus of Variations and Geometric Measure Theory

M. Fornasier - R. March

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

created by march on 21 Dec 2006
modified on 05 Feb 2008


Published Paper

Inserted: 21 dec 2006
Last Updated: 5 feb 2008

Journal: SIAM J. Appl. Math.
Volume: 68
Number: 2
Pages: 437-460
Year: 2007


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 $\Gamma$-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 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.