Multigrid Algorithm for High Order Denoising

Image denoising has been a research topic deeply investigated within the last two decades. Excellent results have been obtained by using such models as the total variation (TV) minimization by Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259-268], which involves solving a second order PDE. In more recent years some effort has been made [Y.-L. You and M. Kaveh, IEEE Trans. Image Process., 9 (2000), pp. 1723-1730; M. Lysaker, S. Osher, and X.-C. Tai, IEEE Trans. Image Process., 13 (2004), pp. 1345-1357; M. Lysaker, A. Lundervold, and X.-C. Tai, IEEE Trans. Image Process., 12 (2003), pp. 1579-1590; Y. Chen, S. Levine, and M. Rao, SIAM J. Appl. Math., 66 (2006), pp. 13831406] in improving these results by using higher order models, particularly to avoid the staircase effect inherent to the solution of the TV model. However, the construction of stable numerical schemes for the resulting PDEs arising from the minimization of such high order models has proved to be very difficult due to high nonlinearity and stiffness. In this paper, we study a curvature, based energy minimizing model [W. Zhu and T. F. Chan, Image Denoising Using Mean Curvature, preprint, http://www.math.nyu.edu/similar to wzhu/], for which one has to solve a fourth order PDE. For this model we develop two new algorithms: a stabilized fixed point method and, based upon this, an efficient nonlinear multigrid (MG) algorithm. We will show numerical experiments to demonstrate the very good performance of our MG algorithm.