Iterative Multigrid Regularization Techniques for Image Matching

Abstract

In this paper, we consider the problem of matching images, i.e., to find a deformation u, which transforms a digital image into another such that the images have nearly equal gray values in every image element. The difference of the two images is measured by their L2 -difference, which should be minimized. This yields a nonlinear ill conditioned inverse problem for u, so the numerical solution is quite difficult. A Tikhonov regularization method is considered to rule out discontinuous and irregular solutions to the minimization problem. An important problem is a proper choice of the regularization parameter $\alpha$. For the practical choice of $\alpha,$ we use iterative regularization methods based on multigrid techniques. To obtain a suitable initial guess, we use an approach similar to the full multigrid (FMG) developed by Brandt [Math. Comp., 31 (1977), pp. 333--390]. The algorithms have optimal complexity: the amount of work is proportional to the number of picture elements. Finally, we present some experimental results for synthetic and real images.