On a nonlinear multigrid algorithm with primal relaxation for the image total variation minimisation

Abstract

Digital image restoration has drawn much attention in the recent years and a lot of research has been done on effective variational partial differential equation models and their theoretical studies. However there remains an urgent need to develop fast and robust iterative solvers, as the underlying problem sizes are large. This paper proposes a fast multigrid method using primal relaxations. The basic primal relaxation is known to get stuck at a Flocal_ non-stationary minimum of the solution, which is usually believed to be Fnonsmooth_. Our idea is to utilize coarse level corrections, overcoming the deadlock of a basic primal relaxation scheme. A further refinement is to allow non-regular coarse levels to correct the solution, which helps to improve the multilevel method. Numerical experiments on both 1D and 2D images are presented.

Fast Multilevel Algorithm for a Minimization Problem in Impulse Noise Removal

“…The answer is no, if the functional J is non-smooth. However, in [7], we found that the wrongly converged solution is only incorrect near flat patches of the solution. The idea of detecting such flat patches during iteration and incorporating new local minimizations based on the patches was suggested in [7].…”

“…We now briefly review the multilevel method proposed in [7] for removing Gaussian noise. One nice advantage about the method is that it can be applied to non-smooth functionals as it does not require their derivatives.…”

“…In this paper, we will develop a multilevel acceleration method for this optimization problem which is based on the method in [7] for Gaussian noise, and is more robust and reliable even for non-smooth regularization functional such as the total-variation norm model [19].…”

“…Standard multigrid [2,11] is difficult to use on the equation since we are solving it on the irregular grid points N and coarsening will be a problem. The method that we are going to use is a multilevel method that solve the minimization problems (1) and (5) directly and has been shown to work well for Gaussian noise removal [7]. Since minimization is not done for the whole index set A, but on N , special modification is needed.…”

“…The plan is to review our recently proposed multilevel method for the Gaussian noise removal [7] in Section 2. Section 3 presents details of implementation of a multilevel method for model (1).…”