A multigrid method for a fourth-order diffusion equation with application to image processing

摘要

In this paper we are concerned with the efficient solution of the elliptic system (alpha L + I)u = f, where L is the biharmonic operator supplemented by higher-order boundary conditions. This is a core task in image processing arising from the minimization of rotation invariant energy functionals. For the purpose of numerical solution we approximate the partial differential equation by finite differences. In order to obtain efficient and robust solution methods we discuss how the system can be treated efficiently by multigrid methods. Finally, experimental results are reported that demonstrate the effectiveness of the multigrid approach.