Abstract

In image registration, the problem is to transform one image to align with another image. One approach is based on the Monge-Kantorovich mass transfer problem. The goal is to find the optimal mapping M which minimizes the Kantorovich-Wasserstein distance. The optimal mapping can be written as M=grad u, where u satisfies the Monge-Ampere equation. In this talk, we will present a monotone discretization scheme for solving the Monge-Ampere equation. Our approach is to reformulate the Monge-Ampere equation as a Hamilton-Jacobi- Bellman (HJB)equation. We will develop a monotone discretization scheme and show that the method is consistent, stable, monotone, and hence convergent to a viscosity solution. We will then present a relaxation scheme which is a very slowly convergent method as a standalone solver but it is very effective for reducing high frequency errors. We will adopt it as a smoother for multigrid and demonstrate its smoothing properties. Finally,numerical results will be presented to illustrate the effectiveness of the method.