A multigrid method for elastic image registration with additional structural constraints

This thesis deals with the solution of a nonlinear inverse problem arising in digital image registration. In image registration one seeks to compute a transformation between two images such that they become more similar in some sense. In the first part, we define the problem as the minimization of a regularized nonlinear least-squares functional, which measures the image difference and smoothness of the transformation. The nonlinear functional is linearized around a current approximation in order to obtain well-posed linear subproblems. The Hessian is replaced by an approximation that leads to an inexact Newton-type method, specifically a regularized Gauss-Newton method. A related gradient descent method is derived in the same framework for the purpose of comparison. In the next part of this thesis we study geometric multigrid methods for the solution of the linear subproblems (inner iteration). The type of regularization employed leads to a system of elliptic partial differential equations. For the regularized Gauss-Newton method the differential operator contains jumping coefficients that cannot be adequately dealt with by standard geometric multigrid methods. Modifications of the multigrid components that improve multigrid convergence and allow for fast and efficient computation are proposed. In the outer iteration a trust region strategy is used and the whole procedure is embedded in a multiresolution framework. Extensive numerical results for the multigrid (inner iteration), outer iteration, and multiresolution framework are given. In the last part a framework for the incorporation of additional structural constraints in the presented registration procedure is proposed. This framework is based on implicit representation of shapes via level sets. Representations for different types of shapes are discussed. Distance functionals of least-square type that can be easily plugged into the existing registration procedure are introduced. Examples for the different representations and the combination with image data are given.