Hyperelastic Image Registration - Theory, Numerical Methods, and Applications

Finding geometrical correspondences between two images, called image registration, is one of the numerous challenging problems in image processing. Commonly, image registration is phrased as a variational problem that is known to be ill-posed. Thus, regularization is used to ensure the existence of solutions, to introduce prior knowledge about the expected solution, and to increase the robustness against noise. In this thesis, we examine a regularization functional based on hyperelasticity. This work gives a comprehensive overview of theory, numerical methods, and applications of hyperelastic regularization functionals in image registration. We transfer existence results for polyconvex functionals from variational calculus to image registration. Thereby, we show that solutions to hyperelastic registration problems are guaranteed to be one-to-one and that the regularizer is well-suited for volume- and local rigidity constraints. We describe a numerical method that combines a Galerkin finite element approach with a multi-level image registration framework. The variational problem is solved in the space of continuous and piecewise linear transformations. Thus, our regularization functional is computed exactly and numerical solutions are one-to-one as guaranteed by the theory. We further give estimates for discretization errors and describe our implementation based on the registration toolbox Flexible Algorithms for Image Registration (FAIR). We outline the great potential of hyperelastic registration methods based on three applications from Positron Emission Tomography (PET), Echo-Planar Imaging (EPI) and Dynamic Contrast Enhanced Magnetic Resonance Imaging (MRI). This thesis establishes techniques from variational calculus and numerical analysis into image registration tools that are useful for applications in medical imaging. Our findings motivate further investigation and wider application of hyperelastic regularization techniques.