Efficient Numerical Methods for Initial Value Control Problems for Diffeomorphic Image Registration

We discuss numerical algorithms for diffeomorphic image registration. We present extensions for a framework for diffeomorphic image registration termed CLAIRE. In diffeomorphic image registration, we seek a diffeomorphic map that establishes spatial correspondence between two views (images) of the same scene or object. In the continuum, this represents an infinite-dimensional, inverse problem. We formulate diffeomorphic image registration as a partial differential equation constrained optimization problem. The partial differential equations enter the formulation as equality constraints. We consider hyperbolic transport equations as well as Euler Poincaré equations associated with the group of diffeomorphisms. The contributions of this thesis are three-fold: (i) We study the performance of the graphics processing unit implementation of CLAIRE for (large-scale) biomedical imaging studies. (ii) We develop numerical algorithms for the solution of a variational optimization problem for diffeomorphic image registration governed by a hyperbolic transport equation and the Euler Poincaré equations associated with the group of diffeomorphisms. This represents the main contribution of this thesis. (iii) We present preliminary results for a computational framework for uncertainty quantification for the latter formulation that exploits the numerical algorithms developed in this thesis.. We report results to assess the rate of convergence, computational performance, reconstruction accuracy, and scalability of our methods.