Optimal mass transport for adaptivity and image registration

Finding a coordinate transformation is a fundamental task in diverse fields of mathematical applications. In this thesis, we propose new techniques of computing coordinate transformations for two different applications. In the first application, the transformations are computed to generate adaptive meshes that are suitable for solving time-dependent partial differential equations in two or more spatial dimensions. For the second application, the transformations are developed for an elastic image registration. Registration means establishing a coordinate transformation between two or more images taken, for example, at different times or from different viewpoints. The transformations are formulated based on solving the optimal mass transport problem, also known as the Monge--Kantorovich problem. This problem concerns finding the best way of moving a pile of material from one site to another with minimum transportation cost. Two different methods are described to compute the transformations for each application. In the first method, the solution of the problem is obtained as the steady-state solution of a parabolic partial differential equation. The second method is a velocity based method in which a velocity field is obtained using the fluid dynamics formulation of the $L^2$~Monge--Kantorovich problem. The optimal mass transport approach for computing coordinate transformations has a number of useful features. The existence and uniqueness of the transformation are guaranteed from the Kantorovich theory. Moreover, it can be characterized as the gradient of a convex function, i.e., it is rotation free. A number of theoretical issues for computing these transformations are addressed. Several numerical experiments are presented to show the performance of the proposed approach for both adaptive grid generation and image registration for medical applications.