Diffeomorphisms have received significant research focus in the medical image registration community over the past 15 years due in part to their desirable mathematical properties: the preservation of image topology and the guaranteed existence of a number of spatial derivatives. The research area began with fundamental mathematical developments detailing how a diffeomorphism can be defined and constructed in the image registration context, and subsequently, algorithms were proposed to implement the continuous domain diffeomorphic theory in the discrete domain. After several iterations in form, the geodesic regression formulation emerged, which can be understood as a natural generalization of Euclidean linear regression to a nonlinear manifold of diffeomorphisms.
Geodesic Regression in Diffeomorphisms (GRiD) is the optimization of an initial momentum field which parameterizes a geodesic flow of diffeomorphisms through a time series of images. The method involves several computational challenges: the optimization of a very high dimensional nonlinear objective function, the integration of several coupled systems of partial differential equations, and the implementation of several fundamental operations including composition of images with deformations, regularization of vector fields, and evaluation of possibly complex image similarity functionals. Additionally, several of these components have free parameters that must be selected carefully to ensure convergence and the biological validity of results.
GRiD theory offers many advantages over standard image registration for the study of image deformations over time; it provides a succinct but comprehensive summary of the primary mode of image deformation over time evident in a time series, all the while guaranteeing desirable mathematical properties of the transformation flow. This powerful theory will be immensely useful in the study of growth, development, and aging in both health and disease. However, the complicated nature of the algorithm has prevented its widespread adoption in the applied medical imaging community.
The goal of this dissertation is to exposit practical and down to earth derivation, implementation, and application of GRiD. Chapters 1-3 cover those topics exactly. Additionally chapters 4 and 5 cover methods for selecting image matching functional and determining some of the model's free parameters. Chapter 6 is a complete large scale study of atrophy in Alzheimer's disease using GRiD. Chapters 7 and 8 discuss a novel extension of the GRiD model wherein multiple GRiD optimizations inform each other simultaneously.