The construction of average models of anatomy, as well as regression analysis of anatomical structures, are key issues in medical research, e.g., in the study of brain development and disease progression. When the underlying anatomical process can be modeled by parameters in a Euclidean space, classical statistical techniques are applicable. However, recent work suggests that attempts to describe anatomical differences using flat Euclidean spaces undermine our ability to represent natural biological variability. In response, this dissertation contributes to the development of a particular nonlinear shape analysis methodology.

This dissertation uses a nonlinear deformable model to measure anatomical change and define geometry-based averaging and regression for anatomical structures represented within medical images. Geometric differences are modeled by coordinate transformations, i.e., deformations, of underlying image coordinates. In order to represent local geometric changes and accommodate large deformations, these transformations are taken to be the group of diffeomorphisms with an associated metric.

A mean anatomical image is defined using this deformation-based metric via the Fréchet mean—the minimizer of the sum of squared distances. Similarly, a new method called manifold kernel regression is presented for estimating systematic changes—as a function of a predictor variable, such as age—from data in nonlinear spaces. It is defined by recasting kernel regression in terms of a kernel-weighted Fréchet mean. This method is applied to determine systematic geometric changes in the brain from a random design dataset of medical images. Finally, diffeomorphic image mapping is extended to accommodate extraneous structures—objects that are present in one image and absent in another and thus change image topology—by deflating them prior to the estimation of geometric change. The method is applied to quantify the motion of the prostate in the presence of transient bowel gas.