This thesis presents the application and development of regularized methods for the statistical analysis of anatomical structures. Focus is on structure-function relationships in the human brain, such as the connection between early onset of Alzheimer’s disease and shape changes of the corpus callosum. One of the comprehensive goals of this type of research is to use non-invasive imaging devices for the detection of diseases which are otherwise difficult to diagnose at an early stage. A more modest but equally interesting goal is to improve the understanding of the brain in relation to body and mind. Statistics represents a quintessential part of such investigations as they are preluded by a clinical hypothesis that must be verified based on observed data. The massive amounts of image data produced in each examination pose an important and interesting statistical challenge, in that there are many more image features (variables) than subjects (observations), making an infinite number of solutions possible. To arrive at a unique and interesting answer, the analysis must be constrained, or regularized, in a sensible manner. This thesis describes such regularization options, discusses efficient algorithms which make the analysis of large data sets feasible, and gives examples of applications.