This chapter describes the computation of large deformation diffeomorphic correspondences based on volume landmarks and their associated imagery. Computational anatomy consists of many diverse facets, including automated construction and mapping of anatomical features such as curves, surfaces, and subvolumes, as well as the statistical characterization of the various anatomical manifolds. The mappings between one brain and another is studied via groups of diffeomorphisms acting on the coordinate systems. Diffeomorphisms, when restricted to the brain substructures, carry the topologies smoothly as well. The principal focus of the chapter is on the development of a complete framework for automated construction of large deformation landmark and image matching diffeomorphic correspondences on these manifolds. Diffeomorphisms allow for the quantitative study of differential constructs by anatomical and clinical experts: Riemannian lengths on curved surfaces, surface area, connectedness of subvolumes, curvature and torsion of fundi and gyri, etc. The small deformation methods have proved to be very powerful and they are of wide application for the study of biological shape.
