Constructing an atlas for the diffeomorphism group of a compact manifold with boundary, with application to the analysis of image registrations

摘要

This paper considers the problem of defining a parameterization (chart) on the group of diffeomorphisms with compact support, motivated primarily by a problem in image registration, where diffeomorphic warps are used to align images. Constructing a chart on the diffeomorphism group will enable the quantitative analysis of these warps to discover the normal and abnormal variation of structures in a population.We construct a chart for particular choices of boundary conditions on the space on which the diffeomorphism acts, and for a particular class of metrics on the diffeomorphism group, which define a class of diffeomorphic interpolating splines. The geodesic equation is computed for this class of metrics, and we show how it can be solved in the spline representation. Furthermore, we demonstrate that the spline representation generates submanifolds of the diffeomorphism group, and we study this mapping. Explicit computational examples are included, showing how this chart can be constructed in practice, and that the use of the geodesic distance allows better classification of variation than those obtained using just a Euclidean metric on the space of warps.