Integral representation of local and global diffeomorphisms

Abstract

The identification of a smooth invertible map between two closed domains in R n is of great importance in neuroimaging and other fields where the domains of images must be nonlinearly transformed so as to align important features. The theory for locally diffeomorphic maps is classic, but much less is known about global diffeomorphisms between closed simply connected sets. These bijectively map interiors onto interiors and boundaries onto boundaries. We use this property and the integral of the exponential map to construct representations of global diffeomorphisms for star-shaped domains. As a simple example, details are provided for maps in two dimensions. Applications are outlined for three types of inverse problems related to image registration, diffeomorphic splines, and dynamical systems.