Brain and Surface Warping via Minimizing Lipschitz Extensions

Based on the notion Minimizing Lipschitz Extensions and its connection with the infinity Laplacian, a computational framework for surface warping and in particular brain warping (the nonlinear registration of brain imaging data) is presented in this paper. The basic concept is to compute a map between surfaces that minimizes a distortion measure based on geodesic distances while respecting the boundary conditions provided. In particular, the global Lipschitz constant of the map is minimized. This framework allows generic boundary conditions to be applied and allows direct surface-to-surface warping. It avoids the need for intermediate maps that flatten the surface onto the plane or sphere, as is commonly done in the literature on surface-based non-rigid brain image registration. The presentation of the framework is complemented with examples on synthetic geometric phantoms and cortical surfaces extracted from human brain MRI scans.