Geometric Surface and Brain Warping via Geodesic Minimizing Lipschitz Extensions

Based on the notion of Minimizing Lipschitz Extensions and its connection with the infinity Laplacian, a theoretical and computational framework for geometric nonrigid surface warping, and in particular 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 provided boundary conditions. In particular, the global Lipschitz constant of the map is minimized. This framework allows generic boundary conditions to be applied and direct surfaceto-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 registration. The presentation of the framework is complemented with examples on synthetic geometric phantoms and cortical surfaces extracted from human brain MRI scans.