Comparing and Warping Shapes in a Metric Framework

April 3, 2006
  • Metric theory
  • 11J83
A geometric framework for comparing manifolds given by point clouds is first presented in this talk. The underlying theory is based on Gromov-Hausdorff distances, leading to isometry invariant and completely geometric comparisons. This theory is embedded in a probabilistic setting as derived from random sampling of manifolds, and then combined with results on matrices of pairwise geodesic distances to lead to a computational implementation of the framework. The theoretical and computational results described are complemented with experiments for real three dimensional shapes. In the second part of the talk, 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. 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. Joint works with F. Memoli and P. Thompson.