Elastic shape analysis of surfaces and images.

Abstract

We describe two Riemannian frameworks for statistical shape analysis of parameterized surfaces. These methods provide tools for registration, comparison, deformation, averaging, statistical modeling, and random sampling of surface shapes. A crucial property of both of these frameworks is that they are invariant to reparameterizations of surfaces. Thus, they result in natural shape comparisons and statistics. The first method we describe is based on a special representation of surfaces termed square-root functions (SRFs). The pullback of the L2 metric from the SRF space results in the Riemannian metric on the space of surfaces. The second method is based on the elastic surface metric. We show that a restriction of this metric, which we call the partial elastic metric, becomes the standard L2 metric under the square-root normal field (SRNF) representation. We show the advantages of these methods by computing geodesic paths between highly articulated surfaces and shape statistics of manually generated surfaces. We also describe applications of this framework to image registration and medical diagnosis.