  In the recent years, Riemannian shape analysis of curves and surfaces has
found several applications in medical image analysis. In this paper we present
a numerical discretization of second order Sobolev metrics on the space of
regular curves in Euclidean space. This class of metrics has several desirable
mathematical properties. We propose numerical solutions for the initial and
boundary value problems of finding geodesics. These two methods are combined in
a Riemannian gradient-based optimization scheme to compute the Karcher mean. We
apply this to a study of the shape variation in HeLa cell nuclei and cycles of
cardiac deformations, by computing means and principal modes of variations.
