  Second order Sobolev metrics on the space of regular unparametrized planar
curves have several desirable completeness properties not present in lower
order metrics, but numerics are still largely missing. In this paper, we
present algorithms to numerically solve the initial and boundary value problems
for geodesics. The combination of these algorithms allows to compute Karcher
means in a Riemannian gradient-based optimization scheme. Our framework has the
advantage that the constants determining the weights of the zero, first, and
second order terms of the metric can be chosen freely. Moreover, due to its
generality, it could be applied to more general spaces of mapping. We
demonstrate the effectiveness of our approach by analyzing a collection of
shapes representing physical objects.
