Statistical shape analysis can be done in a Riemannian framework by endowing the set of shapes with a Riemannian metric. Sobolev metrics of order two and higher on shape spaces of parametrized or unparametrized curves have several desirable properties not present in lower order metrics, but their discretization is still largely missing. In this paper, we present algorithms to numerically solve the geodesic initial and boundary value problems for these metrics. The combination of these algorithms enables one to compute Karcher means in a Riemannian gradient-based optimization scheme and perform principal component analysis and clustering. Our framework is sufficiently general to be applicable to a wide class of metrics. We demonstrate the effectiveness of our approach by analyzing a collection of shapes representing HeLa cell nuclei.

This paper extends parts of the results from [P. W. Michor and D. Mumford, Appl. Comput. Harmon. Anal., 23 (2007), pp. 74–113] for plane curves to the case of hypersurfaces in $\mathbb R^n$. Let M be a compact connected oriented $n-1$ dimensional manifold without boundary like the sphere or the torus. Then shape space is either the manifold of submanifolds of $\mathbb R^n$ of type M or the orbifold of immersions from M to $\mathbb R^n$ modulo the group of diffeomorphisms of M. We investigate almost local Riemannian metrics on shape space. These are induced by metrics of the following form on the space of immersions: $G_f(h,k) = \int_{M} \Phi(Vol(f),Tr(L))\bar{g}(h, k) vol(f^*\bar{g})$, where $\bar{g}$ is the Euclidean metric on $\mathbb R^n$, $f^*\bar{g}$ is the induced metric on M, $h,k\in C^\infty(M,\mathbb R^n)$ are tangent vectors at f to the space of embeddings or immersions, where $\Phi:\mathbb R^2\to \mathbb R_{>0}$ is a suitable smooth function, $Vol(f) = \int_M vol(f^*\bar{g})$ is the total hypersurface volume of $f(M)$, and the trace $Tr(L)$ of the Weingarten mapping is the mean curvature. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, the conserved momenta arising from the obvious symmetries, and the sectional curvature. For special choices of $\Phi$ we give complete formulas for the sectional curvature. Numerical experiments illustrate the behavior of these metrics.