Almost local metrics on shape space of hypersurfaces in n-space

Details Almost local metrics on shape space of hypersurfaces in n-space Almost local metrics on shape space of hypersurfaces in n-space

2010-01-05

arxiv.org/abs/1001.0717v4

SIAM J. Imaging Sci. 5 (2012), pp. 244-310

Mathematics

Differential Geometry

67 pages, revised version: some mistakes corrected, much better numerics, AMPL code added

Scientific paper

10.1137/100807983

This paper extends parts of the results from [17] 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 $S^2$ or the torus $S^1\times S^1$. 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: $$ G_f(h,k) = \int_{M} \Phi(\operatorname{Vol}(M),\operatorname{Tr}(L))\bar g(h, k) \operatorname{vol}(f^*\bar{g})$$ where $\bar g$ is the standard 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 $\Ph:\mathbb R^2\to \mathbb R_{>0}$ is a suitable smooth function, $\on{Vol}(M) = \int_M\on{vol}(f^*\bar g)$ is the total hypersurface volume of $f(M)$, and where the trace $\on{Tr}(L)$ 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 $\Ph$ we give complete formulas for the sectional curvature. Numerical experiments illustrate the behavior of these metrics.

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