34 research outputs found
On Completeness of Groups of Diffeomorphisms
We study completeness properties of the Sobolev diffeomorphism groups
endowed with strong right-invariant Riemannian metrics when
the underlying manifold is or compact without boundary. The
main result is that for , the group is
geodesically and metrically complete with a surjective exponential map. We then
present the connection between the Sobolev diffeomorphism group and the large
deformation matching framework in order to apply our results to diffeomorphic
image matching.Comment: 43 pages, revised versio
Riemannian geometry for shape analysis and computational anatomy
Shape analysis and compuational anatomy both make use of sophisticated tools
from infinite-dimensional differential manifolds and Riemannian geometry on
spaces of functions. While comprehensive references for the mathematical
foundations exist, it is sometimes difficult to gain an overview how
differential geometry and functional analysis interact in a given problem. This
paper aims to provide a roadmap to the unitiated to the world of
infinite-dimensional Riemannian manifolds, spaces of mappings and Sobolev
metrics: all tools used in computational anatomy and shape analysis.Comment: 20 page
Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics
We show for a certain class of operators and holomorphic functions
that the functional calculus is holomorphic. Using this result
we are able to prove that fractional Laplacians depend real
analytically on the metric in suitable Sobolev topologies. As an
application we obtain local well-posedness of the geodesic equation for
fractional Sobolev metrics on the space of all Riemannian metrics.Comment: 31 page
On Completeness of Groups of Diffeomorphisms
We study completeness properties of the Sobolev diffeomorphism groups Ds(M) endowed with strong right-invariant Riemannian metrics when the underlying manifold M is ℝd or compact without boundary. The main result is that for dimM/2 + 1, the group Ds (M) is geodesically and metrically complete with a surjective exponential map. We also extend the result to its closed subgroups, in particular the group of volume preserving diffeomorphisms and the group of symplectomorphisms. We then present the connection between the Sobolev diffeomorphism group and the large deformation matching framework in order to apply our results to diffeomorphic image matching
Well-posedness of the EPDiff equation with a pseudo-differential inertia operator
In this article we study the class of right-invariant, fractional order Sobolev-type metrics on groups of diffeomorphisms of a compact manifold M. Our main result concerns well-posedness properties for the corresponding Euler-Arnold equations, also called the EPDiff equations, which are of importance in mathematical physics and in the field of shape analysis and template registration. Depending on the order of the metric, we will prove both local and global well-posedness results for these equations. As a result of our analysis we will also obtain new commutator estimates for elliptic pseudo-differential operators