43 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
Uniqueness of the Fisher-Rao metric on the space of smooth densities
MB was supported by ‘Fonds zur F¨orderung der wissenschaftlichen Forschung, Projekt P 24625’
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
A relaxed approach for curve matching with elastic metrics
In this paper we study a class of Riemannian metrics on the space of
unparametrized curves and develop a method to compute geodesics with given
boundary conditions. It extends previous works on this topic in several
important ways. The model and resulting matching algorithm integrate within one
common setting both the family of -metrics with constant coefficients and
scale-invariant -metrics on both open and closed immersed curves. These
families include as particular cases the class of first-order elastic metrics.
An essential difference with prior approaches is the way that boundary
constraints are dealt with. By leveraging varifold-based similarity metrics we
propose a relaxed variational formulation for the matching problem that avoids
the necessity of optimizing over the reparametrization group. Furthermore, we
show that we can also quotient out finite-dimensional similarity groups such as
translation, rotation and scaling groups. The different properties and
advantages are illustrated through numerical examples in which we also provide
a comparison with related diffeomorphic methods used in shape registration.Comment: 27 page