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Optimal Reparametrizations in the Square Root Velocity Framework
It is shown that the square root velocity transform is a homeomorphism and that the action of the reparametrisation semigroup is continuous, and that given two $C^1$-curves, there exist optimal reparamETrisations realising the minimal distance between the unparametrised curves represented by them.
Abstract We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and
Completeness properties of Sobolev metrics on the space of curves
We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order $n\geq 2$ are metrically
Overview of the Geometries of Shape Spaces and Diffeomorphism Groups
This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of
Mixture of Kernels and Iterated Semidirect Product of Diffeomorphisms Groups
A variational approach for multiscale analysis of diffeomorphisms is developed in detail to generalize to several scales the semidirect product representation, and to illustrate the resulting diffeomorphic decomposition on synthetic and real images.
On Completeness of Groups of Diffeomorphisms
We study completeness properties of the Sobolev diffeomorphism groups $\mathcal D^s(M)$ endowed with strong right-invariant Riemannian metrics when the underlying manifold $M$ is $\mathbb R^d$ or
The Momentum Map Representation of Images
A theorem is proved showing that Large Deformation Diffeomorphic Matching methods may be designed by using the actions of diffeomorphisms on the image data structure to define their associated momentum representations as (cotangent-lift) momentum maps.
Geometry of Image Registration: The Diffeomorphism Group and Momentum Maps
These lecture notes explain the geometry and discuss some of the analytical questions underlying image registration within the framework of large deformation diffeomorphic metric mapping (LDDMM) used
Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group
We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diffc(M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S1, the geodesic