Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two manifolds M and N, a differentiable map f \colon M \rightarrow N is called a diffeomorphism if it is a bijection and its inverse f^{-1} \colon N \rightarrow M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. They are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times conti

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Mixture Of Kernels And Iterated Semidirect Product Of Diffeomorphisms Groups

Martins Bruveris

In the framework of large deformation diffeomorphic metric mapping (LDDMM), a multiscale theory for the diffeomorphism group is developed based on previous works of the authors. The purpose of this paper is (1) to develop in detail a variational approach for multiscale analysis of diffeomorphisms, (2) to generalize to several scales the the semidirect product representation, and (3) to illustrate the resulting diffeomorphic decomposition on synthetic and real images. We also show that the so-called kernel bundle method and the mixture of kernels are equivalent.

Siam Publications2012

Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

Martin Bauer, Martins Bruveris

We study Sobolev-type metrics of fractional order s a parts per thousand yen 0 on the group Diff (c) (M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S (1), the geodesic distance on Diff (c) (S (1)) vanishes if and only if . For other manifolds, we obtain a partial characterization: the geodesic distance on Diff (c) (M) vanishes for and for , with N being a compact Riemannian manifold. On the other hand, the geodesic distance on Diff (c) (M) is positive for and dim(M) a parts per thousand yen 2, s a parts per thousand yen 1. For , we discuss the geodesic equations for these metrics. For n = 1, we obtain some well-known PDEs of hydrodynamics: Burgers' equation for s = 0, the modified Constantin-Lax-Majda equation for , and the Camassa-Holm equation for s = 1.

Springer Verlag2013

Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II

Martin Bauer, Martins Bruveris

The geodesic distance vanishes on the group of compactly supported diffeomorphisms of a Riemannian manifold of bounded geometry, for the right invariant weak Riemannian metric which is induced by the Sobolev metric of order on the Lie algebra of vector fields with compact support.

Springer Verlag2013

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Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for sho

Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be desc

Lie group

In mathematics, a Lie group (pronounced liː ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract conce