An Overview of the Riemannian Metrics on Spaces of Curves Using the Hamiltonian Approach

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An Overview of the Riemannian Metrics on Spaces of Curves Using the Hamiltonian Approach

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Title: An Overview of the Riemannian Metrics on Spaces of Curves Using the Hamiltonian Approach
Author: Michor, Peter W.; Mumford, David Bryant

Note: Order does not necessarily reflect citation order of authors.

Citation: Michor, Peter W., and David Bryant Mumford. 2007. An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Applied and Computational Harmonic Analysis 23(1): 74-113.
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Abstract: Here shape space is either the manifold of simple closed smooth unparameterized curves in View the MathML source or is the orbifold of immersions from S1 to View the MathML source modulo the group of diffeomorphisms of S1. We investigate several Riemannian metrics on shape space: L2-metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two length-weighted metrics. Sobolev metrics of order n on curves are described. Here the horizontal projection of a tangent field is given by a pseudo-differential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on View the MathML source is treated. Although the quotient metrics are all given by pseudo-differential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.
Published Version: doi:10.1016/j.acha.2006.07.004
Other Sources: http://www.mat.univie.ac.at/~michor/curves-hamiltonian.pdf
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:3637113

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  • FAS Scholarly Articles [7078]
    Peer reviewed scholarly articles from the Faculty of Arts and Sciences of Harvard University
 
 

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