dc.contributor.author Michor, Peter W. dc.contributor.author Mumford, David Bryant dc.date.accessioned 2010-02-12T19:47:14Z dc.date.issued 2007 dc.identifier.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. en_US dc.identifier.issn 1063-5203 en_US dc.identifier.uri http://nrs.harvard.edu/urn-3:HUL.InstRepos:3637113 dc.description.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. en_US 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. dc.description.sponsorship Mathematics en_US dc.language.iso en_US en_US dc.publisher Elsevier en_US dc.relation.isversionof doi:10.1016/j.acha.2006.07.004 en_US dc.relation.hasversion http://www.mat.univie.ac.at/~michor/curves-hamiltonian.pdf en_US dash.license LAA dc.title An Overview of the Riemannian Metrics on Spaces of Curves Using the Hamiltonian Approach en_US dc.type Journal Article en_US dc.description.version Version of Record en_US dc.relation.journal Applied and Computational Harmonic Analysis en_US dash.depositing.author Mumford, David Bryant dc.date.available 2010-02-12T19:47:14Z dc.identifier.doi 10.1016/j.acha.2006.07.004 * dash.contributor.affiliated Mumford, David
﻿