Please use this identifier to cite or link to this item: http://bura.brunel.ac.uk/handle/2438/15781
Title: Constructing reparameterization invariant metrics on spaces of plane curves
Authors: Bruveris, M
Marsland, S
Michor, PW
Issue Date: 2014
Citation: Differential Geometry and its Application, 2014, 34 pp. 139 - 165
Abstract: Metrics on shape spaces are used to describe deformations that take one shape to another, and to define a distance between shapes. We study a family of metrics on the space of curves, which includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics can be easily computed. This family consists of Sobolev-type Riemannian metrics of order one on the space Imm(S1,R2) of parameterized plane curves and the quotient space Imm(S1,R2)/Diff(S1) of unparameterized curves. For the space of open parameterized curves we find an explicit formula for the geodesic distance and show that the sectional curvatures vanish on the space of parameterized open curves and are non-negative on the space of unparameterized open curves. For one particular metric we provide a numerical algorithm that computes geodesics between unparameterized, closed curves, making use of a constrained formulation that is implemented numerically using the RATTLE algorithm. We illustrate the algorithm with some numerical tests between shapes. © 2014 Elsevier B.V.
URI: http://bura.brunel.ac.uk/handle/2438/15781
DOI: http://dx.doi.org/10.1016/j.difgeo.2014.04.008
ISSN: 0926-2245
Appears in Collections:Dept of Electronic and Electrical Engineering Research Papers

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