# Constructing reparametrization invariant metrics on spaces of plane curves

@article{Bauer2012ConstructingRI, title={Constructing reparametrization invariant metrics on spaces of plane curves}, author={Martin Bauer and Martins Bruveris and Stephen R. Marsland and Peter W. Michor}, journal={arXiv: Differential Geometry}, year={2012} }

## 70 Citations

Reparameterization Invariant Metric on the Space of Curves

- MathematicsGSI
- 2015

This paper uses the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions M' = Imm([0,1], M), and observes that such a natural choice of Riemannian metric on TM' induces a first-order Sobolev metric on M' with an extra term involving the origins, and leads to a distance.

Computing distances and geodesics between manifold-valued curves in the SRV framework

- Mathematics
- 2016

This paper focuses on the study of open curves in a Riemannian manifold M, and proposes a reparametrization invariant metric on the space of such paths. We use the square root velocity function…

Metric registration of curves and surfaces using optimal control

- MathematicsHandbook of Numerical Analysis
- 2019

Simplifying Transforms for General Elastic Metrics on the Space of Plane Curves

- MathematicsSIAM J. Imaging Sci.
- 2020

This paper extends the transformations appearing in the existing literature to a family of isometries, which take any elastic metric to the flat L 2 metric, and extends the transforms to treat piecewise linear curves and demonstrates the existence of optimal matchings over the diffeomorphism group in this setting.

Metrics with prescribed horizontal bundle on spaces of curve

- Mathematics
- 2015

We study metrics on the shape space of curves that induce a prescribed splitting of the tangent bundle. More specifically, we consider reparametrization invariant metrics $G$ on the space…

Homogeneous Sobolev Metric of Order One on Diffeomorphism Groups on Real Line

- MathematicsJ. Nonlinear Sci.
- 2014

It is proved that the spaceequipped with the homogeneous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat L2-metric.

Reparameterization invariant distance on the space of curves in the hyperbolic plane

- Mathematics
- 2015

This paper focuses on the study of time-varying paths in the two-dimensional hyperbolic space, and its aim is to define a reparameterization invariant distance on the space of such paths. We adapt…

Shape Analysis of Framed Space Curves

- MathematicsJournal of Mathematical Imaging and Vision
- 2019

This paper is able to generalize the square root transform by using quaternionic arithmetic and properties of the Hopf fibration to describe geodesics in framed curve space explicitly and to compute means for collections of space curves and perform statistical analysis of circular DNA molecule shapes.

Proceedings Of Math On The Rocks Shape Analysis Workshop In Grundsund

- Mathematics
- 2015

We study metrics on the shape space of curves that induce a prescribed splitting of the tangent bundle. More speciﬁcally, we consider reparametrization invariant metrics G on the space Imm( S 1 , R 2…

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