Soliton solutions for the elastic metric on spaces of curves

Martin Bauer; Martins Bruveris; Philipp Harms; Peter W. Michor

doi:10.3934/dcds.2018049

Article Contents

2018, Volume 38, Issue 3: 1161-1185. Doi: 10.3934/dcds.2018049

This issue Previous Article Uniform hyperbolicity in nonflat billiards Next Article Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system

Soliton solutions for the elastic metric on spaces of curves

1.
Florida State University, Department of Mathematics, 1017 Academic Way, Tallahassee, FL 32304, USA
2.
Brunel University London, Department of Mathematics, Uxbridge UB8 3PH, United Kingdom
3.
University of Freiburg, Department of Mathematics, Eckerstraße 1,79104 Freiburg, Germany
4.
University of Vienna, Department of Mathematics, Oskar-Morgenstern-Platz 1,1090 Wien, Austria

* Corresponding author: Martin Bauer
* Corresponding author: Martin Bauer

Received: February 2017

Revised: July 2017

Published: July 2018

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Abstract

In this article we investigate a first order reparametrization-invariant Sobolev metric on the space of immersed curves. Motivated by applications in shape analysis where discretizations of this infinite-dimensional space are needed, we extend this metric to the space of Lipschitz curves, establish the wellposedness of the geodesic equation thereon, and show that the space of piecewise linear curves is a totally geodesic submanifold. Thus, piecewise linear curves are natural finite elements for the discretization of the geodesic equation. Interestingly, geodesics in this space can be seen as soliton solutions of the geodesic equation, which were not known to exist for reparametrization-invariant Sobolev metrics on spaces of curves.

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Mathematics Subject Classification: Primary: 58E10; Secondary: 49M25, 58B20, 58D10.

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Figure 1. A closed geodesic in $P\mathcal I^1_0$ which has self-intersection and changes its winding number (c.f. Example 4.9). See here or here for an animation

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Figure 2. A closed geodesic in $P\mathcal I^1_0$ which has self-intersection and changes its winding number (c.f. Example 4.9). See here or here for an animation

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Figure 3. The kernel of the $H^1$ metric (left) compared to a Gaussian kernel (middle) at a specific landmark (right) on the space $\text{Land}$. Dark colors correspond to large values of the kernel. See Remark 6.4 for an interpretation.

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Figure 4. A geodesic with respect to the LDDMM metric with the same initial condition as in Figure 2. Note that the LDDMM metric avoids landmark collisions; the landmarks never touch. See here or here for an animation

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