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Soliton solutions for the elastic metric on spaces of curves

  • * Corresponding author: Martin Bauer

    * Corresponding author: Martin Bauer 
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  • 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.

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

    Citation:

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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

    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

    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.

    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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