March  2018, 38(3): 1161-1185. doi: 10.3934/dcds.2018049

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

Received  February 2017 Revised  July 2017 Published  December 2017

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.

Citation: Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor. Soliton solutions for the elastic metric on spaces of curves. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1161-1185. doi: 10.3934/dcds.2018049
References:
[1]

M. BauerP. Harms and P. W. Michor, Almost local metrics on shape space of hypersurfaces in $ n $-space, SIAM J. Imaging Sci., 5 (2012), 244-310.  doi: 10.1137/100807983.  Google Scholar

[2]

M. BauerM. BruverisP. Harms and J. Moller-Andersen, A numerical framework for Sobolev metrics on the space of curves, SIAM J. Imaging Sci., 10 (2017), 47-73.  doi: 10.1137/16M1066282.  Google Scholar

[3]

M. BauerM. BruverisS. Marsland and P. W. Michor, Constructing reparameterization invariant metrics on spaces of plane curves, Differential Geom. Appl., 34 (2014), 139-165.  doi: 10.1016/j.difgeo.2014.04.008.  Google Scholar

[4]

M. BauerM. Bruveris and P. W. Michor, Overview of the geometries of shape spaces and diffeomorphism groups, Journal of Mathematical Imaging and Vision, 50 (2014), 60-97.  doi: 10.1007/s10851-013-0490-z.  Google Scholar

[5]

M. F. BegM. I. MillerA. Trouvé and L. Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms, International Journal of Computer Vision, 61 (2005), 139-157.  doi: 10.1023/B:VISI.0000043755.93987.aa.  Google Scholar

[6]

F. L. Bookstein, The study of shape transformation after d'Arcy Thompson, Mathematical Biosciences, 34 (1977), 177-219.  doi: 10.1016/0025-5564(77)90101-8.  Google Scholar

[7]

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, New York, 1982.  Google Scholar

[8]

N. Dunford and J. T. Schwartz, Linear Operators, Part 1, John Wiley & Sons, Inc., New York, 1988.  Google Scholar

[9]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid., Ann. of Math. (2), 92 (1970), 102-163.  doi: 10.2307/1970699.  Google Scholar

[10]

M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds, The Visual Computer, 31 (2015), 1179-1190.  doi: 10.1007/s00371-014-1001-y.  Google Scholar

[11]

A. Frölicher and A. Kriegl, Linear Spaces and Differentiation Theory, John Wiley & Sons Ltd., 1988.  Google Scholar

[12]

S. C. Joshi and M. I. Miller, Landmark matching via large deformation diffeomorphisms, Image Processing, IEEE Transactions on, 9 (2000), 1357-1370.  doi: 10.1109/83.855431.  Google Scholar

[13]

D. G. Kendall, Shape manifolds, Procrustean metrics, and complex projective spaces, Bull. London Math. Soc., 16 (1984), 81-121.  doi: 10.1112/blms/16.2.81.  Google Scholar

[14]

E. KlassenA. SrivastavaM. Mio and S. Joshi, Analysis of planar shapes using geodesic paths on shape spaces, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 26 (2004), 372-383.  doi: 10.1109/TPAMI.2004.1262333.  Google Scholar

[15]

H. LagaS. KurtekA. Srivastava and S. J. Miklavcic, Landmark-free statistical analysis of the shape of plant leaves, J. Theoret. Biol., 363 (2014), 41-52.  doi: 10.1016/j.jtbi.2014.07.036.  Google Scholar

[16]

S. LahiriD. Robinson and E. Klassen, Precise matching of PL curves in $ \mathbb{R}^N $ in the square root velocity framework, Geom. Imaging Comput., 2 (2015), 133-186.  doi: 10.4310/GIC.2015.v2.n3.a1.  Google Scholar

[17]

M. MicheliP. W. Michor and D. Mumford, Sectional curvature in terms of the cometric, with applications to the Riemannian manifolds of landmarks, SIAM J. Imaging Sci., 5 (2012), 394-433.  doi: 10.1137/10081678X.  Google Scholar

[18]

P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS), 8 (2006), 1-48.   Google Scholar

[19]

P. W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Appl. Comput. Harmon. Anal., 23 (2007), 74-113.  doi: 10.1016/j.acha.2006.07.004.  Google Scholar

[20]

P. W. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the hamiltonian approach, in Phase space analysis of partial differential equations, Springer, 69 (2006), 133–215.  Google Scholar

[21]

D. Mumford and P. W. Michor, On Euler's equation and 'EPDiff', J. Geom. Mech., 5 (2013), 319-344.  doi: 10.3934/jgm.2013.5.319.  Google Scholar

[22]

M. Salvai, Geodesic paths of circles in the plane, Rev. Mat. Complut., 24 (2011), 211-218.  doi: 10.1007/s13163-010-0036-5.  Google Scholar

[23]

A. SrivastavaE. KlassenS. H. Joshi and I. H. Jermyn, Shape analysis of elastic curves in Euclidean spaces, IEEE T. Pattern Anal., 33 (2011), 1415-1428.  doi: 10.1109/TPAMI.2010.184.  Google Scholar

[24]

G. SundaramoorthiA. MennucciS. Soatto and A. Yezzi, A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering, SIAM J. Imaging Sci., 4 (2011), 109-145.  doi: 10.1137/090781139.  Google Scholar

[25]

A. Yezzi and A. Mennucci, Conformal metrics and true "gradient flows" for curves, in Proceedings of the Tenth IEEE International Conference on Computer Vision, 1 (2005), 913–919. doi: 10.1109/ICCV.2005.60.  Google Scholar

[26]

L. YounesP. W. MichorJ. Shah and D. Mumford, A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25-57.   Google Scholar

[27]

L. Younes, Shapes and Diffeomorphisms, vol. 171 of Applied Mathematical Sciences, Springer-Verlag, Berlin, 2010.  Google Scholar

[28]

N. J. Zabusky and M. D. Kruskal, Interaction of "solitons" in a collisionless plasma and the recurrence of initial states, Physical review letters, 15 (1965), p240. doi: 10.1103/PhysRevLett.15.240.  Google Scholar

show all references

References:
[1]

M. BauerP. Harms and P. W. Michor, Almost local metrics on shape space of hypersurfaces in $ n $-space, SIAM J. Imaging Sci., 5 (2012), 244-310.  doi: 10.1137/100807983.  Google Scholar

[2]

M. BauerM. BruverisP. Harms and J. Moller-Andersen, A numerical framework for Sobolev metrics on the space of curves, SIAM J. Imaging Sci., 10 (2017), 47-73.  doi: 10.1137/16M1066282.  Google Scholar

[3]

M. BauerM. BruverisS. Marsland and P. W. Michor, Constructing reparameterization invariant metrics on spaces of plane curves, Differential Geom. Appl., 34 (2014), 139-165.  doi: 10.1016/j.difgeo.2014.04.008.  Google Scholar

[4]

M. BauerM. Bruveris and P. W. Michor, Overview of the geometries of shape spaces and diffeomorphism groups, Journal of Mathematical Imaging and Vision, 50 (2014), 60-97.  doi: 10.1007/s10851-013-0490-z.  Google Scholar

[5]

M. F. BegM. I. MillerA. Trouvé and L. Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms, International Journal of Computer Vision, 61 (2005), 139-157.  doi: 10.1023/B:VISI.0000043755.93987.aa.  Google Scholar

[6]

F. L. Bookstein, The study of shape transformation after d'Arcy Thompson, Mathematical Biosciences, 34 (1977), 177-219.  doi: 10.1016/0025-5564(77)90101-8.  Google Scholar

[7]

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, New York, 1982.  Google Scholar

[8]

N. Dunford and J. T. Schwartz, Linear Operators, Part 1, John Wiley & Sons, Inc., New York, 1988.  Google Scholar

[9]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid., Ann. of Math. (2), 92 (1970), 102-163.  doi: 10.2307/1970699.  Google Scholar

[10]

M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds, The Visual Computer, 31 (2015), 1179-1190.  doi: 10.1007/s00371-014-1001-y.  Google Scholar

[11]

A. Frölicher and A. Kriegl, Linear Spaces and Differentiation Theory, John Wiley & Sons Ltd., 1988.  Google Scholar

[12]

S. C. Joshi and M. I. Miller, Landmark matching via large deformation diffeomorphisms, Image Processing, IEEE Transactions on, 9 (2000), 1357-1370.  doi: 10.1109/83.855431.  Google Scholar

[13]

D. G. Kendall, Shape manifolds, Procrustean metrics, and complex projective spaces, Bull. London Math. Soc., 16 (1984), 81-121.  doi: 10.1112/blms/16.2.81.  Google Scholar

[14]

E. KlassenA. SrivastavaM. Mio and S. Joshi, Analysis of planar shapes using geodesic paths on shape spaces, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 26 (2004), 372-383.  doi: 10.1109/TPAMI.2004.1262333.  Google Scholar

[15]

H. LagaS. KurtekA. Srivastava and S. J. Miklavcic, Landmark-free statistical analysis of the shape of plant leaves, J. Theoret. Biol., 363 (2014), 41-52.  doi: 10.1016/j.jtbi.2014.07.036.  Google Scholar

[16]

S. LahiriD. Robinson and E. Klassen, Precise matching of PL curves in $ \mathbb{R}^N $ in the square root velocity framework, Geom. Imaging Comput., 2 (2015), 133-186.  doi: 10.4310/GIC.2015.v2.n3.a1.  Google Scholar

[17]

M. MicheliP. W. Michor and D. Mumford, Sectional curvature in terms of the cometric, with applications to the Riemannian manifolds of landmarks, SIAM J. Imaging Sci., 5 (2012), 394-433.  doi: 10.1137/10081678X.  Google Scholar

[18]

P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS), 8 (2006), 1-48.   Google Scholar

[19]

P. W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Appl. Comput. Harmon. Anal., 23 (2007), 74-113.  doi: 10.1016/j.acha.2006.07.004.  Google Scholar

[20]

P. W. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the hamiltonian approach, in Phase space analysis of partial differential equations, Springer, 69 (2006), 133–215.  Google Scholar

[21]

D. Mumford and P. W. Michor, On Euler's equation and 'EPDiff', J. Geom. Mech., 5 (2013), 319-344.  doi: 10.3934/jgm.2013.5.319.  Google Scholar

[22]

M. Salvai, Geodesic paths of circles in the plane, Rev. Mat. Complut., 24 (2011), 211-218.  doi: 10.1007/s13163-010-0036-5.  Google Scholar

[23]

A. SrivastavaE. KlassenS. H. Joshi and I. H. Jermyn, Shape analysis of elastic curves in Euclidean spaces, IEEE T. Pattern Anal., 33 (2011), 1415-1428.  doi: 10.1109/TPAMI.2010.184.  Google Scholar

[24]

G. SundaramoorthiA. MennucciS. Soatto and A. Yezzi, A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering, SIAM J. Imaging Sci., 4 (2011), 109-145.  doi: 10.1137/090781139.  Google Scholar

[25]

A. Yezzi and A. Mennucci, Conformal metrics and true "gradient flows" for curves, in Proceedings of the Tenth IEEE International Conference on Computer Vision, 1 (2005), 913–919. doi: 10.1109/ICCV.2005.60.  Google Scholar

[26]

L. YounesP. W. MichorJ. Shah and D. Mumford, A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25-57.   Google Scholar

[27]

L. Younes, Shapes and Diffeomorphisms, vol. 171 of Applied Mathematical Sciences, Springer-Verlag, Berlin, 2010.  Google Scholar

[28]

N. J. Zabusky and M. D. Kruskal, Interaction of "solitons" in a collisionless plasma and the recurrence of initial states, Physical review letters, 15 (1965), p240. doi: 10.1103/PhysRevLett.15.240.  Google Scholar

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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