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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

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

[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.

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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