# American Institute of Mathematical Sciences

June  2017, 9(2): 227-256. doi: 10.3934/jgm.2017010

## Quotient elastic metrics on the manifold of arc-length parameterized plane curves

 1 Laboratoire Paul Painlevé, CNRS U.M.R. 8524, 59 655 Villeneuve d'Ascq Cedex, France 2 Department of Mathematics, Brooklyn College and CUNY Graduate Center, New York, USA

* Corresponding author: Alice B. Tumpach

Received  January 2016 Revised  March 2017 Published  May 2017

We study the pull-back of the 2-parameter family of quotient elastic metrics introduced in [13] on the space of arc-length parameterized curves. This point of view has the advantage of concentrating on the manifold of arc-length parameterized curves, which is a very natural manifold when the analysis of un-parameterized curves is concerned, pushing aside the tricky quotient procedure detailed in [12] of the preshape space of parameterized curves by the reparameterization (semi-)group. In order to study the problem of finding geodesics between two given arc-length parameterized curves under these quotient elastic metrics, we give a precise computation of the gradient of the energy functional in the smooth case as well as a discretization of it, and implement a path-straightening method. This allows us to have a better understanding of how the landscape of the energy functional varies with respect to the parameters.

Citation: Alice B. Tumpach, Stephen C. Preston. Quotient elastic metrics on the manifold of arc-length parameterized plane curves. Journal of Geometric Mechanics, 2017, 9 (2) : 227-256. doi: 10.3934/jgm.2017010
##### References:
 [1] M. Bauer, M. Bruveris, S. Marsland and P. W. Michor, Constructing reparametrization invariant metrics on spaces of plane curves, Differential Geometry and its Applications, 34 (2014), 139-165. doi: 10.1016/j.difgeo.2014.04.008. Google Scholar [2] M. Bauer, M. 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 [3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011. Google Scholar [4] M. Bruveris, Optimal reparametrizations in the square root velocity framework, SIAM Journal on Mathematical Analysis, 48 (2016), 4335-4354. doi: 10.1137/15M1014693. Google Scholar [5] J. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4731-5. Google Scholar [6] V. Cervera, F. Mascaró and P. W. Michor, The action of the diffeomorphism group on the space of immersions, Differential Geometry and its Applications, 1 (1991), 391-401. doi: 10.1016/0926-2245(91)90015-2. Google Scholar [7] T. Diez, Slice theorem for Fréchet group actions and covariant symplectic field theory, 2014, arXiv: 1405.2249Google Scholar [8] F. Dubeau and J. Savoie, A remark on cyclic tridiagonal matrices, Zastosowania Matematyki Applicationes Mathematicae, 21 (1991), 253-256. Google Scholar [9] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bulletin (New series) of the American Mathematical Society, 7 (1982), 65-222. doi: 10.1090/S0273-0979-1982-15004-2. Google Scholar [10] N. J. Higham, Accuracy and Stability in Numerical Algorithms: Second Edition, SIAM, 2002. doi: 10.1137/1.9780898718027. Google Scholar [11] E. Klassen, A. Srivastava, W. Mio and S. H. Joshi, Analysis of planar shapes using geodesic paths on shape spaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26 (2004), 372-383. doi: 10.1109/TPAMI.2004.1262333. Google Scholar [12] S. Lahiri, D. 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, arXiv:1501.00577. doi: 10.4310/GIC.2015.v2.n3.a1. Google Scholar [13] W. Mio, A. Srivastava and S. H. Joshi, On shape of plane elastic curves, International Journal of Computer Vision, 73 (2007), 307-324. doi: 10.1007/s11263-006-9968-0. Google Scholar [14] S. C. Preston, The geometry of whips, Annals of Global Analysis Geometry, 41 (2012), 281-305. doi: 10.1007/s10455-011-9283-z. Google Scholar [15] A. Srivastava, E. Klassen, S. H. Joshi and I. H. Jermyn, Shape analysis of elastic curves in Euclidean spaces, IEEE Trans. PAMI, 33 (2011), 1415-1428. doi: 10.1109/TPAMI.2010.184. Google Scholar [16] C. Temperton, Algorithms for the solution of cyclic tridiagonal systems, Journal of Computational Physics, 19 (1975), 317-323. doi: 10.1016/0021-9991(75)90081-9. Google Scholar [17] A. B. Tumpach, H. Drira, M. Daoudi and A. Srivastava, Gauge invariant framework for shape analysis of surfaces, IEEE Trans Pattern Anal Mach Intell., 38 (2016), 46-59. doi: 10.1109/TPAMI.2015.2430319. Google Scholar [18] L. Younes, P. W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics, Matematica e Applicazioni, 19 (2008), 25-57. doi: 10.4171/RLM/506. Google Scholar

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##### References:
 [1] M. Bauer, M. Bruveris, S. Marsland and P. W. Michor, Constructing reparametrization invariant metrics on spaces of plane curves, Differential Geometry and its Applications, 34 (2014), 139-165. doi: 10.1016/j.difgeo.2014.04.008. Google Scholar [2] M. Bauer, M. 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 [3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011. Google Scholar [4] M. Bruveris, Optimal reparametrizations in the square root velocity framework, SIAM Journal on Mathematical Analysis, 48 (2016), 4335-4354. doi: 10.1137/15M1014693. Google Scholar [5] J. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4731-5. Google Scholar [6] V. Cervera, F. Mascaró and P. W. Michor, The action of the diffeomorphism group on the space of immersions, Differential Geometry and its Applications, 1 (1991), 391-401. doi: 10.1016/0926-2245(91)90015-2. Google Scholar [7] T. Diez, Slice theorem for Fréchet group actions and covariant symplectic field theory, 2014, arXiv: 1405.2249Google Scholar [8] F. Dubeau and J. Savoie, A remark on cyclic tridiagonal matrices, Zastosowania Matematyki Applicationes Mathematicae, 21 (1991), 253-256. Google Scholar [9] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bulletin (New series) of the American Mathematical Society, 7 (1982), 65-222. doi: 10.1090/S0273-0979-1982-15004-2. Google Scholar [10] N. J. Higham, Accuracy and Stability in Numerical Algorithms: Second Edition, SIAM, 2002. doi: 10.1137/1.9780898718027. Google Scholar [11] E. Klassen, A. Srivastava, W. Mio and S. H. Joshi, Analysis of planar shapes using geodesic paths on shape spaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26 (2004), 372-383. doi: 10.1109/TPAMI.2004.1262333. Google Scholar [12] S. Lahiri, D. 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, arXiv:1501.00577. doi: 10.4310/GIC.2015.v2.n3.a1. Google Scholar [13] W. Mio, A. Srivastava and S. H. Joshi, On shape of plane elastic curves, International Journal of Computer Vision, 73 (2007), 307-324. doi: 10.1007/s11263-006-9968-0. Google Scholar [14] S. C. Preston, The geometry of whips, Annals of Global Analysis Geometry, 41 (2012), 281-305. doi: 10.1007/s10455-011-9283-z. Google Scholar [15] A. Srivastava, E. Klassen, S. H. Joshi and I. H. Jermyn, Shape analysis of elastic curves in Euclidean spaces, IEEE Trans. PAMI, 33 (2011), 1415-1428. doi: 10.1109/TPAMI.2010.184. Google Scholar [16] C. Temperton, Algorithms for the solution of cyclic tridiagonal systems, Journal of Computational Physics, 19 (1975), 317-323. doi: 10.1016/0021-9991(75)90081-9. Google Scholar [17] A. B. Tumpach, H. Drira, M. Daoudi and A. Srivastava, Gauge invariant framework for shape analysis of surfaces, IEEE Trans Pattern Anal Mach Intell., 38 (2016), 46-59. doi: 10.1109/TPAMI.2015.2430319. Google Scholar [18] L. Younes, P. W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics, Matematica e Applicazioni, 19 (2008), 25-57. doi: 10.4171/RLM/506. Google Scholar
Some parameterized closed immersions $\gamma$ in the plane
Toy example: initial path joining a circle to the same circle via an ellipse. The 5 first shapes at the left correspond to the path at time $t = 0$, $t = 0.25$, $t = 0.5$, $t = 0.75$ and $t = 1$. The right picture shows the entire path, with color varying from red ($t=0$) to blue ($t = 0.5$) to red again ($t=1$)
Straightening of the path illustrated in Fig. 2, with $a=100$ and $b=1$. The first line corresponds to the initial path, the second line to the path after 3500 iterations, and the third line corresponds to the path after 7000 iterations. Underneath, the evolution of the energy with respect to the number of iterations is depicted
Negative gradient of the energy functional at the middle of the path depicted in Fig. 2 for $b=1$ and different values of the parameter $a/b$.
Negative gradient of the energy functional at the middle of the path connecting a circle to the same circle via an ellipse for different values of the eccentricity of the middle ellipse. The first line corresponds to the values of parameters $a =0.01$ and $b=1$. The second line corresponds to $a = 100$ and $b=1$
Negative gradient of the energy functional along the path depicted in Fig. 2 for $a=1$ (upper line), $a = 5$ (middle line) and $a = 50$ (lower line) and $b = 1$
$2$-parameter family of variations of the middle shape of a path connecting a circle to the same circle
Energy functional for the $2$-parameter family of paths whose middle shape is one of the shapes depicted in Fig. 7. The left upper picture corresponds to $a= 0.01$, $b =1$ and the right upper picture to $a = 100$, $b =1$. The lower picture shows the plots of both energy functionals with equal axis
Different paths connecting a Mickey Mouse hand to the same hand with a missing finger
Energy of the paths depicted in Fig.9
 parameters lin. interpol. 1 lin. interpol. 2 path 3 path 4 path 5 $a=0.01$, $b = 1$ 32.3749 27.45 25.3975 26.2504 28.3768 $a = 0.25$, $b =1$ 63.1326 52.4110 47.8818 47.5037 48.2284 $a =100$, $b=1$ 77.6407 66.6800 63.4840 60.9704 57.4557
 parameters lin. interpol. 1 lin. interpol. 2 path 3 path 4 path 5 $a=0.01$, $b = 1$ 32.3749 27.45 25.3975 26.2504 28.3768 $a = 0.25$, $b =1$ 63.1326 52.4110 47.8818 47.5037 48.2284 $a =100$, $b=1$ 77.6407 66.6800 63.4840 60.9704 57.4557
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