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The group of diffeomorphisms of the circle: Reproducing kernels and analogs of spherical functions
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 |
We study the pull-back of the 2-parameter family of quotient elastic metrics introduced in [
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. |
[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. |
[3] |
H. Brezis,
Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011. |
[4] |
M. Bruveris,
Optimal reparametrizations in the square root velocity framework, SIAM Journal on Mathematical Analysis, 48 (2016), 4335-4354.
doi: 10.1137/15M1014693. |
[5] |
J. Brylinski,
Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser Boston, Inc., Boston, MA, 2008.
doi: 10.1007/978-0-8176-4731-5. |
[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. |
[7] |
T. Diez, Slice theorem for Fréchet group actions and covariant symplectic field theory, 2014, arXiv: 1405.2249 |
[8] |
F. Dubeau and J. Savoie,
A remark on cyclic tridiagonal matrices, Zastosowania Matematyki Applicationes Mathematicae, 21 (1991), 253-256.
|
[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. |
[10] |
N. J. Higham,
Accuracy and Stability in Numerical Algorithms: Second Edition, SIAM, 2002.
doi: 10.1137/1.9780898718027. |
[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. |
[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. |
[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. |
[14] |
S. C. Preston,
The geometry of whips, Annals of Global Analysis Geometry, 41 (2012), 281-305.
doi: 10.1007/s10455-011-9283-z. |
[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. |
[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. |
[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. |
[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. |
show all references
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. |
[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. |
[3] |
H. Brezis,
Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011. |
[4] |
M. Bruveris,
Optimal reparametrizations in the square root velocity framework, SIAM Journal on Mathematical Analysis, 48 (2016), 4335-4354.
doi: 10.1137/15M1014693. |
[5] |
J. Brylinski,
Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser Boston, Inc., Boston, MA, 2008.
doi: 10.1007/978-0-8176-4731-5. |
[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. |
[7] |
T. Diez, Slice theorem for Fréchet group actions and covariant symplectic field theory, 2014, arXiv: 1405.2249 |
[8] |
F. Dubeau and J. Savoie,
A remark on cyclic tridiagonal matrices, Zastosowania Matematyki Applicationes Mathematicae, 21 (1991), 253-256.
|
[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. |
[10] |
N. J. Higham,
Accuracy and Stability in Numerical Algorithms: Second Edition, SIAM, 2002.
doi: 10.1137/1.9780898718027. |
[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. |
[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. |
[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. |
[14] |
S. C. Preston,
The geometry of whips, Annals of Global Analysis Geometry, 41 (2012), 281-305.
doi: 10.1007/s10455-011-9283-z. |
[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. |
[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. |
[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. |
[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. |







parameters | lin. interpol. 1 | lin. interpol. 2 | path 3 | path 4 | path 5 |
32.3749 | 27.45 | 25.3975 | 26.2504 | 28.3768 | |
63.1326 | 52.4110 | 47.8818 | 47.5037 | 48.2284 | |
77.6407 | 66.6800 | 63.4840 | 60.9704 | 57.4557 |
parameters | lin. interpol. 1 | lin. interpol. 2 | path 3 | path 4 | path 5 |
32.3749 | 27.45 | 25.3975 | 26.2504 | 28.3768 | |
63.1326 | 52.4110 | 47.8818 | 47.5037 | 48.2284 | |
77.6407 | 66.6800 | 63.4840 | 60.9704 | 57.4557 |
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