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Dual pairs for matrix groups
Riemannian cubics and elastica in the manifold $ \operatorname{SPD}(n) $ of all $ n\times n $ symmetric positive-definite matrices
Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia |
Left Lie reduction is a technique used in the study of curves in bi-invariant Lie groups [
References:
[1] |
L. Abrunheiro, M. Camarinha, J. Clemente-Gallardo, J. C. Cuch$\acute{i}$ and P. Santos, A general framework for quantum splines, International Journal of Geometric Methods in Modern Physics, 15 (2018).
doi: 10.1142/S0219887818501475. |
[2] |
M. Assif, R. Banavar, A. Bloch, M. Camarinha and L. Colombo, Variational collision avoidance problems on Riemannian manifolds, preprint, arXiv: 1804.00122. Google Scholar |
[3] |
P. Balseiro, T. J. Stuchi, A. Cabrera and J. Koiller,
About simple variational splines from the Hamiltonian viewpoint, Journal of Geometric Mechanics, 9 (2017), 257-290.
doi: 10.3934/jgm.2017011. |
[4] |
J. Batista, K. Krakowski and F. S. Leite, Exploring quasi-geodesics on Stiefel manifolds in order to smooth interpolate between domains, 2017 IEEE 56th Annual Conference on Decision and Control (CDC), (2017).
doi: 10.1109/CDC.2017.8264624. |
[5] |
J. D. Benamou, T. Gallouet and F. X. Vialard, Second order models for optimal transport and cubic splines on the Wasserstein space, preprint, arXiv: 1801.04144. Google Scholar |
[6] |
A. Bloch, M. Camarinha and L. Colombo, Variational obstacle avoidance problem on Riemannian manifolds, 2017 IEEE 56th Annual Conference on Decision and Control (CDC), (2017).
doi: 10.1109/CDC.2017.8263657. |
[7] |
A. Bloch, M. Camarinha and L. Colombo, Dynamic interpolation for obstacle avoidance on Riemannian manifolds, preprint, arXiv: 1809.03168. Google Scholar |
[8] |
D. C. Brody, D. D. Holm and D. M. Meier, Quantum splines, Physical Review Letters, 109 (2012), 100501.
doi: 10.1103/PhysRevLett.109.100501. |
[9] |
R. Bryant and P. Griffiths,
Reduction for constrained variational problems and $\int\kappa^2/2 ds$, American Journal of Mathematics, 108 (1986), 525-570.
doi: 10.2307/2374654. |
[10] |
C. Burnett, D. Holm and D. Meier, Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469 (2013), 20130249, 24pp.
doi: 10.1098/rspa.2013.0249. |
[11] |
S. S. Chern, W. H. Chen and K. S. Lam, Lectures on Differential Geometry, World Scientific, Singapore, 1999. Google Scholar |
[12] |
P. Chossat and O. Faugeras, Hyperbolic planforms in relation to visual edges and textures perception, PLoS Computational Biology, 5 (2009), e1000625, 16pp.
doi: 10.1371/journal.pcbi.1000625. |
[13] |
P. Crouch and F. S. Leite,
The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces, Journal of Dynamical and Control Systems, 1 (1995), 177-202.
doi: 10.1007/BF02254638. |
[14] |
M. P. Do Carmo, Differential Geometry Of Curves and Surfaces: Revised and Upadated Second Edition, 2$^{nd}$ edition, Courier Dover Publications, New York, 2016. |
[15] |
S. Fiori,
Learning the Frchet mean over the manifold of symmetric positive-definite matrices, Cognitive Computation, 1 (2009), 279-291.
doi: 10.1007/s12559-009-9026-7. |
[16] |
S. A. Gabriel and J. T. Kajiya, Spline interpolation in curved manifolds, Unpublished manuscript, 1985. Google Scholar |
[17] |
F. Gay-Balmaz, D. Holm and T. Ratiu, Geometric dynamics of optimization, Commun. Math. Sci., 11 (2013), 163–231, arXiv: 0912.2989.
doi: 10.4310/CMS.2013.v11.n1.a6. |
[18] |
F. Gay-Balmaz, D. Holm and T. Ratiu,
Higher order Lagrange-Poincar$\acute{e}$ and Hamilton-Poincar$\acute{e}$ reductions, Bulletin of the Brazilian Mathematical Society, 42 (2011), 579-606.
doi: 10.1007/s00574-011-0030-7. |
[19] |
F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F. X. Vialard,
Invariant higher-order variational problems, Communications in Mathematical Physics, 309 (2012), 413-458.
doi: 10.1007/s00220-011-1313-y. |
[20] |
F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F. X. Vialard,
Invariant higher-order variational problems Ⅱ, Journal of Nonlinear Science, 22 (2012), 553-597.
doi: 10.1007/s00332-012-9137-2. |
[21] |
P. A. Griffiths, Exterior Differential Systems and the Calculus of Variations, Progress in Mathematics, 25. Birkh?user, Boston, Mass., 1983.
doi: 10.1007/978-1-4615-8166-6. |
[22] |
J. Hinkle, P. T. Fletcher and S. Joshi,
Intrinsic polynomials for regression on Riemannian manifolds, Journal of Mathematical Imaging and Vision, 50 (2014), 32-52.
doi: 10.1007/s10851-013-0489-5. |
[23] |
D. Holm, L. Noakes and J. Vankerschaver, Relative geodesics in the special Euclidean group, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469 (2013), 20130297, 21pp.
doi: 10.1098/rspa.2013.0297. |
[24] |
S. Jayasumana, R. Hartley, M. Salzmann, H. Li and M. Harandi, Kernel methods on the Riemannian manifold of symmetric positive definite matrices, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2013), 73–80. Google Scholar |
[25] |
V. Jurdjevic,
Non-Euclidean elastica, American Journal of Mathematics, 117 (1995), 93-124.
doi: 10.2307/2375037. |
[26] |
J. Langer and D. A. Singer,
The total squared curvature of closed curves, Journal of Differential Geometry, 20 (1984), 1-22.
doi: 10.4310/jdg/1214438990. |
[27] |
J. Langer and D. A. Singer,
Curves in the hyperbolic plane and mean curvature of tori in $3$-space, Bulletin of the London Mathematical Society, 16 (1984), 531-534.
doi: 10.1112/blms/16.5.531. |
[28] |
R. Levien, The elastica: a mathematical history, University of California at Berkeley, 2008. Available from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.226.2020&rep=rep1&type=pdf. Google Scholar |
[29] |
L. Noakes,
Null cubics and Lie quadratics, Journal of Mathematical Physics, 44 (2003), 1436-1448.
doi: 10.1063/1.1537461. |
[30] |
L. Noakes,
Non-null Lie quadratics in $\mathbb{E}^3$, Journal of Mathematical Physics, 45 (2004), 4334-4351.
doi: 10.1063/1.1803609. |
[31] |
L. Noakes,
Asymptotics of null Lie quadratics in $\mathbb{E}^3$, SIAM Journal on Applied Dynamical Systems, 7 (2008), 437-460.
doi: 10.1137/070686755. |
[32] |
L. Noakes,
Duality and Riemannian cubics, Advances in Computational Mathematics, 25 (2006), 195-209.
doi: 10.1007/s10444-004-7621-4. |
[33] |
L. Noakes and T. Popiel,
Quadratures and cubics in $SO(3)$ and $SO(1, 2)$, IMA Journal of Mathematical Control and Information, 23 (2006), 463-473.
doi: 10.1093/imamci/dni069. |
[34] |
L. Noakes, G. Heinzinger and B. Paden,
Cubic splines on curved spaces, IMA Journal of Mathematical Control and Information, 6 (1989), 465-473.
doi: 10.1093/imamci/6.4.465. |
[35] |
L. Noakes and T. Popiel,
Geometry for robot path planning, Robotica, 25 (2007), 691-701.
doi: 10.1017/S0263574707003669. |
[36] |
M. Pauley and L. Noakes,
Cubics and negative curvature, Differential Geometry and its Applications, 30 (2012), 694-701.
doi: 10.1016/j.difgeo.2012.09.004. |
[37] |
X. Pennec, Statistical computing on manifolds for computational anatomy, Ph.D thesis, Universit$\acute{e}$ Nice Sophia Antipolis, 2006. Available from: https://tel.archives-ouvertes.fr/tel-00633163/document. Google Scholar |
[38] |
X. Pennec and N. Ayache,
Uniform distribution, distance and expectation problems for geometric features processing, Journal of Mathematical Imaging and Vision, 9 (1998), 49-67.
doi: 10.1023/A:1008270110193. |
[39] |
X. Pennec, P. Fillard and N. Ayache,
A Riemannian framework for tensor computing, International Journal of Computer Vision, 66 (2006), 41-66.
doi: 10.1007/s11263-005-3222-z. |
[40] |
T. Popiel and L. Noakes,
Elastica in $SO(3)$, Journal of the Australian Mathematical Society, 83 (2007), 105-124.
doi: 10.1017/S1446788700036417. |
[41] |
I. U. Rahman, I. Drori, V. C. Stodden, D. L. Donoho and P. Schr$\ddot{o}$der,
Multiscale representations for manifold-valued data, Multiscale Modeling & Simulation, 4 (2005), 1201-1232.
doi: 10.1137/050622729. |
[42] |
N. Singh, F. X. Vialard and M. Niethammer,
Splines for diffeomorphisms, Medical Image Analysis, 25 (2015), 56-71.
doi: 10.1016/j.media.2015.04.012. |
[43] |
L. T. Skovgaard,
A Riemannian geometry of the multivariate normal model, Scandinavian Journal of Statistics, 11 (1984), 211-223.
|
[44] |
T. Stuchi, P. Balseiro, A. Cabrera and J. Koiller,
Minimal time splines on the sphere, Sao Paulo Journal of Mathematical Sciences, 12 (2018), 82-107.
doi: 10.1007/s40863-017-0078-4. |
[45] |
M. Zefran and V. Kumar,
Planning of smooth motions on $SE(3)$, Proceedings of IEEE International Conference on Robotics and Automation, 1 (1996), 121-126.
doi: 10.1109/ROBOT.1996.503583. |
[46] |
E. Zhang and L. Noakes,
Left Lie reduction for curves in homogeneous spaces, Advances in Computational Mathematics, 44 (2018), 1673-1686.
doi: 10.1007/s10444-018-9601-0. |
show all references
References:
[1] |
L. Abrunheiro, M. Camarinha, J. Clemente-Gallardo, J. C. Cuch$\acute{i}$ and P. Santos, A general framework for quantum splines, International Journal of Geometric Methods in Modern Physics, 15 (2018).
doi: 10.1142/S0219887818501475. |
[2] |
M. Assif, R. Banavar, A. Bloch, M. Camarinha and L. Colombo, Variational collision avoidance problems on Riemannian manifolds, preprint, arXiv: 1804.00122. Google Scholar |
[3] |
P. Balseiro, T. J. Stuchi, A. Cabrera and J. Koiller,
About simple variational splines from the Hamiltonian viewpoint, Journal of Geometric Mechanics, 9 (2017), 257-290.
doi: 10.3934/jgm.2017011. |
[4] |
J. Batista, K. Krakowski and F. S. Leite, Exploring quasi-geodesics on Stiefel manifolds in order to smooth interpolate between domains, 2017 IEEE 56th Annual Conference on Decision and Control (CDC), (2017).
doi: 10.1109/CDC.2017.8264624. |
[5] |
J. D. Benamou, T. Gallouet and F. X. Vialard, Second order models for optimal transport and cubic splines on the Wasserstein space, preprint, arXiv: 1801.04144. Google Scholar |
[6] |
A. Bloch, M. Camarinha and L. Colombo, Variational obstacle avoidance problem on Riemannian manifolds, 2017 IEEE 56th Annual Conference on Decision and Control (CDC), (2017).
doi: 10.1109/CDC.2017.8263657. |
[7] |
A. Bloch, M. Camarinha and L. Colombo, Dynamic interpolation for obstacle avoidance on Riemannian manifolds, preprint, arXiv: 1809.03168. Google Scholar |
[8] |
D. C. Brody, D. D. Holm and D. M. Meier, Quantum splines, Physical Review Letters, 109 (2012), 100501.
doi: 10.1103/PhysRevLett.109.100501. |
[9] |
R. Bryant and P. Griffiths,
Reduction for constrained variational problems and $\int\kappa^2/2 ds$, American Journal of Mathematics, 108 (1986), 525-570.
doi: 10.2307/2374654. |
[10] |
C. Burnett, D. Holm and D. Meier, Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469 (2013), 20130249, 24pp.
doi: 10.1098/rspa.2013.0249. |
[11] |
S. S. Chern, W. H. Chen and K. S. Lam, Lectures on Differential Geometry, World Scientific, Singapore, 1999. Google Scholar |
[12] |
P. Chossat and O. Faugeras, Hyperbolic planforms in relation to visual edges and textures perception, PLoS Computational Biology, 5 (2009), e1000625, 16pp.
doi: 10.1371/journal.pcbi.1000625. |
[13] |
P. Crouch and F. S. Leite,
The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces, Journal of Dynamical and Control Systems, 1 (1995), 177-202.
doi: 10.1007/BF02254638. |
[14] |
M. P. Do Carmo, Differential Geometry Of Curves and Surfaces: Revised and Upadated Second Edition, 2$^{nd}$ edition, Courier Dover Publications, New York, 2016. |
[15] |
S. Fiori,
Learning the Frchet mean over the manifold of symmetric positive-definite matrices, Cognitive Computation, 1 (2009), 279-291.
doi: 10.1007/s12559-009-9026-7. |
[16] |
S. A. Gabriel and J. T. Kajiya, Spline interpolation in curved manifolds, Unpublished manuscript, 1985. Google Scholar |
[17] |
F. Gay-Balmaz, D. Holm and T. Ratiu, Geometric dynamics of optimization, Commun. Math. Sci., 11 (2013), 163–231, arXiv: 0912.2989.
doi: 10.4310/CMS.2013.v11.n1.a6. |
[18] |
F. Gay-Balmaz, D. Holm and T. Ratiu,
Higher order Lagrange-Poincar$\acute{e}$ and Hamilton-Poincar$\acute{e}$ reductions, Bulletin of the Brazilian Mathematical Society, 42 (2011), 579-606.
doi: 10.1007/s00574-011-0030-7. |
[19] |
F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F. X. Vialard,
Invariant higher-order variational problems, Communications in Mathematical Physics, 309 (2012), 413-458.
doi: 10.1007/s00220-011-1313-y. |
[20] |
F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F. X. Vialard,
Invariant higher-order variational problems Ⅱ, Journal of Nonlinear Science, 22 (2012), 553-597.
doi: 10.1007/s00332-012-9137-2. |
[21] |
P. A. Griffiths, Exterior Differential Systems and the Calculus of Variations, Progress in Mathematics, 25. Birkh?user, Boston, Mass., 1983.
doi: 10.1007/978-1-4615-8166-6. |
[22] |
J. Hinkle, P. T. Fletcher and S. Joshi,
Intrinsic polynomials for regression on Riemannian manifolds, Journal of Mathematical Imaging and Vision, 50 (2014), 32-52.
doi: 10.1007/s10851-013-0489-5. |
[23] |
D. Holm, L. Noakes and J. Vankerschaver, Relative geodesics in the special Euclidean group, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469 (2013), 20130297, 21pp.
doi: 10.1098/rspa.2013.0297. |
[24] |
S. Jayasumana, R. Hartley, M. Salzmann, H. Li and M. Harandi, Kernel methods on the Riemannian manifold of symmetric positive definite matrices, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2013), 73–80. Google Scholar |
[25] |
V. Jurdjevic,
Non-Euclidean elastica, American Journal of Mathematics, 117 (1995), 93-124.
doi: 10.2307/2375037. |
[26] |
J. Langer and D. A. Singer,
The total squared curvature of closed curves, Journal of Differential Geometry, 20 (1984), 1-22.
doi: 10.4310/jdg/1214438990. |
[27] |
J. Langer and D. A. Singer,
Curves in the hyperbolic plane and mean curvature of tori in $3$-space, Bulletin of the London Mathematical Society, 16 (1984), 531-534.
doi: 10.1112/blms/16.5.531. |
[28] |
R. Levien, The elastica: a mathematical history, University of California at Berkeley, 2008. Available from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.226.2020&rep=rep1&type=pdf. Google Scholar |
[29] |
L. Noakes,
Null cubics and Lie quadratics, Journal of Mathematical Physics, 44 (2003), 1436-1448.
doi: 10.1063/1.1537461. |
[30] |
L. Noakes,
Non-null Lie quadratics in $\mathbb{E}^3$, Journal of Mathematical Physics, 45 (2004), 4334-4351.
doi: 10.1063/1.1803609. |
[31] |
L. Noakes,
Asymptotics of null Lie quadratics in $\mathbb{E}^3$, SIAM Journal on Applied Dynamical Systems, 7 (2008), 437-460.
doi: 10.1137/070686755. |
[32] |
L. Noakes,
Duality and Riemannian cubics, Advances in Computational Mathematics, 25 (2006), 195-209.
doi: 10.1007/s10444-004-7621-4. |
[33] |
L. Noakes and T. Popiel,
Quadratures and cubics in $SO(3)$ and $SO(1, 2)$, IMA Journal of Mathematical Control and Information, 23 (2006), 463-473.
doi: 10.1093/imamci/dni069. |
[34] |
L. Noakes, G. Heinzinger and B. Paden,
Cubic splines on curved spaces, IMA Journal of Mathematical Control and Information, 6 (1989), 465-473.
doi: 10.1093/imamci/6.4.465. |
[35] |
L. Noakes and T. Popiel,
Geometry for robot path planning, Robotica, 25 (2007), 691-701.
doi: 10.1017/S0263574707003669. |
[36] |
M. Pauley and L. Noakes,
Cubics and negative curvature, Differential Geometry and its Applications, 30 (2012), 694-701.
doi: 10.1016/j.difgeo.2012.09.004. |
[37] |
X. Pennec, Statistical computing on manifolds for computational anatomy, Ph.D thesis, Universit$\acute{e}$ Nice Sophia Antipolis, 2006. Available from: https://tel.archives-ouvertes.fr/tel-00633163/document. Google Scholar |
[38] |
X. Pennec and N. Ayache,
Uniform distribution, distance and expectation problems for geometric features processing, Journal of Mathematical Imaging and Vision, 9 (1998), 49-67.
doi: 10.1023/A:1008270110193. |
[39] |
X. Pennec, P. Fillard and N. Ayache,
A Riemannian framework for tensor computing, International Journal of Computer Vision, 66 (2006), 41-66.
doi: 10.1007/s11263-005-3222-z. |
[40] |
T. Popiel and L. Noakes,
Elastica in $SO(3)$, Journal of the Australian Mathematical Society, 83 (2007), 105-124.
doi: 10.1017/S1446788700036417. |
[41] |
I. U. Rahman, I. Drori, V. C. Stodden, D. L. Donoho and P. Schr$\ddot{o}$der,
Multiscale representations for manifold-valued data, Multiscale Modeling & Simulation, 4 (2005), 1201-1232.
doi: 10.1137/050622729. |
[42] |
N. Singh, F. X. Vialard and M. Niethammer,
Splines for diffeomorphisms, Medical Image Analysis, 25 (2015), 56-71.
doi: 10.1016/j.media.2015.04.012. |
[43] |
L. T. Skovgaard,
A Riemannian geometry of the multivariate normal model, Scandinavian Journal of Statistics, 11 (1984), 211-223.
|
[44] |
T. Stuchi, P. Balseiro, A. Cabrera and J. Koiller,
Minimal time splines on the sphere, Sao Paulo Journal of Mathematical Sciences, 12 (2018), 82-107.
doi: 10.1007/s40863-017-0078-4. |
[45] |
M. Zefran and V. Kumar,
Planning of smooth motions on $SE(3)$, Proceedings of IEEE International Conference on Robotics and Automation, 1 (1996), 121-126.
doi: 10.1109/ROBOT.1996.503583. |
[46] |
E. Zhang and L. Noakes,
Left Lie reduction for curves in homogeneous spaces, Advances in Computational Mathematics, 44 (2018), 1673-1686.
doi: 10.1007/s10444-018-9601-0. |


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