Riemannian cubics and elastica in the manifold <inline-formula><tex-math id="M1">\begin{document}$ \operatorname{SPD}(n) $\end{document}</tex-math></inline-formula> of all <inline-formula><tex-math id="M2">\begin{document}$ n\times n $\end{document}</tex-math></inline-formula> symmetric positive-definite matrices

Erchuan Zhang; Lyle Noakes

doi:10.3934/jgm.2019015

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June 2019, 11(2): 277-299. doi: 10.3934/jgm.2019015

Riemannian cubics and elastica in the manifold $ \operatorname{SPD}(n) $ of all $ n\times n $ symmetric positive-definite matrices

Erchuan Zhang and Lyle Noakes ^,

Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia

^* Corresponding author

To Darryl Holm, with warm greetings and deep respect, on his 70th birthday

Received March 2018 Revised February 2019 Published May 2019

Left Lie reduction is a technique used in the study of curves in bi-invariant Lie groups [32, 33, 40]. Although the manifold $ \operatorname{SPD}(n) $ of all $ n\times n $ symmetric positive-definite matrices is not a Lie group with respect to the standard matrix multiplication, it is a symmetric space with a left action of $ GL(n) $ and an isotropy group $ SO(n) $ leaving the identity matrix fixed. The main purpose of this paper is to extend the method of left Lie reduction to $ \operatorname{SPD}(n) $ and use it to study two second order variational curves: Riemannian cubics and elastica. Riemannian cubics in $ \operatorname{SPD}(n) $ are reduced to so-called Lie quadratics in the Lie algebra $ \mathfrak{gl}(n) $ and geometric analyses are presented. Besides, by using the Frenet-Serret frames and the extended left Lie reduction separately, we investigate elastica in the manifold $ \operatorname{SPD}(n) $. The latter presents a comparatively simple form of the equations for elastica in $ \operatorname{SPD}(n) $.

Keywords: Riemannian cubic, elastica, symmetric positive-definite matrix, extended left Lie reduction, Lie quadratic, Frenet-Serret frame.

Mathematics Subject Classification: Primary: 58E50, 58E40; Secondary: 53C35, 53A35.

Citation: Erchuan Zhang, Lyle Noakes. Riemannian cubics and elastica in the manifold $ \operatorname{SPD}(n) $ of all $ n\times n $ symmetric positive-definite matrices. Journal of Geometric Mechanics, 2019, 11 (2) : 277-299. doi: 10.3934/jgm.2019015

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

Figure 1. On the left, the red curve represents $ u $, blue denotes $ \nu $ and green one shows $ \theta $. On the right we plot coordinates $ (x_1, x_2, x_3) $ of the non-null Riemannian cubic $ x $

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Figure 2. On the left, the blue curve represents $ \kappa^2 $ and the red one shows $ \tau^2 $. On the right we plot coordinates $ (x_1, x_2, x_3) $ of the elastic curve $ x = [x_1 x_3;x_3 x_2] $

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Riemannian cubics and elastica in the manifold $ \operatorname{SPD}(n) $ of all $ n\times n $ symmetric positive-definite matrices

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Riemannian cubics and elastica in the manifold $ \operatorname{SPD}(n) $ of all $ n\times n $ symmetric positive-definite matrices

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