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

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) $.

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:
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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.  Google Scholar

[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. BalseiroT. J. StuchiA. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

F. Gay-BalmazD. 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.  Google Scholar

[19]

F. Gay-BalmazD. D. HolmD. M. MeierT. 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.  Google Scholar

[20]

F. Gay-BalmazD. D. HolmD. M. MeierT. 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.  Google Scholar

[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.  Google Scholar

[22]

J. HinkleP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

L. Noakes, Non-null Lie quadratics in $\mathbb{E}^3$, Journal of Mathematical Physics, 45 (2004), 4334-4351.  doi: 10.1063/1.1803609.  Google Scholar

[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.  Google Scholar

[32]

L. Noakes, Duality and Riemannian cubics, Advances in Computational Mathematics, 25 (2006), 195-209.  doi: 10.1007/s10444-004-7621-4.  Google Scholar

[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.  Google Scholar

[34]

L. NoakesG. 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.  Google Scholar

[35]

L. Noakes and T. Popiel, Geometry for robot path planning, Robotica, 25 (2007), 691-701.  doi: 10.1017/S0263574707003669.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

X. PennecP. 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.  Google Scholar

[40]

T. Popiel and L. Noakes, Elastica in $SO(3)$, Journal of the Australian Mathematical Society, 83 (2007), 105-124.  doi: 10.1017/S1446788700036417.  Google Scholar

[41]

I. U. RahmanI. DroriV. C. StoddenD. 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.  Google Scholar

[42]

N. SinghF. X. Vialard and M. Niethammer, Splines for diffeomorphisms, Medical Image Analysis, 25 (2015), 56-71.  doi: 10.1016/j.media.2015.04.012.  Google Scholar

[43]

L. T. Skovgaard, A Riemannian geometry of the multivariate normal model, Scandinavian Journal of Statistics, 11 (1984), 211-223.   Google Scholar

[44]

T. StuchiP. BalseiroA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. BalseiroT. J. StuchiA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

F. Gay-BalmazD. 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.  Google Scholar

[19]

F. Gay-BalmazD. D. HolmD. M. MeierT. 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.  Google Scholar

[20]

F. Gay-BalmazD. D. HolmD. M. MeierT. 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.  Google Scholar

[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.  Google Scholar

[22]

J. HinkleP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

L. Noakes, Non-null Lie quadratics in $\mathbb{E}^3$, Journal of Mathematical Physics, 45 (2004), 4334-4351.  doi: 10.1063/1.1803609.  Google Scholar

[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.  Google Scholar

[32]

L. Noakes, Duality and Riemannian cubics, Advances in Computational Mathematics, 25 (2006), 195-209.  doi: 10.1007/s10444-004-7621-4.  Google Scholar

[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.  Google Scholar

[34]

L. NoakesG. 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.  Google Scholar

[35]

L. Noakes and T. Popiel, Geometry for robot path planning, Robotica, 25 (2007), 691-701.  doi: 10.1017/S0263574707003669.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

X. PennecP. 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.  Google Scholar

[40]

T. Popiel and L. Noakes, Elastica in $SO(3)$, Journal of the Australian Mathematical Society, 83 (2007), 105-124.  doi: 10.1017/S1446788700036417.  Google Scholar

[41]

I. U. RahmanI. DroriV. C. StoddenD. 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.  Google Scholar

[42]

N. SinghF. X. Vialard and M. Niethammer, Splines for diffeomorphisms, Medical Image Analysis, 25 (2015), 56-71.  doi: 10.1016/j.media.2015.04.012.  Google Scholar

[43]

L. T. Skovgaard, A Riemannian geometry of the multivariate normal model, Scandinavian Journal of Statistics, 11 (1984), 211-223.   Google Scholar

[44]

T. StuchiP. BalseiroA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 $
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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