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June  2017, 9(2): 131-156. doi: 10.3934/jgm.2017005

Computing distances and geodesics between manifold-valued curves in the SRV framework

Alice Le Brigant 1,2,

1. 

Institut Mathématique de Bordeaux, UMR 5251, Université de Bordeaux and CNRS, France
2. 

Thales Air Systems, Surface Radar Domain, Technical Directorate, Voie Pierre-Gilles de Gennes, 91470 Limours, France

Received  January 2016 Revised  November 2016 Published  May 2017

This paper focuses on the study of open curves in a Riemannian manifold $M$, and proposes a reparameterization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [29] to define a Riemannian metric on the space of immersions $\mathcal{M}=\text{Imm}([0,1],M)$ by pullback of a natural metric on the tangent bundle $\text{T}\mathcal{M}$. This induces a first-order Sobolev metric on $\mathcal{M}$ and leads to a distance which takes into account the distance between the origins in $M$ and the $L^2$-distance between the SRV representations of the curves. The geodesic equations for this metric are given and exploited to define an exponential map on $\mathcal M$. The optimal deformation of one curve into another can then be constructed using geodesic shooting, which requires to characterize the Jacobi fields of $\mathcal M$. The particular case of curves lying in the hyperbolic half-plane $\mathbb H$ is considered as an example, in the setting of radar signal processing.

Keywords: Shape analysis, manifold-valued curves, square root velocity.
Mathematics Subject Classification: Primary: 58B20, 58D10, 49Q10; Secondary: 62B10.
Citation: Alice Le Brigant. Computing distances and geodesics between manifold-valued curves in the SRV framework. Journal of Geometric Mechanics, 2017, 9 (2) : 131-156. doi: 10.3934/jgm.2017005
References:
[1]

S. I. Amari, O. E. Barndorff-Nielsen, R. E. Kass, S. L. Lauritzen and C. R. Rao, Differential Geometry in Statistical Inference, Institute of Mathematical Statistics, Lecture Notes-Monograph Series, 1987.  Google Scholar

[2]

J. Angulo and S. Velasco-Forero, Morphological processing of univariate Gaussian distribution-valued images based on Poincaré upper-half plane representation, Geometric Theory of Information, Frank Nielsen, Springer International Publ., (2014), 331-366.  Google Scholar

[3]

M. Arnaudon, F. Barbaresco and L. Yang, Riemannian medians and means with applications to radar signal processing, Journal of Selected Topics in Signal Processing, 7 (2013), 595-604.  doi: 10.1109/JSTSP.2013.2261798.  Google Scholar

[4]

F. Barbaresco, Interactions between symmetric cone and information geometries: Bruhat-Tits and Siegel Spaces Models for High Resolution Autoregressive Doppler Imagery, Emerging Trends in Visual Computing, Lecture notes in Computer Science, 5416 (2009), 124-163. doi: 10.1007/978-3-642-00826-9_6.  Google Scholar

[5]

F. Barbaresco, Information Geometry of Covariance Matrix: Cartan-Siegel homogeneous bounded domains, Mostow-Berger fibration and Fréchet median, Springer, 9 (2013), 199-255. doi: 10.1007/978-3-642-30232-9_9.  Google Scholar

[6]

F. Barbaresco, Koszul information geometry and souriau geometric temperature/capacity of lie group thermodynamics, Entropy, 16 (2014), 4521-4565.  doi: 10.3390/e16084521.  Google Scholar

[7]

M. Bauer, M. Bruveris and P. W. Michor, Why use Sobolev metrics on the space of curves, Riemannian Computing in Computer Vision (eds. P. K. Turaga and A. Srivastava), SpringerVerlag, (2016), 233-255.  Google Scholar

[8]

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

[9]

J. Burbea and C. R. Rao, Entropy differential metric, distance and divergence measures in probability spaces: A unified approach, Journal of Multivariate Analysis, 12 (1982), 575-596.  doi: 10.1016/0047-259X(82)90065-3.  Google Scholar

[10]

J. P. Burg, Maximum Entropy Spectral Analysis, Dissertation, Stanford University, 1975. Google Scholar

[11]

E. Celledoni, M. Eslitzbichler and A. Schmeding, Shape analysis on Lie groups with applications in computer animation, Journal of Geometric Mechanics, 8 (2016), 273-304.  doi: 10.3934/jgm.2016008.  Google Scholar

[12]

S. I. R. Costa, S. A. Santos and J. E. Strapasson, Fisher information distance: A geometrical reading, Discrete Applied Mathematics, 197 (2015), 59-69.  doi: 10.1016/j.dam.2014.10.004.  Google Scholar

[13]

M. P. do Carmo, Riemannian Geometry, 1st Edition, Birkhauser, 1992. doi: 10.1007/978-1-4757-2201-7.  Google Scholar

[14]

M. Fréchet, Sur l'extension de certaines évaluations statistiques au cas de petits echantillons, Revue de l'Institut International de Statistique, 11 (1943), 182-205.   Google Scholar

[15]

A. Kriegl and P. W. Michor, Aspects of the theory of infinite dimensional manifolds, Differential Geometry and its Applications, 1 (1991), 159-176.  doi: 10.1016/0926-2245(91)90029-9.  Google Scholar

[16]

H. Laga, S. Kurtek, A. Srivastava and S. J. Miklavcic, Landmark-free statistical analysis of the shape of plant leaves, Journal of Theoretical Biology, 363 (2014), 41-52.  doi: 10.1016/j.jtbi.2014.07.036.  Google Scholar

[17]

A. Le Brigant, M. Arnaudon and F. Barbaresco, Reparameterization invariant distance on the space of curves in the hyperbolic plane AIP Conference Proceedings, 1641 (2015), p504. doi: 10.1063/1.4906016.  Google Scholar

[18]

A. Le Brigant, F. Barbaresco and M. Arnaudon, Geometric barycenters of time/Doppler spectra for the recognition of non-stationary targets, 17th International Radar Symposium Krakow, (2016), 1-6. doi: 10.1109/IRS.2016.7497368.  Google Scholar

[19]

A. Le Brigant, A discrete framework to find the optimal matching between manifold-valued curves, preprint, arXiv: 1703.05107. Google Scholar

[20]

A. C. Mennucci, A. Yezzi and G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves, Interfaces and Free Boundaries, 10 (2008), 423-445.  doi: 10.4171/IFB/196.  Google Scholar

[21]

P. W. Michor, Manifolds of Differentiable Mappings, in vol. 3 of Shiva Mathematics Series (Shiva Publ.), Orpington, 1980.  Google Scholar

[22]

P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Documenta Mathematica, 10 (2005), 217-245.   Google Scholar

[23]

P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves, Journal of the European Mathematical Society, 8 (2006), 1-48.  doi: 10.4171/JEMS/37.  Google Scholar

[24]

P. W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Applied and Computational Harmonic Analysis, 23 (2007), 74-113.  doi: 10.1016/j.acha.2006.07.004.  Google Scholar

[25]

P. W. Michor, Topics in Differential Geometry, in volume 93 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/093.  Google Scholar

[26]

M. Pilté and F. Barbaresco, Tracking quality monitoring based on information geometry and geodesic shooting, 17th International Radar Symposium, (2016), 1-6. Google Scholar

[27]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Mathematical Journal, 10 (1958), 338-354.  doi: 10.2748/tmj/1178244668.  Google Scholar

[28]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds Ⅱ, Tohoku Mathematical Journal, 14 (1962), 146-155.  doi: 10.2748/tmj/1178244169.  Google Scholar

[29]

A. Srivastava, E. Klassen, S. H. Joshi and I. H. Jermyn, Shape analysis of elastic curves in Euclidean spaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 1415-1428.  doi: 10.1109/TPAMI.2010.184.  Google Scholar

[30]

J. Su, S. Kurtek, E. Klassen and A. Srivastava, Statistical analysis of trajectories on Riemannian manifolds: Bird migration, hurricane tracking and video surveillance, Annals of Applied Statistics, 8 (2014), 530-552.  doi: 10.1214/13-AOAS701.  Google Scholar

[31]

W. F. Trench, An algorithm for the inversion of finite Toeplitz matrices, Journal of the Society for Industrial and Applied Mathematics, 12 (1964), 515-522.  doi: 10.1137/0112045.  Google Scholar

[32]

S. Verblunsky, On positive harmonic functions: A contribution to the algebra of Fourier series, Proceedings London Mathematical Society, s2-38 (1935), 125-157.  doi: 10.1112/plms/s2-38.1.125.  Google Scholar

[33]

L. Younes, Computable elastic distances between shapes, SIAM Journal on Applied Mathematics, 58 (1998), 565-586.  doi: 10.1137/S0036139995287685.  Google Scholar

[34]

L. Younes, P. W. Michor, J. Shah and D. Mumford, A Metric on shape space with explicit geodesics, Rendiconti Lincei Matematica e Applicazioni, 19 (2008), 25-57.  doi: 10.4171/RLM/506.  Google Scholar

[35]

Z. Zhang, J. Su, E. Klassen, H. Le and A. Srivastava, Video-based action recognition using rate-invariant analysis of covariance trajectories, preprint, arXiv: 1503.06699. Google Scholar

show all references

References:
[1]

S. I. Amari, O. E. Barndorff-Nielsen, R. E. Kass, S. L. Lauritzen and C. R. Rao, Differential Geometry in Statistical Inference, Institute of Mathematical Statistics, Lecture Notes-Monograph Series, 1987.  Google Scholar

[2]

J. Angulo and S. Velasco-Forero, Morphological processing of univariate Gaussian distribution-valued images based on Poincaré upper-half plane representation, Geometric Theory of Information, Frank Nielsen, Springer International Publ., (2014), 331-366.  Google Scholar

[3]

M. Arnaudon, F. Barbaresco and L. Yang, Riemannian medians and means with applications to radar signal processing, Journal of Selected Topics in Signal Processing, 7 (2013), 595-604.  doi: 10.1109/JSTSP.2013.2261798.  Google Scholar

[4]

F. Barbaresco, Interactions between symmetric cone and information geometries: Bruhat-Tits and Siegel Spaces Models for High Resolution Autoregressive Doppler Imagery, Emerging Trends in Visual Computing, Lecture notes in Computer Science, 5416 (2009), 124-163. doi: 10.1007/978-3-642-00826-9_6.  Google Scholar

[5]

F. Barbaresco, Information Geometry of Covariance Matrix: Cartan-Siegel homogeneous bounded domains, Mostow-Berger fibration and Fréchet median, Springer, 9 (2013), 199-255. doi: 10.1007/978-3-642-30232-9_9.  Google Scholar

[6]

F. Barbaresco, Koszul information geometry and souriau geometric temperature/capacity of lie group thermodynamics, Entropy, 16 (2014), 4521-4565.  doi: 10.3390/e16084521.  Google Scholar

[7]

M. Bauer, M. Bruveris and P. W. Michor, Why use Sobolev metrics on the space of curves, Riemannian Computing in Computer Vision (eds. P. K. Turaga and A. Srivastava), SpringerVerlag, (2016), 233-255.  Google Scholar

[8]

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

[9]

J. Burbea and C. R. Rao, Entropy differential metric, distance and divergence measures in probability spaces: A unified approach, Journal of Multivariate Analysis, 12 (1982), 575-596.  doi: 10.1016/0047-259X(82)90065-3.  Google Scholar

[10]

J. P. Burg, Maximum Entropy Spectral Analysis, Dissertation, Stanford University, 1975. Google Scholar

[11]

E. Celledoni, M. Eslitzbichler and A. Schmeding, Shape analysis on Lie groups with applications in computer animation, Journal of Geometric Mechanics, 8 (2016), 273-304.  doi: 10.3934/jgm.2016008.  Google Scholar

[12]

S. I. R. Costa, S. A. Santos and J. E. Strapasson, Fisher information distance: A geometrical reading, Discrete Applied Mathematics, 197 (2015), 59-69.  doi: 10.1016/j.dam.2014.10.004.  Google Scholar

[13]

M. P. do Carmo, Riemannian Geometry, 1st Edition, Birkhauser, 1992. doi: 10.1007/978-1-4757-2201-7.  Google Scholar

[14]

M. Fréchet, Sur l'extension de certaines évaluations statistiques au cas de petits echantillons, Revue de l'Institut International de Statistique, 11 (1943), 182-205.   Google Scholar

[15]

A. Kriegl and P. W. Michor, Aspects of the theory of infinite dimensional manifolds, Differential Geometry and its Applications, 1 (1991), 159-176.  doi: 10.1016/0926-2245(91)90029-9.  Google Scholar

[16]

H. Laga, S. Kurtek, A. Srivastava and S. J. Miklavcic, Landmark-free statistical analysis of the shape of plant leaves, Journal of Theoretical Biology, 363 (2014), 41-52.  doi: 10.1016/j.jtbi.2014.07.036.  Google Scholar

[17]

A. Le Brigant, M. Arnaudon and F. Barbaresco, Reparameterization invariant distance on the space of curves in the hyperbolic plane AIP Conference Proceedings, 1641 (2015), p504. doi: 10.1063/1.4906016.  Google Scholar

[18]

A. Le Brigant, F. Barbaresco and M. Arnaudon, Geometric barycenters of time/Doppler spectra for the recognition of non-stationary targets, 17th International Radar Symposium Krakow, (2016), 1-6. doi: 10.1109/IRS.2016.7497368.  Google Scholar

[19]

A. Le Brigant, A discrete framework to find the optimal matching between manifold-valued curves, preprint, arXiv: 1703.05107. Google Scholar

[20]

A. C. Mennucci, A. Yezzi and G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves, Interfaces and Free Boundaries, 10 (2008), 423-445.  doi: 10.4171/IFB/196.  Google Scholar

[21]

P. W. Michor, Manifolds of Differentiable Mappings, in vol. 3 of Shiva Mathematics Series (Shiva Publ.), Orpington, 1980.  Google Scholar

[22]

P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Documenta Mathematica, 10 (2005), 217-245.   Google Scholar

[23]

P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves, Journal of the European Mathematical Society, 8 (2006), 1-48.  doi: 10.4171/JEMS/37.  Google Scholar

[24]

P. W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Applied and Computational Harmonic Analysis, 23 (2007), 74-113.  doi: 10.1016/j.acha.2006.07.004.  Google Scholar

[25]

P. W. Michor, Topics in Differential Geometry, in volume 93 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/093.  Google Scholar

[26]

M. Pilté and F. Barbaresco, Tracking quality monitoring based on information geometry and geodesic shooting, 17th International Radar Symposium, (2016), 1-6. Google Scholar

[27]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Mathematical Journal, 10 (1958), 338-354.  doi: 10.2748/tmj/1178244668.  Google Scholar

[28]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds Ⅱ, Tohoku Mathematical Journal, 14 (1962), 146-155.  doi: 10.2748/tmj/1178244169.  Google Scholar

[29]

A. Srivastava, E. Klassen, S. H. Joshi and I. H. Jermyn, Shape analysis of elastic curves in Euclidean spaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 1415-1428.  doi: 10.1109/TPAMI.2010.184.  Google Scholar

[30]

J. Su, S. Kurtek, E. Klassen and A. Srivastava, Statistical analysis of trajectories on Riemannian manifolds: Bird migration, hurricane tracking and video surveillance, Annals of Applied Statistics, 8 (2014), 530-552.  doi: 10.1214/13-AOAS701.  Google Scholar

[31]

W. F. Trench, An algorithm for the inversion of finite Toeplitz matrices, Journal of the Society for Industrial and Applied Mathematics, 12 (1964), 515-522.  doi: 10.1137/0112045.  Google Scholar

[32]

S. Verblunsky, On positive harmonic functions: A contribution to the algebra of Fourier series, Proceedings London Mathematical Society, s2-38 (1935), 125-157.  doi: 10.1112/plms/s2-38.1.125.  Google Scholar

[33]

L. Younes, Computable elastic distances between shapes, SIAM Journal on Applied Mathematics, 58 (1998), 565-586.  doi: 10.1137/S0036139995287685.  Google Scholar

[34]

L. Younes, P. W. Michor, J. Shah and D. Mumford, A Metric on shape space with explicit geodesics, Rendiconti Lincei Matematica e Applicazioni, 19 (2008), 25-57.  doi: 10.4171/RLM/506.  Google Scholar

[35]

Z. Zhang, J. Su, E. Klassen, H. Le and A. Srivastava, Video-based action recognition using rate-invariant analysis of covariance trajectories, preprint, arXiv: 1503.06699. Google Scholar

Figure 1.  Illustration of the distance between two curves $c_0$ and $c_1$ in the space of curves $\mathcal{M}$
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Figure 2.  Geodesic shooting in the space of curves $\mathcal M$
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Figure 3.  Steps of the first iteration of the geodesic shooting algorithm applied to a pair of geodesics of the upper half-plane $\mathbb H$
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Figure 4.  Optimal deformations between pairs of geodesics (in black) of the upper half-plane $\mathbb H$, for our metric (in blue) and for the $L^2$-metric (in green). The orientation of the right-hand curve is inverted in the second image compared to the first, and in the fourth compared to the third
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Figure 5.  Geodesics of the hyperbolic half-plane
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Figure 6.  Computation of the mean curve (in black) for 4 sets of 11 curves in the hyperbolic half-plane, constructed from simulated helicopter radar data
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