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

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

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.

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.

[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.

[3]

M. ArnaudonF. 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.

[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.

[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.

[6]

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

[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.

[8]

M. BauerM. BruverisS. 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.

[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.

[10]

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

[11]

E. CelledoniM. 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.

[12]

S. I. R. CostaS. 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.

[13]

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

[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. 

[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.

[16]

H. LagaS. KurtekA. 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.

[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.

[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.

[19]

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

[20]

A. C. MennucciA. 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.

[21]

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

[22]

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

[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.

[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.

[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.

[26]

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

[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.

[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.

[29]

A. SrivastavaE. KlassenS. 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.

[30]

J. SuS. KurtekE. 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.

[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.

[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.

[33]

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

[34]

L. YounesP. W. MichorJ. 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.

[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.

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.

[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.

[3]

M. ArnaudonF. 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.

[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.

[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.

[6]

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

[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.

[8]

M. BauerM. BruverisS. 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.

[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.

[10]

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

[11]

E. CelledoniM. 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.

[12]

S. I. R. CostaS. 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.

[13]

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

[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. 

[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.

[16]

H. LagaS. KurtekA. 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.

[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.

[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.

[19]

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

[20]

A. C. MennucciA. 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.

[21]

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

[22]

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

[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.

[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.

[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.

[26]

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

[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.

[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.

[29]

A. SrivastavaE. KlassenS. 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.

[30]

J. SuS. KurtekE. 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.

[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.

[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.

[33]

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

[34]

L. YounesP. W. MichorJ. 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.

[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.

Figure 1.  Illustration of the distance between two curves $c_0$ and $c_1$ in the space of curves $\mathcal{M}$
Figure 2.  Geodesic shooting in the space of curves $\mathcal M$
Figure 3.  Steps of the first iteration of the geodesic shooting algorithm applied to a pair of geodesics of the upper half-plane $\mathbb H$
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
Figure 5.  Geodesics of the hyperbolic half-plane
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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