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Preface
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 |
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 [
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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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 |






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