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 [
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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
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.
![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() |
|
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.![]() ![]() |
|
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.![]() ![]() ![]() |
|
F. Barbaresco
, Koszul information geometry and souriau geometric temperature/capacity of lie group thermodynamics, Entropy, 16 (2014)
, 4521-4565.
doi: 10.3390/e16084521.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
J. P. Burg,
Maximum Entropy Spectral Analysis, Dissertation, Stanford University, 1975.
![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
M. P. do Carmo,
Riemannian Geometry, 1st Edition, Birkhauser, 1992.
doi: 10.1007/978-1-4757-2201-7.![]() ![]() ![]() |
|
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.
![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() |
|
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.![]() ![]() |
|
A. Le Brigant, A discrete framework to find the optimal matching between manifold-valued curves, preprint, arXiv: 1703.05107.
![]() |
|
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.![]() ![]() ![]() |
|
P. W. Michor,
Manifolds of Differentiable Mappings, in vol. 3 of Shiva Mathematics Series (Shiva Publ.), Orpington, 1980.
![]() ![]() |
|
P. W. Michor
and D. Mumford
, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Documenta Mathematica, 10 (2005)
, 217-245.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
M. Pilté and F. Barbaresco, Tracking quality monitoring based on information geometry and geodesic shooting, 17th International Radar Symposium, (2016), 1-6.
![]() |
|
S. Sasaki
, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Mathematical Journal, 10 (1958)
, 338-354.
doi: 10.2748/tmj/1178244668.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() |
|
L. Younes
, Computable elastic distances between shapes, SIAM Journal on Applied Mathematics, 58 (1998)
, 565-586.
doi: 10.1137/S0036139995287685.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() |
Illustration of the distance between two curves
Geodesic shooting in the space of curves
Steps of the first iteration of the geodesic shooting algorithm applied to a pair of geodesics of the upper half-plane
Optimal deformations between pairs of geodesics (in black) of the upper half-plane
Geodesics of the hyperbolic half-plane
Computation of the mean curve (in black) for 4 sets of 11 curves in the hyperbolic half-plane, constructed from simulated helicopter radar data