American Institute of Mathematical Sciences

Advanced
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.

[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.

[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. 

[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.

[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.

[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.

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.

[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. 

[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.

[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.

[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.

Figure 1.  Illustration of the distance between two curves $c_0$ and $c_1$ in the space of curves $\mathcal{M}$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Geodesic shooting in the space of curves $\mathcal M$
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Geodesics of the hyperbolic half-plane
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Ronny Bergmann, Raymond H. Chan, Ralf Hielscher, Johannes Persch, Gabriele Steidl. Restoration of manifold-valued images by half-quadratic minimization. Inverse Problems and Imaging, 2016, 10 (2) : 281-304. doi: 10.3934/ipi.2016001
[2]

Vincenzo Ambrosio, Giovanni Molica Bisci, Dušan Repovš. Nonlinear equations involving the square root of the Laplacian. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 151-170. doi: 10.3934/dcdss.2019011
[3]

Partha Sharathi Dutta, Soumitro Banerjee. Period increment cascades in a discontinuous map with square-root singularity. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 961-976. doi: 10.3934/dcdsb.2010.14.961
[4]

Theresa Lange, Wilhelm Stannat. Mean field limit of Ensemble Square Root filters - discrete and continuous time. Foundations of Data Science, 2021, 3 (3) : 563-588. doi: 10.3934/fods.2021003
[5]

Alice B. Tumpach, Stephen C. Preston. Quotient elastic metrics on the manifold of arc-length parameterized plane curves. Journal of Geometric Mechanics, 2017, 9 (2) : 227-256. doi: 10.3934/jgm.2017010
[6]

S. Maier-Paape, Ulrich Miller. Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1137-1153. doi: 10.3934/dcds.2006.15.1137
[7]

Tyrus Berry, Timothy Sauer. Consistent manifold representation for topological data analysis. Foundations of Data Science, 2019, 1 (1) : 1-38. doi: 10.3934/fods.2019001
[8]

Elena Celledoni, Markus Eslitzbichler, Alexander Schmeding. Shape analysis on Lie groups with applications in computer animation. Journal of Geometric Mechanics, 2016, 8 (3) : 273-304. doi: 10.3934/jgm.2016008
[9]

Alex Wright. Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces. Journal of Modern Dynamics, 2012, 6 (3) : 405-426. doi: 10.3934/jmd.2012.6.405
[10]

Derek H. Justice, H. Joel Trussell, Mette S. Olufsen. Analysis of Blood Flow Velocity and Pressure Signals using the Multipulse Method. Mathematical Biosciences & Engineering, 2006, 3 (2) : 419-440. doi: 10.3934/mbe.2006.3.419
[11]

Martin Bauer, Markus Eslitzbichler, Markus Grasmair. Landmark-guided elastic shape analysis of human character motions. Inverse Problems and Imaging, 2017, 11 (4) : 601-621. doi: 10.3934/ipi.2017028
[12]

Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453
[13]

Yu-Ting Lin, John Malik, Hau-Tieng Wu. Wave-shape oscillatory model for nonstationary periodic time series analysis. Foundations of Data Science, 2021, 3 (2) : 99-131. doi: 10.3934/fods.2021009
[14]

Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations and Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022
[15]

Atul Kumar, R. R. Yadav. Analytical approach of one-dimensional solute transport through inhomogeneous semi-infinite porous domain for unsteady flow: Dispersion being proportional to square of velocity. Conference Publications, 2013, 2013 (special) : 457-466. doi: 10.3934/proc.2013.2013.457
[16]

Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157
[17]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
[18]

Chiara Corsato, Colette De Coster, Franco Obersnel, Pierpaolo Omari, Alessandro Soranzo. A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 213-256. doi: 10.3934/dcdss.2018013
[19]

Shi Jin, Yingda Li. Local sensitivity analysis and spectral convergence of the stochastic Galerkin method for discrete-velocity Boltzmann equations with multi-scales and random inputs. Kinetic and Related Models, 2019, 12 (5) : 969-993. doi: 10.3934/krm.2019037
[20]

Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial and Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (218)
  • HTML views (76)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy