Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Shape splines and stochastic shape evolutions: A second order point of view

Authors: Alain Trouvé and François-Xavier Vialard
Journal: Quart. Appl. Math. 70 (2012), 219-251
MSC (2010): Primary 65D07, 62J02, 37K65, 34F05, 93E14
DOI: https://doi.org/10.1090/S0033-569X-2012-01250-4
Published electronically: February 3, 2012
MathSciNet review: 2953101
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This article presents a new mathematical framework to perform statistical analysis on time-indexed sequences of 2D or 3D shapes. At the core of this statistical analysis is the task of time interpolation of such data. Current models in use can be compared to linear interpolation for one-dimensional data. We develop a spline interpolation method which is directly related to cubic splines on a Riemannian manifold. Our strategy consists of introducing a control variable on the Hamiltonian equations of the geodesics. Motivated by statistical modeling of spatiotemporal data, we also design a stochastic model to deal with random shape evolutions. This model is closely related to the spline model since the control variable previously introduced is set as a random force perturbing the evolution.

Although we focus on the finite-dimensional case of landmarks, our models can be extended to infinite-dimensional shape spaces, and they provide a first step for a nonparametric growth model for shapes taking advantage of the widely developed framework of large deformations by diffeomorphisms.

References [Enhancements On Off] (What's this?)

  • 1. A. A. Agrachev and Y. L. Sachkov.
    Control theory from the geometric viewpoint, volume 87 of Encyclopaedia of Mathematical Sciences.
    Springer-Verlag, Berlin, 2004,
    Control Theory and Optimization, II. MR 2062547 (2005b:93002)
  • 2. J. Ahlberg, E. Nilson, and J. Walsh.
    The theory of splines and their applications.
    Mathematics in Science and Engineering, 38, 1967. MR 0239327 (39:684)
  • 3. S. Allassonnière, Y. Amit, and A. Trouvé.
    Towards a coherent statistical framework for dense deformable template estimation.
    J. R. Statist. Soc. B, 69(1):3-29, 2007. MR 2301497 (2008m:62105)
  • 4. S. Allassonnière, A. Trouvé, and L. Younes.
    Geodesic shooting and diffeomorphic matching via textured meshes.
    In EMMCVPR05, pages 365-381, 2005.
  • 5. M. Camarinha, F. S. Leite, and P. Crouch.
    Splines of class $ \mathcal {C}^k$ on non-Euclidean spaces.
    IMA Journal of Mathematical Control & Information, 12:399-410, 1995. MR 1363321 (96i:41004)
  • 6. P. Crouch and F. S. Leite.
    The dynamic interpolation problem: On Riemannian manifolds, Lie groups and symmetric spaces.
    Journal of Dynamical & Control Systems, 1:177-202, 1995. MR 1333770 (96e:58048)
  • 7. B. Davis, P. Fletcher, E. Bullitt, and S. Joshi.
    Population shape regression from random design data.
    In Computer Vision, 2007. ICCV 2007. IEEE 11th International Conference, pages 1-7, Oct. 2007.
  • 8. I. L. Dryden and K. V. Mardia.
    Statistical Shape Analysis.
    Wiley Series in Probability and Statistics, 1998. MR 1646114 (2000b:60022)
  • 9. S. Durrleman, X. Pennec, G. Gerig, A. Trouvé, and N. Ayache.
    Spatiotemporal atlas estimation for developmental delay detection in longitudinal datasets.
    In Medical Image Computing and Computer Assisted Intervention, September 2009.
  • 10. Edwin T.
    Jaynes. Information theory and statistical mechanics.
    Physical Review, 106(4):620-630, 1957. MR 0087305 (19:335b)
  • 11. R. Giambo and F. Giannoni.
    An analytical theory for Riemannian cubic polynomials.
    IMA Journal of Mathematical Control & Information, 19:445-460, 2002. MR 1949013 (2003k:58015)
  • 12. J. Glaunes, A. Trouvé, and L. Younes.
    Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching.
    In Computer Vision and Pattern Recognition, volume 2, 2004.
  • 13. J. Glaunès, A. Trouvé, and L. Younes.
    Modeling planar shape variation via Hamiltonian flows of curves.
    In H. Krim and A. Yezzi, editors, Statistics and Analysis of Shapes, pages 335-361. Springer Birkhäuser, 2006. MR 2274202 (2007i:53073)
  • 14. U. Grenander, A. Srivastava, and S. Saini.
    Characterization of biological growth using iterated diffeomorphisms.
    In ISBI, pages 1136-1139, 2006.
  • 15. U. Grenander, A. Srivastava, and S. Saini.
    A pattern-theoretic characterization of biological growth.
    IEEE Trans. Med. Imaging, 26(5):648-659, 2007.
  • 16. D. R. Holm, A. Trouvé, and L. Younes.
    The Euler-Poincaré theory of metamorphosis.
    Quarterly of Applied Mathematics, 67(4):661-685, 2009. MR 2588229 (2010m:58031)
  • 17. R. V. Iyer, R. Holsapple, and D. Doman.
    Optimal control problems on parallelizable Riemannian manifolds: Theory and applications.
    ESAIM. COCV, 12:1-11, 2006. MR 2192065 (2006h:49046)
  • 18. J. Jackson.
    Dynamic interpolation and application to flight control.
    Ph.D. thesis, Arizona State University, 1990. MR 2685853
  • 19. S. Joshi and M. Miller.
    Landmark matching via large deformation diffeomorphisms.
    International Journal of Computer Vision, 2000. MR 1808275 (2001k:37138)
  • 20. J. Kapur.
    Maximum-entropy models in science and engineering.
    Wiley-Interscience, 1989. MR 1079544 (92b:00017)
  • 21. D. G. Kendall.
    The diffusion of shape.
    Advances in Applied Probability, vol. 9:pp. 428-430, 1977.
  • 22. A. Khan and M. Beg.
    Representation of time-varying shapes in the large deformation diffeomorphic framework.
    In Biomedical Imaging: From Nano to Macro, 2008. ISBI 2008. 5th IEEE International Symposium, pages 1521-1524, May 2008.
  • 23. H. Kunita.
    Stochastic flows and Stochastic Differential Equations.
    Cambridge Studies in Advanced Mathematics, 1997. MR 1472487 (98e:60096)
  • 24. J. Macki and A. Strauss.
    Introduction to optimal control theory.
    Springer, 1982. MR 638591 (84d:49001)
  • 25. J. Marsden, T. Ratiu, and Maisser.
    Introduction to mechanics and symmetry.
    Springer, 1999. MR 1723696 (2000i:70002)
  • 26. 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, 23:74, 2007. MR 2333829 (2008g:37072)
  • 27. M. I. Miller, A. Trouvé, and L. Younes.
    On the metrics and Euler-Lagrange equations of computational anatomy.
    Annual Review of Biomedical Engineering, 4:375-405, 2002.
  • 28. M. I. Miller, A. Trouvé, and L. Younes.
    Geodesic shooting for computational anatomy.
    J. Math. Imaging Vis., 24(2):209-228, 2006. MR 2227097 (2007e:68064)
  • 29. L. Noakes, G. Heinzinger, and B. Paden.
    Cubic splines on curved spaces.
    IMA Journal of Mathematical Control & Information, 6:465-473, 1989. MR 1036158 (91k:58026)
  • 30. N. Portman, U. Grenander, and E. R. Vrscay.
    Direct estimation of biological growth properties from image data using the ``grid'' model.
    In ICIAR, pages 832-843, 2009.
  • 31. I. Schoenberg.
    Contributions to the problem of approximation of equidistant data by analytic functions.
    Quart. Appl. Math. 4, 45-99 (Part A), 112-141 (Part B), 1946.
  • 32. A. Srivastava, S. Saini, Z. Ding, and U. Grenander.
    Maximum-likelihood estimation of biological growth variables.
    pages 107-118, 2005.
  • 33. A. Trouvé and L. Younes.
    Local geometry of deformable templates.
    Siam Journal of Mathematical Analysis, 2005. MR 2176922 (2006g:58010)
  • 34. F.-X. Vialard.
    Hamiltonian Approach to Shape Spaces in a Diffeomorphic Framework: From the Discontinuous Image Matching Problem to a Stochastic Growth Model.
    Ph.D. thesis, ENS Cachan, 2009.
  • 35. L. Younes, F. Arrate, and M. I. Miller.
    Evolution equations in computational anatomy.
    NeuroImage, 45(1, Supplement 1):S40 - S50, 2009.
    Mathematics in Brain Imaging.

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 65D07, 62J02, 37K65, 34F05, 93E14

Retrieve articles in all journals with MSC (2010): 65D07, 62J02, 37K65, 34F05, 93E14

Additional Information

Alain Trouvé
Affiliation: CMLA, Ecole Normale Supérieure de Cachan, CNRS, UniverSud, 61, Avenue du Président Wilson, F-94 235 Cachan Cedex, France
Email: alain.trouve@cmla.ens-cachan.fr

François-Xavier Vialard
Affiliation: Institute for Mathematical Science, Imperial College London, 53 Prince’s Gate, SW7 2PG, London, United Kingdom
Email: francois.xavier.vialard@normalesup.org

DOI: https://doi.org/10.1090/S0033-569X-2012-01250-4
Received by editor(s): March 19, 2010
Published electronically: February 3, 2012
Article copyright: © Copyright 2012 Brown University

American Mathematical Society