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

Quarterly of Applied Mathematics

Volumn 70, Issue 2, 2012, Pages 219-251

  Trouvé, Alain ^a   Vialard, François Xavier ^b  

^a UNIVERSITÉ PARIS SACLAY   (France)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

3-D SHAPE; CONTROL VARIABLE; CUBIC SPLINE; CURRENT MODELS; DIFFEOMORPHISMS; GROWTH MODELS; HAMILTONIAN EQUATIONS; LARGE DEFORMATIONS; LINEAR INTERPOLATION; MATHEMATICAL FRAMEWORKS; NON-PARAMETRIC; RIEMANNIAN MANIFOLD; SECOND ORDERS; SHAPE EVOLUTION; SHAPE SPACE; SPATIO-TEMPORAL DATA; SPLINE INTERPOLATION; STATISTICAL MODELING; TIME INTERPOLATION;

GEOMETRY; STATISTICAL METHODS;

INTERPOLATION;

EID: 84864925886     PISSN: 0033569X     EISSN: 15524485     Source Type: Journal    
DOI: 10.1090/S0033-569X-2012-01250-4     Document Type: Article

References (35)

1. A. A. Agrachev and Y. L. Sachkov. Control theory from the geometric viewpoint,volume87of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2004, Control Theory and Opti- mization, II.
2. J. Ahlberg, E. Nilson, and J. Walsh. The theory of splines and their applications. Mathematics in Science and Engineering, 38, 1967.
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.
4. S. Allassonnière, A. Trouvé, and L. Younes. Geodesic shooting and diffeomorphic matching via textured meshes. In EMMCVPR05, pages 365-381, 2005.
5. k on non-Euclidean spaces. IMA Journal of Mathematical Control & Information, 12:399-410, 1995.
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.
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.
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.
11. R. Giambo and F. Giannoni. An analytical theory for Riemannian cubic polynomials. IMA Journal of Mathematical Control & Information, 19:445-460, 2002.
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.
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.
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.
18. J. Jackson. Dynamic interpolation and application to flight control. Ph.D. thesis, Arizona State University, 1990.
19. S. Joshi and M. Miller. Landmark matching via large deformation diffeomorphisms. International Journal of Computer Vision, 2000.
20. J. Kapur. Maximum-entropy models in science and engineering. Wiley-Interscience, 1989.
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.
24. J. Macki and A. Strauss. Introduction to optimal control theory. Springer, 1982.
25. J. Marsden, T. Ratiu, and Maisser. Introduction to mechanics and symmetry. Springer, 1999.
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.
27. M. I. Miller, A. Trouvé, and L. Younes. On the metrics and Euler-Lagrange equations of computa- tional 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.
29. L. Noakes, G. Heinzinger, and B. Paden. Cubic splines on curved spaces. IMA Journal of Mathe- matical Control & Information, 6:465-473, 1989.
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 func- tions. 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.
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, ENSCachan, 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.