Jacobi fields in groups of diffeomorphisms and applications

Younes, Laurent

doi:10.1090/S0033-569X-07-01027-5

Author: Laurent Younes
Journal: Quart. Appl. Math. 65 (2007), 113-134
MSC (2000): Primary 58E50; Secondary 53C22
DOI: https://doi.org/10.1090/S0033-569X-07-01027-5
Published electronically: February 15, 2007
MathSciNet review: 2313151
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper presents a series of applications of the Jacobi evolution equations along geodesics in groups of diffeomorphisms. We describe, in particular, how they can be used to perform implementable gradient descent algorithms for image matching, in several situations, and illustrate this with 2D and 3D experiments. We also discuss parallel translation in the group, and its projection on shape manifolds, and focus in particular on an implementation of these equations using iterated Jacobi fields.

References

V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391
V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295

V. I. Arnold, Sur un principe variationnel pour les ecoulements stationnaires des liquides parfaits et ses applications aux problèmes de stabilité non linéaires, J. Mécanique 5 (1966), 29–43.

B. Avants and J. Gee, Shape averaging with diffeomorphic flows for atlas creation, Proceedings of ISBI 2004, 2004.

R. Bajcsy and C. Broit, Matching of deformed images, The 6th international conference in pattern recognition, 1982, pp. 351–353.

M. F. Beg, M. I. Miller, A. Trouvé, and L. Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms, Int. J. Comp. Vis. 61 (2005), no. 2, 139–157.

Roberto Camassa and Darryl D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), no. 11, 1661–1664. MR 1234453, DOI https://doi.org/10.1103/PhysRevLett.71.1661

Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207

E. G. Christensen, D. R. Rabbitt, and I. M. Miller, Deformable templates using large deformation kinematics, IEEE Trans. Image Proc. (1996).

Paul Dupuis, Ulf Grenander, and Michael I. Miller, Variational problems on flows of diffeomorphisms for image matching, Quart. Appl. Math. 56 (1998), no. 3, 587–600. MR 1632326, DOI https://doi.org/10.1090/qam/1632326

Ulf Grenander and Michael I. Miller, Computational anatomy: an emerging discipline, Quart. Appl. Math. 56 (1998), no. 4, 617–694. Current and future challenges in the applications of mathematics (Providence, RI, 1997). MR 1668732, DOI https://doi.org/10.1090/qam/1668732

Darryl D. Holm, Jerrold E. Marsden, and Tudor S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. 137 (1998), no. 1, 1–81. MR 1627802, DOI https://doi.org/10.1006/aima.1998.1721

D. D. Holm, T. J. Ratnanather, A. Trouvé, and L. Younes, Soliton dynamics in computational anatomy, Neuroimage 23 (2004), S170–S178.

S. Joshi, Large deformation diffeomorphisms and Gaussian random fields for statistical characterization of brain sub-manifolds, Ph.D. thesis, Sever Institute of Technology, Washington University, 1997.

Jerrold E. Marsden and Tudor S. Ratiu, Introduction to mechanics and symmetry, 2nd ed., Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1999. A basic exposition of classical mechanical systems. MR 1723696

P. Michor, Some geometric equations arising as geodesic equations on groups of diffeomorphisms and spaces of plane curves including the hamiltonian approach, Lecture Course, University of Wien.

I. M. Miller, A. Trouvé, and L. Younes, On the metrics and Euler-Lagrange equations of computational anatomy, Annual Review of Biomedical Engineering 4 (2002), 375–405.

---, Geodesic shooting for computational anatomy, J. Math. Image and Vision (2005).

I. M. Miller and L. Younes, Group action, diffeomorphism and matching: a general framework, Int. J. Comp. Vis 41 (2001), 61–84 (Originally published in electronic form in: Proceeding of SCTV 99, http://www.cis.ohio-state.edu/ szhu/SCTV99.html).

Barrett O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469. MR 200865

William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, Numerical recipes in C, 2nd ed., Cambridge University Press, Cambridge, 1992. The art of scientific computing. MR 1201159

J.-P. Thirion, Image matching as a diffusion process: An analogy with Maxwell’s demons, Medical Image Analysis 2 (1998), no. 3, 243–260.

---, Diffusing models and applications, Brain Warping (A. W. Toga, ed.), 1999, pp. 144–155.

Alain Trouvé and Laurent Younes, Local geometry of deformable templates, SIAM J. Math. Anal. 37 (2005), no. 1, 17–59. MR 2176922, DOI https://doi.org/10.1137/S0036141002404838

Alain Trouvé and Laurent Younes, Metamorphoses through Lie group action, Found. Comput. Math. 5 (2005), no. 2, 173–198. MR 2149415, DOI https://doi.org/10.1007/s10208-004-0128-z

Alain Trouvé, Action de groupe de dimension infinie et reconnaissance de formes, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 8, 1031–1034 (French, with English and French summaries). MR 1360567

---, Diffeomorphism groups and pattern matching in image analysis, Int. J. of Comp. Vis. 28 (1998), no. 3, 213–221.