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Journal of Mathematical Imaging and Vision

, Volume 25, Issue 3, pp 329–340 | Cite as

Geodesic Matching with Free Extremities

  • Laurent Garcin
  • Laurent Younes
Article

Abstract

In this paper, we describe how to use geodesic energies defined on various sets of objects to solve several distance related problems. We first present the theory of metamorphoses and the geodesic distances it induces on a Riemannian manifold, followed by classical applications in landmark and image matching. We then explain how to use the geodesic distance for new issues, which can be embedded in a general framework of matching with free extremities. This is illustrated by results on image and shape averaging and unlabeled landmark matching.

Keywords

geodesic distance image matching shape matching deformable templates Kärcher mean 

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References

  1. 1.
    N. Arad, N. Dyn, D. Reisfeld, and Y. Yeshurun, “Image warping by radial basis functions: Application to facial expressions,” CVGIP: Graphical Models and Image Processing, Vol. 56, No. 2, pp. 161–172, 1994.CrossRefGoogle Scholar
  2. 2.
    R. Bajcsy and C. Broit, “matching of deformed images,” in the 6th International Conference in Pattern Recognition, pp. 351–353, 1982.Google Scholar
  3. 3.
    M.F. Beg, M.I. Miller, A. Trouvé, and L. Younes, “Computing large deformation metrics mappings via geodesic flows of Diffeomorphisms,” International Journal of Computer Vision, Vol. 61, pp. 139–157, 2004.CrossRefGoogle Scholar
  4. 4.
    C. Bernard, “Fast optic flow computation with discrete wavelets,” Technical Report RI365, Centre de Mathématiques Appliquées, École Polytechnique, 1997.Google Scholar
  5. 5.
    C.P. Bernard, “Discrete wavelet analysis for fast optic flow computation,” Applied and Computational Harmonic Analysis, 11, No. 1, pp. 32–63, 2001.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    F.L. Bookstein, “Principal warps: Thin plate splines and the decomposition of deformations,” IEEE Transactions PAMI, Vol. 11, No. 6, pp. 567–585, 1989.MATHGoogle Scholar
  7. 7.
    J.M. Buhmann and T. Hoffman, “A maximum entropy approach to pairwise data clustering,” in Proceedings of the International Conference on Pattern Recognition, IEEE Computer Society Press, Vol. II, pp. 207–212, 1994.Google Scholar
  8. 8.
    G. Charpiat, O. Faugeras, and R. Keriven, “Approximations of shape metrics and application to shape warping and empirical shape statistics,” Technical Report 4820, INRIA, INRIA, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis, France, 2003.Google Scholar
  9. 9.
    G.E. Christensen, R.D. Rabbit, and M.I. Miller, “Deformable templates using large deformation kinematics,” IEEE Transactions on Image Processing, 1996.Google Scholar
  10. 10.
    H. Chui and A. Rangarajan, “Learning an atlas from unlabeled point-sets,” in IEEE Workshop on Mathematical Methods in Biomedical Image Analysis, IEEE Press, pp. 58–65, 2001.Google Scholar
  11. 11.
    H. Chui, L. Win, R. Schultz, J. Duncan, and A. Rangarajan, “Unsupervised learning of an atlas from unlabeled point sets,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 2004.Google Scholar
  12. 12.
    T.F. Cootes, C.J. Taylor, D.H. Cooper, and J. Graham, “Active shape models: Their training and application,” Computer Vision and Image Understanding, Vol. 61, No. 1, pp. 38–59, 1995.CrossRefGoogle Scholar
  13. 13.
    I. Daubechies, Ten Lectures on Wavelets, Philadelphia, PA: SIAM, 1992.MATHGoogle Scholar
  14. 14.
    I.L. Dryden and V.K. Mardia, Statistical Shape Analysis, John Wiley, 1998.Google Scholar
  15. 15.
    P. Dupuis and U. Grenander, “Variational problems on flows of diffeomorphisms for image matching,” 1998.Google Scholar
  16. 16.
    P.T. Fletcher, C. Lu, M. Pizer, and S. Joshi, “Principal geodesic analysis for the study of nonlinear statistics of shape,” IEEE Transactions Medical Imaging, Vol. 23, No. 8, pp. 995–1005, 2004.CrossRefGoogle Scholar
  17. 17.
    L. Garcin, “Techniques de mise en correspondance et détection de changements,” Ph.D. thesis, Ecole Normale Supérieure de Cachan, 2004.Google Scholar
  18. 18.
    L. Garcin, A. Rangarajan, and L. Younes, “Non rigid registration of shapes via diffeomorphic point matching and clustering,” Preprint, 2004.Google Scholar
  19. 19.
    J.C. Gee, R. Haynor, D.L. Le Briquer, and Z. Bajcsy, “Advances in elastic matching theory and its implementation,” in P. Cinquin, R. Kikinis, and D. Lavalée (eds.), CVRMed-MRCAS’97, Springer Verlag, 1997.Google Scholar
  20. 20.
    C.A. Glasbey and V. Mardia, “A review of image-warping methods,” Journal of Applied Statistics, Vol. 25, No. 2, pp. 155–171, 1998.MATHCrossRefGoogle Scholar
  21. 21.
    U. Grenander, General Pattern Theory, Oxford University Press, 1993.Google Scholar
  22. 22.
    H. Guo, A. Rangarajan, S. Joshi, and L. Younes, “Non-Rigid Registration of Shapes via Diffeomorphic Point-Matching,” in ISBI 04, 2004.Google Scholar
  23. 23.
    T. Hofmann and J.M. Buhmann, “Pairwise data clustering by deterministic annealing,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 19, No. 1, pp. 1–14, 1997.CrossRefGoogle Scholar
  24. 24.
    S. Joshi and M. Miller, “Landmark matching via large deformation diffeomorphisms,” IEEE Transactions in Image Processing, Vol. 9, No. 8, pp. 1357–1370, 2000.MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    H. Karcher, “Riemann center of mass and mollifier smoothing,” Communication Pure Applied Mathematics, Vol. 30, pp. 509–541, 1977.MATHMathSciNetGoogle Scholar
  26. 26.
    D.G. Kendall, “Shape manifolds, procrustean metrics and complex projective spaces,” Bulletin of the London Mathematical Society, Vol. 16, pp. 81–121, 1984.MATHMathSciNetGoogle Scholar
  27. 27.
    D.G. Kendall, D. Barden, T.K. Carne, and H. Le, Shape and Shape Theory, Wiley Series in Probability and Statistics: Wiley, 1999.Google Scholar
  28. 28.
    E. Klassen, A. Srivastava, M. Mio, and S. Joshi, “Analysis of planar shapes using geodesic paths on shape spaces,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 26, No. 3, pp. 372–383, 2004.CrossRefGoogle Scholar
  29. 29.
    H. Le, “Mean Size-and-shapes and mean shapes: A geometric point of view,” Advanced Applied Probabilities, Vol. 27, pp. 44–55, 1995.MATHCrossRefGoogle Scholar
  30. 30.
    S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation.,” IEEE Trans. Pat. Anal. Mach. Intell., Vol. 11, pp. 674–693, 1989.MATHCrossRefGoogle Scholar
  31. 31.
    S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1998.Google Scholar
  32. 32.
    M.I. Miller, A. Trouvé, and L. Younes, “On the metrics and Euler-Lagrange equations of computational anatomy,” Annual Review of biomedical Engineering, Vol. 4, pp. 375–405, 2002.CrossRefGoogle Scholar
  33. 33.
    M.I. Miller and L. Younes, “Group action, diffeomorphisms and matching: A general framework,” International Journal of Computer Vision, Vol. 41, pp. 61–84, 2001.MATHCrossRefGoogle Scholar
  34. 34.
    P. Olver, Equivalence, Invariants and Symmetry, Cambridge University Press, 1995.Google Scholar
  35. 35.
    R.D. Rabbit, J.A. Weiss, G.E. Christensen, and M.I. Miller, “Mapping of hyperelastic deformable templates using the finite element method,” in Proceeding of San Diego’s SPIE Conference, 1995.Google Scholar
  36. 36.
    C.G. Small, The Statistical Theory of Shape, Springer Series in Statistics: Springer, 1996.Google Scholar
  37. 37.
    J.P. Thirion, “Diffusing models and applications,” in: W. Toga, A (ed.), Brain Warping, pp. 144–155, 1999.Google Scholar
  38. 38.
    P.M. Thompson and A.W. Toga, “Detection, visualization and animation of abnormal anatomic structure with a deformable probabilistic brain atlas based on random vector field transformations,” Medical Image Analysis, Vol. 1, No. 4, pp. 271–294, 1996/7.CrossRefGoogle Scholar
  39. 39.
    A.W. Toga (ed.), Brain warping, Academic Press, 1999.Google Scholar
  40. 40.
    A. Trouvé, “Diffeomorphism groups and pattern matching in image analysis,” International Journal of Computer Vision, Vol. 28, No. 3, pp. 213–221, 1998.CrossRefGoogle Scholar
  41. 41.
    A. Trouvé and L. Younes, “Metamorphoses through lie group action,” Foundations of Computational Mathematics, 2004.Google Scholar
  42. 42.
    L. Wiskott and C. von der Malsburg, “A neural system for the recognition of partially occulted objects in cluttered scenes,” International Journal Pattern Recognition and Artificial Intelligence, Vol. 7, pp. 935–948, 1993.CrossRefGoogle Scholar
  43. 43.
    L. Younes and V. Camion, “Geodesic interpolating Splines,” In Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 513–527, Springer, 2001.Google Scholar
  44. 44.
    L. Younes, A. Trouvé, and J. Glaunès, “Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching,” in Proceedings of CVPR, Vol. 2, pp. 712–718, 2004.Google Scholar
  45. 45.
    A.L. Yuille, “Generalized deformable models, statistical physics, and matching problems,” Neural Computation, Vol. 2, No. 1, pp. 1–24, 1990.MathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Laurent Garcin
    • 1
  • Laurent Younes
    • 2
  1. 1.Laboratoire MATISInstitut Géographique NationalSaint Mande Cedex
  2. 2.Center for Imaging Science, Department of Appplied Mathematics and StatisticsJohns Hopkins UniversityBaltimore

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