International Journal of Computer Vision

, Volume 105, Issue 2, pp 128–143 | Cite as

Flexible Shape Matching with Finite Element Based LDDMM

Article

Abstract

The Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework acting on currents is a conceptually powerful tool for matching highly varying shapes. In the classical approach, the numerical treatment is based on currents representing individual particles, and couples the discretization of shape and deformation. This design restricts the capabilities of LDDMM. In this work, we propose to decouple current and deformation discretization by using conforming adaptive finite elements. We show how to efficiently (a) compute the temporal evolution of discrete \(m\)-current attributes for any \(m\), and (b) incorporate multiple scales into the matching process. This effectively leads to more flexibility, which is demonstrated in several numerical experiments on anatomical shapes.

Keywords

Large deformation Diffeomorphic registration Matching Currents Adaptive finite elements 

Mathematics Subject Classification (2000)

58A25 37E30 58J72 49J20 65N30 

References

  1. Adams, R. (1975). Sobolev spaces. New York: Academic Press.MATHGoogle Scholar
  2. Beg, M. F., Miller, M. I., Trouvé, A., & Younes, L. (2005). Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal of Computer Vision, 61(2), 139–157. doi:10.1023/B:VISI.0000043755.93987.aa.CrossRefGoogle Scholar
  3. Burger, M., Ruthotto, L., & Modersitzki, J. (2011). A hyperelastic regularization energy for image registration. Münster: WWU Münster.Google Scholar
  4. Camion, V., & Younes, L. (2001). Geodesic interpolating splines energy minimization methods in computer vision and pattern recognition. Berlin: Springer.Google Scholar
  5. Cao, Y., Miller, M. I., Winslow, R. L., & Younes, L. (2005). Large deformation diffeomorphic metric mapping of vector fields. IEEE Transactions on Medical Imaging, 24(9), 1216–1230.CrossRefGoogle Scholar
  6. Ciarlet, P. (1987). The finite element method for elliptic problems. Amsterdam: North-Holland.Google Scholar
  7. Cotter, C. (2008). The variational particle-mesh method for matching curves. Journal of Physics A: Mathematical and Theoretical, 41(34): 344003.Google Scholar
  8. Deuflhard, P., & Weiser, M. (2012). Adaptive numerical solution of PDEs. Berlin: de Gruyter.MATHCrossRefGoogle Scholar
  9. Dupuis, P., Grenander, U., & Miller, M. I. (1998). Variational problems on flows of diffeomorphisms for image matching. Quarterly of Applied Mathematics, 56(3), 587–600.Google Scholar
  10. Durrleman, S. (2010). Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution. Thèse de sciences (phd thesis), Université de Nice-Sophia Antipolis, Cedex.Google Scholar
  11. Durrleman, S., Pennec, X., Trouvé, A., & Ayache, N. (2009). Statistical models of sets of curves and surfaces based on currents. Medical Image Analysis, 13(5), 793–808. doi:10.1016/j.media.2009.07.007; includes special section on the 12th international conference on medical imaging and computer assisted intervention.
  12. Federer, H. (1969). Geometric measure theory. Report of the 1969. Berlin: Springer-Verlag.Google Scholar
  13. Glaunès, J. (2005). Transport par difféomorphismes de points, de mesures et de courants pour la comparaison de formes et l’anatomie numérique. PhD thesis, Université Paris, Paris.Google Scholar
  14. Glaunès. J., Trouvé A., Younes, L. (2004). Diffeomorphic matching of distributions: a new approach for unlabelled point-sets and sub-manifolds matching. In Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Vol. 2 (pp. 712–718). Washington, DC: CVPR. doi:10.1109/CVPR.2004.1315234.
  15. Glaunès, J., Qiu, A., Miller, M. I., & Younes, L. (2008). Large deformation diffeomorphic metric curve mapping. International Journal of Computer Vision, 80(3), 317–336. doi:10.1007/s11263-008-0141-9.CrossRefGoogle Scholar
  16. Günther, A., Lamecker H., Weiser, M., (2011). Direct lddmm of discrete currents with adaptive finite elements. In X. Pennec, S. Joshi, & M. Nielsen (Eds.), Proceedings of 3rd MICCAI Workshop on Mathematical Foundations of Computational Anatomy (pp. 1–15). Beijing: MICCAI .Google Scholar
  17. Haber, E., Heldmann, S., & Modersitzki, J. (2008). Adaptive mesh refinement for nonparametric image registration. SIAM Journal on Scientific Computing, 30(6), 3012–3027. doi:10.1137/070687724.Google Scholar
  18. Joshi, S. C., & Miller, M. I. (2000). Landmark matching via large deformation diffeomorphisms. IEEE Transactions on Image Processing, 9(8), 1357–1370. doi:10.1109/83.855431.MathSciNetMATHCrossRefGoogle Scholar
  19. Kirk, B. S., Peterson, J. W., Stogner, R. H., & Carey, G. F. (2006). libMesh: A C++ library for parallel adaptive mesh refinement/coarsening simulation. Engineering with Computers, 22(3–4), 237–254.Google Scholar
  20. Marsland, S., & Twining, C. (2004). Constructing diffeomorphic representations for the groupwise analysis of non-rigid registrations of medical images. IEEE Transactions on Medical Imaging, 23(8), 1006–1020.CrossRefGoogle Scholar
  21. Mattheij, R. M., & Molenaar, J. (2002). Ordinary differential equations in theory and practice. Reprint of (1996) original. In Classics in applied mathematics. Philadelphia: SIAM. Google Scholar
  22. Morgan, F. (2009). Geometric measure theory: A beginner’s guide (4th ed.). New York: Elsevier.Google Scholar
  23. Morita, S. (2001). Geometry of differential forms. New York: American Mathematical Society.MATHGoogle Scholar
  24. Risser, L., Vialard, F. X., Wolz, R., Holm, D. D., & Rueckert, D. (2010). Simultaneous fine and coarse diffeomorphic registration: Application to atrophy measurement in Alzheimer’s disease. In T. Jiang, N. Navab, J. Pluim, M. Viergever (Eds.), Medical image computing and computer-assisted intervention. Lecture notes in computer science, Vol. 6362 (pp. 610–617). Berlin/Heidelberg: Springer. doi:10.1007/978-3-642-15745-5_75.
  25. Sommer, S. H., Nielsen, M., Lauze, F. B., & Pennec, X. (2011). A multi-scale kernel bundle for LDDMM: Towards sparse deformation description across space and scales. In G. Szkely & H. Hahn (Eds.), Information processing in medical imaging. Lecture notes in computer science, Vol. 6801 (pp. 624–635). Berlin/Heidelberg: Springer. doi:10.1007/978-3-642-22092-0_51.
  26. Trouvé, A. (1995). An infinite dimensional group approach for physics based models in pattern recognition. Technical report. Baltimore: Johns Hopkins University.Google Scholar
  27. Vaillant, M., & Glaunès, J. (2005). Surface matching via currents. In G. Christensen & M. Sonka (Eds.), Information processing in medical imaging. Lecture notes in computer science, Vol. 3565 (pp. 1–5). Berlin/Heidelberg: Springer. doi:10.1007/11505730_32.
  28. Younes, L. (2010). Shapes and diffeomorphisms. In Applied mathematical sciences, Vol. 171. Berlin: Springer. doi:10.1007/978-3-642-12055-8.

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Andreas Günther
    • 1
  • Hans Lamecker
    • 1
  • Martin Weiser
    • 1
  1. 1.Zuse InstituteBerlinGermany

Personalised recommendations