A Log-Euclidean Framework for Statistics on Diffeomorphisms

  • Vincent Arsigny
  • Olivier Commowick
  • Xavier Pennec
  • Nicholas Ayache
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4190)

Abstract

In this article, we focus on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm. Remarkably, this logarithm is a simple 3D vector field, and is well-defined for diffeomorphisms close enough to the identity. This allows to perform vectorial statistics on diffeomorphisms, while preserving the invertibility constraint, contrary to Euclidean statistics on displacement fields. We also present here two efficient algorithms to compute logarithms of diffeomorphisms and exponentials of vector fields, whose accuracy is studied on synthetic data. Finally, we apply these tools to compute the mean of a set of diffeomorphisms, in the context of a registration experiment between an atlas an a database of 9 T1 MR images of the human brain.

References

  1. 1.
    Arsigny, V., Commowick, O., Pennec, X., Ayache, N.: A Log-Euclidean polyaffine framework for locally rigid or affine registration. In: Pluim, J.P.W., Likar, B., Gerritsen, F.A. (eds.) WBIR 2006. LNCS, vol. 4057, pp. 120–127. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Fast and simple calculus on tensors in the Log-Euclidean framework. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3749, pp. 115–122. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Beg, M.F., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. Jour. Comp. Vis. 61(2), 139–157 (2005)CrossRefGoogle Scholar
  4. 4.
    Camion, V., Younes, L.: Geodesic interpolating splines. In: Figueiredo, M., Zerubia, J., Jain, A.K. (eds.) EMMCVPR 2001. LNCS, vol. 2134, pp. 513–527. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    Chefd’hotel, C., Hermosillo, G., Faugeras, O.: Flows of diffeomorphisms for multimodal image registration. In: Proc. of ISBI (2002)Google Scholar
  6. 6.
    Commowick, O., Stefanescu, R., Fillard, P., Arsigny, V., Ayache, N., Pennec, X., Malandain, G.: Incorporating Statistical Measures of Anatomical Variability in Atlas-to-Subject Registration for Conformal Brain Radiotherapy. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3750, pp. 927–934. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Higham, N.J.: The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26(4), 1179–1193 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Marsland, S., Twining, C.J.: Constructing diffeomorphic representations for the groupwise analysis of nonrigid registrations of medical images. IEEE Trans. Med. Imaging 23(8), 1006–1020 (2004)CrossRefGoogle Scholar
  9. 9.
    Pennec, X., Stefanescu, R., Arsigny, V., Fillard, P., Ayache, N.: Riemannian Elasticity: A Statistical Regularization Framework for Non-linear Registration. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3750, pp. 943–950. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Rueckert, D., Frangi, A.F., Schnabel, J.A.: Automatic construction of 3D statistical deformation models of the brain using non-rigid registration. IEEE TMI 22(8), 1014–1025 (2003)Google Scholar
  11. 11.
    Stefanescu, R., Pennec, X., Ayache, N.: Grid powered nonlinear image registration with locally adaptive regularization. MedI. A 88(3), 325–342 (2004)Google Scholar
  12. 12.
    Sternberg, S.: Lectures on Differential Geometry. Prentice Hall Mathematics Series. Prentice Hall Inc., Englewood Cliffs (1964)MATHGoogle Scholar
  13. 13.
    Tenenbaum, M., Pollard, H.: Ordinary Differential Equations. Dover (1985)Google Scholar
  14. 14.
    Trouvé, A.: Diffeomorphisms groups and pattern matching in image analysis. International Journal of Computer Vision 28(3), 213–221 (1998)CrossRefGoogle Scholar
  15. 15.
    Vaillant, M., Miller, M.I., Younes, L., Trouvé, A.: Statistics on diffeomorphisms via tangent space representations. NeuroImage 23, S161–S169 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Vincent Arsigny
    • 1
  • Olivier Commowick
    • 1
    • 2
  • Xavier Pennec
    • 1
  • Nicholas Ayache
    • 1
  1. 1.INRIA Sophia – Epidaure ProjectSophia AntipolisFrance
  2. 2.DOSISoft S.A.CachanFrance

Personalised recommendations