Journal of Mathematical Imaging and Vision

, Volume 33, Issue 2, pp 222–238 | Cite as

A Fast and Log-Euclidean Polyaffine Framework for Locally Linear Registration

  • Vincent Arsigny
  • Olivier Commowick
  • Nicholas Ayache
  • Xavier PennecEmail author


In this article, we focus on the parameterization of non-rigid geometrical deformations with a small number of flexible degrees of freedom. In previous work, we proposed a general framework called polyaffine to parameterize deformations with a finite number of rigid or affine components, while guaranteeing the invertibility of global deformations. However, this framework lacks some important properties: the inverse of a polyaffine transformation is not polyaffine in general, and the polyaffine fusion of affine components is not invariant with respect to a change of coordinate system. We present here a novel general framework, called Log-Euclidean polyaffine, which overcomes these defects.

We also detail a simple algorithm, the Fast Polyaffine Transform, which allows to compute very efficiently Log-Euclidean polyaffine transformations and their inverses on regular grids. The results presented here on real 3D locally affine registration suggest that our novel framework provides a general and efficient way of fusing local rigid or affine deformations into a global invertible transformation without introducing artifacts, independently of the way local deformations are first estimated.


Locally affine transformations Medical imaging ODE Diffeomorphisms Polyaffine transformations Log-Euclidean Non-rigid registration 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arsigny, V.: Processing data in lie groups: an algebraic approach. Application to non-linear registration and diffusion tensor MRI. Thèse de sciences (PhD Thesis), École polytechnique, November (2006) Google Scholar
  2. 2.
    Arsigny, V., Pennec, X., Ayache, N.: Polyrigid and polyaffine transformations: a novel geometrical tool to deal with non-rigid deformations—application to the registration of histological slices. Med. Image Anal. 9(6), 507–523 (2005) CrossRefGoogle Scholar
  3. 3.
    Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56(2), 411–421 (2006) CrossRefGoogle Scholar
  4. 4.
    Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. (2007, in press) Google Scholar
  5. 5.
    Commowick, O., Arsigny, V., Costa, J., Ayache, N., Malandain, G.: An efficient locally affine framework for the registration of anatomical structures. In: Proceedings of the Third IEEE International Symposium on Biomedical Imaging (ISBI 2006), pp. 478–481. Crystal Gateway Marriott, Arlington, Virginia, USA, April (2006) Google Scholar
  6. 6.
    Commowick, O., Arsigny, V., Costa, J., Ayache, N., Malandain, G.: An efficient locally affine framework for the registration of anatomical structures. Med. Image Anal. 12(4), 427–441 (2008) CrossRefGoogle Scholar
  7. 7.
    Cuzol, A., Hellier, P., Mémin, E.: A novel parametric method for non-rigid image registration. In: Christensen, G., Sonka, M. (eds.) Proc. of IPMI’05, Glenwood, Colorado, USA, July 2005. LNCS, vol. 3565, pp. 456–467. Springer, Berlin (2005) Google Scholar
  8. 8.
    Gallier, J.: Logarithms and square roots of real matrices (2008). arXiv:0805.0245v1
  9. 9.
    Higham, N.J.: The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26(4), 1179–1193 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hun Cheng, S., Higham, N.J., Kenney, C.S., Laub, A.J.: Approximating the logarithm of a matrix to specified accuracy. SIAM J. Matrix Anal. Appl. 22(4), 1112–1125 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lambert, J.D.: Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. Wiley, New York (1991) zbMATHGoogle Scholar
  12. 12.
    Little, J.A., Hill, D.L.G., Hawkes, D.J.: Deformations incorpotationg rigid structures. Comput. Vis. Imaging Underst. 66(2), 223–232 (1996) CrossRefGoogle Scholar
  13. 13.
    Maintz, J.B.A., Viergever, M.A.: A survey of medical registration. Med. Image Anal. 2(1), 1–36 (1998) CrossRefGoogle Scholar
  14. 14.
    Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix. SIAM J. Matrix Anal. Appl. 20(4), 801–836 (1978) zbMATHGoogle Scholar
  15. 15.
    Narayanan, R., Fessler, J.A., Park, H., Meyer, C.R.: Diffeomorphic nonlinear transformations: A local parametric approach for image registration. In: Proceedings of IPMI’05. LNCS, vol. 3565, pp. 174–185. Springer, Berlin (2005) Google Scholar
  16. 16.
    Papademetris, X., Dione, D.P., Dobrucki, L.W., Staib, L.H., Sinusas, A.J.: Articulated rigid registration for serial lower-limb mouse imaging. In: Proc. of MICCAI’05 (Part 2). LNCS, vol. 3750, pp. 919–926. Springer, Berlin (2005) Google Scholar
  17. 17.
    Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66(1), 41–66 (2006). A preliminary version appeared as INRIA Research Report 5255, July 2004 CrossRefMathSciNetGoogle Scholar
  18. 18.
    Pitiot, A., Bardinet, E., Thompson, P.M., Malandain, G.: Piecewise affine registration of biological images for volume reconstruction. Med. Image Anal. 10(3), 465–483 (2006) CrossRefGoogle Scholar
  19. 19.
    Rueckert, D., Sonoda, L.I., Hayes, C., Hill, D.L.G., Leach, M.O., Hawkes, D.J.: Non-rigid registration using free-form deformations: Application to breast MR images. IEEE Trans. Med. Imaging 18(8), 712–721 (1999) CrossRefGoogle Scholar
  20. 20.
    Sheppard, D.: A two-dimensionnal interpolation function for irregularly spaced data. In: 23rd National Conference of the ACM, pp. 517–524 (1968) Google Scholar
  21. 21.
    Stefanescu, R., Pennec, X., Ayache, N.: Grid powered nonlinear image registration with locally adaptive regularization. Med. Image Anal. 8(3), 325–342 (2004) CrossRefGoogle Scholar
  22. 22.
    Sternberg, S.: Lectures on Differential Geometry. Prentice Hall Mathematics Series. Prentice Hall, New York (1964) zbMATHGoogle Scholar
  23. 23.
    Tenenbaum, M., Pollard, H.: Ordinary Differential Equations. Dover, New York (1985) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Vincent Arsigny
    • 1
  • Olivier Commowick
    • 1
  • Nicholas Ayache
    • 1
  • Xavier Pennec
    • 1
    Email author
  1. 1.Asclepios Project-TeamINRIA Sophia-Antipolis MediterraneeSophia Antipolis CedexFrance

Personalised recommendations