A Log-Euclidean Polyaffine Framework for Locally Rigid or Affine Registration

  • Vincent Arsigny
  • Olivier Commowick
  • Xavier Pennec
  • Nicholas Ayache
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4057)


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 small 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 a regular grid. 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.


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  1. 1.
    Arsigny, V., Commowick, O., Pennec, X., Ayache, N.: A fast and Log-Euclidean polyaffine framework for locally affine registration. Research report RR-5865, INRIA (March 2006)Google 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.
    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. Im. Anal. 9(6), 507–523 (2005)CrossRefGoogle Scholar
  4. 4.
    Cheng, S.H., 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
  5. 5.
    Commowick, O., Arsigny, V., Costa, J., Malandain, G., Ayache, N.: An efficient multi-affine framework for the registration of anatomical structures. In: Proceedings of ISBI 2006. IEEE, Los Alamitos (to appear, 2006)Google Scholar
  6. 6.
    Cuzol, A., Hellier, P., Mémin, E.: A novel parametric method for non-rigid image registration. In: Christensen, G.E., Sonka, M. (eds.) IPMI 2005. LNCS, vol. 3565, pp. 456–467. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Little, J.A., Hill, D.L.G., Hawkes, D.J.: Deformations incorpotationg rigid structures. CVIU 66(2), 223–232 (1996)Google Scholar
  9. 9.
    Maintz, J.B.A., Viergever, M.A.: A survey of medical registration. Medical image analysis 2(1), 1–36 (1998)CrossRefGoogle Scholar
  10. 10.
    Narayanan, R., Fessler, J.A., Park, H., Meyer, C.R.: Diffeomorphic nonlinear transformations: A local parametric approach for image registration. In: Christensen, G.E., Sonka, M. (eds.) IPMI 2005. LNCS, vol. 3565, pp. 174–185. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Papademetris, X., Dione, D.P., Dobrucki, L.W., Staib, L.H., Sinusas, A.J.: Articulated Rigid Registration for Serial Lower-Limb Mouse Imaging. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3750, pp. 919–926. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Pitiot, A., Bardinet, E., Thompson, P.M., Malandain, G.: Piecewise affine registration of biological images for volume reconstruction. Med. Im. Anal.(accepted for publication, 2005)Google Scholar
  13. 13.
    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. Medecal Imaging 18(8), 712–721 (1999)CrossRefGoogle Scholar
  14. 14.
    Sheppard, D.: A two-dimensionnal interpolation function for irregularly spaced data. In: 23rd National Conference of the ACM, pp. 517–524 (1968)Google Scholar
  15. 15.
    Stefanescu, R., Pennec, X., Ayache, N.: Grid powered nonlinear image registration with locally adaptive regularization. Med. Im. Anal. 8(3), 325–342 (2004)CrossRefGoogle Scholar
  16. 16.
    Tenenbaum, M., Pollard, H.: Ordinary Differential Equations. Dover (1985)Google 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. 1.2004 Route des LuciolesINRIA Sophia – Epidaure ProjectSophia AntipolisFrance
  2. 2.DOSISoft S.A.CachanFrance

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