Brownian Warps for Non-Rigid Registration

  • Mads Nielsen
  • Peter Johansen
  • Andrew D. Jackson
  • Benny Lautrup
  • Søren HaubergEmail author


A Brownian motion model in the group of diffeomorphisms has been introduced as inducing a least committed prior on warps. This prior is source-destination symmetric, fulfills a natural semi-group property for warps, and with probability 1 creates invertible warps. Using this as a least committed prior, we formulate a Partial Differential Equation for obtaining the maximally likely warp given matching constraints derived from the images. We solve for the free boundary conditions, and the bias toward smaller areas in the finite domain setting. Furthermore, we demonstrate the technique on 2D images, and show that the obtained warps are also in practice source-destination symmetric and in an example on X-ray spine registration provides extrapolations from landmark point superior to those of spline solutions.


Non-rigid registration Brownian motion Central limit theorem Invariance 


  1. 1.
    Amini, A.A., Curwen, R.W., Gore, J.C.: Snakes and splines for tracking non-rigid heart motion. In: ECCV96, pp. II:251–261 (1996) Google Scholar
  2. 2.
    Andresen, P.R., Nielsen, M.: Non-rigid registration by geometry-constrained diffusion. Med. Image Anal. 6, 81–88 (2000) Google Scholar
  3. 3.
    Bajcsy, R., Kovacic, S.: Multiresolution elastic matching. CVGIP 46, 1–21 (1989) Google Scholar
  4. 4.
    Faisal Beg, M., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vis. 61(2), 139–157 (2005) CrossRefGoogle Scholar
  5. 5.
    Besl, P.J., McKay, N.D.: A method for registration of 3-d shapes. IEEE Trans. Pattern Anal. Mach. Intell. 14(2), 239–256 (1992) CrossRefGoogle Scholar
  6. 6.
    Bishop, C.M.: Pattern Recognition and Machine Learning. Springer, Berlin (2006) zbMATHGoogle Scholar
  7. 7.
    Bookstein, F.L.: Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge University Press, Cambridge (1991) zbMATHGoogle Scholar
  8. 8.
    Bro-Nielsen, M., Gramkow, C.: Fast fluid registration of medical images. In: Proc. Visualization in Biomedical Imaging (VBC’96), pp. 267–276 (1996) Google Scholar
  9. 9.
    Christensen, G.E., He, J.: Consistent nonlinear elastic image registration. In: MMBIA01, p. 37 (2001) Google Scholar
  10. 10.
    Christensen, G.E., Miller, M.I., Vannier, M.: A 3d deformable magnetic resonance textbook based on elasticity. In: AAAI Spring Symposion Series, pp. 153–156. Standford University Press, Standford (1994) Google Scholar
  11. 11.
    Faugeras, O., Hermosillo, G.: Well-posedness of eight problems of multi-modal statistical image-matching. Technical Report, INRIA, August 2001. Research Report 4235 Google Scholar
  12. 12.
    Gill, R.D., Johansen, S.: A survey of product-integration with a view toward application un survival analysis. Ann. Stat. 18(4), 1501–1555 (1990) CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Högnäs, G., Mukherjea, A.: Probability Measures on Semigroups. Plenum, New York (1995) zbMATHGoogle Scholar
  14. 14.
    Jackson, A.D., Lautrup, B., Johansen, P., Nielsen, M.: Products of random matrices. Phys. Rev. E 66(6), 5 (2002). Technical Report, article 66124 CrossRefMathSciNetGoogle Scholar
  15. 15.
    Joshi, S., Miller, M.M.: Landmark matching via large deformation diffeomorphisms. IEEE Trans. Image Process. 9(8), 1357–1370 (2000) CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Maintz, J., Viergever, M.: A survey of medical image registration. Med. Image Anal. 2(1), 1–36 (1998) CrossRefGoogle Scholar
  17. 17.
    Markussen, B.: Large deformation diffeomorphisms with application to optic flow. Comput. Vis. Image Underst. 106, 97–105 (2007) CrossRefGoogle Scholar
  18. 18.
    Nielsen, M.: Evaluation of Brownian warps for shape alignment. In: Proc. of SPIE Medical Imaging 2007 (2007) Google Scholar
  19. 19.
    Nielsen, M., Johansen, P., Jackson, A.D., Lautrup, B.: Brownian warps: A least committed prior for non-rigid registration. In: MICCAI (2), pp. 557–564 (2002) Google Scholar
  20. 20.
    Nielsen, M., Markussen, B.: From Bayes to PDEs in image warping. In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Mathematical Models in Computer Vision: The Handbook. Springer, Berlin (2005) Google Scholar
  21. 21.
    Rissanen, J.: Stochastic Complexity in Statistical Enquiry. World Scientific, Singapore (1989) Google Scholar
  22. 22.
    Rohr, K.: Landmark-Based Image Analysis: Using Geometric and Intensity Models. Kluwer Academic, Dordrecht (2001) zbMATHGoogle Scholar
  23. 23.
    Rueckert, D., Sonoda, L.I., Hayes, C., Hill, D.L.G., Leach, M.O., Hawkes, D.J.: Nonrigid registration using free-form deformations: application to breast MR images. Med. Imaging 18(8), 712–721 (1999) CrossRefGoogle Scholar
  24. 24.
    Viola, P.A.: Alignment by maximization of mutual information. Technical Report AITR-1548 (1995) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Mads Nielsen
    • 1
    • 2
  • Peter Johansen
    • 1
  • Andrew D. Jackson
    • 3
  • Benny Lautrup
    • 3
  • Søren Hauberg
    • 1
    Email author
  1. 1.Department of Computer ScienceUniversity of CopenhagenCopenhagenDenmark
  2. 2.Nordic Bioscience A/SHerlevDenmark
  3. 3.Niels Bohr InstituteUniversity of CopenhagenCopenhagenDenmark

Personalised recommendations