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Sparse Multi-Scale Diffeomorphic Registration: The Kernel Bundle Framework

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Abstract

In order to detect small-scale deformations during disease propagation while allowing large-scale deformation needed for inter-subject registration, we wish to model deformation at multiple scales and represent the deformation compactly at the relevant scales only. This paper presents the kernel bundle extension of the LDDMM framework that allows multiple kernels at multiple scales to be incorporated in the registration. We combine sparsity priors with the kernel bundle resulting in compact representations across scales, and we present the mathematical foundation of the framework with derivation of the KB-EPDiff evolution equations. Through examples, we illustrate the influence of the kernel scale and show that the method achieves the important property of sparsity across scales. In addition, we demonstrate on a dataset of annotated lung CT images how the kernel bundle framework with a compact representation reaches the same accuracy as the standard method optimally tuned with respect to scale.

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Notes

  1. The Bochner integral extends the Lebesgue integral to functions taking values in Banach spaces. The Banach space norm allows definition of L p-spaces of Banach valued functions. In particular, the L 2-spaces of functions taking values in Hilbert spaces are themselves Hilbert spaces [7].

  2. With a functional f∈V ∗ and a vector v∈V, the notation (f|v) denotes f evaluated on v, i.e. (f|v)=f(v).

  3. See e.g. http://www.di.ens.fr/~mschmidt/Software/minFunc.html.

References

  1. Beg, M.F., 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)

    Article  Google Scholar 

  2. Bruveris, M., Gay-Balmaz, F., Holm, D.D., Ratiu, T.S.: The momentum map representation of images. J. Nonlinear Sci. 21(1), 115–150 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bruveris, M., Risser, L., Vialard, F.: Mixture of kernels and iterated Semi-Direct product of diffeomorphism groups. Multiscale Model. Simul. 10(4), 1344–1368 (2011)

    Article  MathSciNet  Google Scholar 

  4. Candès, E.J., Wakin, M.B., Boyd, S.P.: Enhancing sparsity by reweighted l1 minimization. J. Fourier Anal. Appl. 14, 877–905 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Castillo, R., Castillo, E., Guerra, R., Johnson, V.E., McPhail, T., Garg, A.K., Guerrero, T.: A framework for evaluation of deformable image registration spatial accuracy using large landmark point sets. Phys. Med. Biol. 54(7), 1849–1870 (2009)

    Article  Google Scholar 

  6. Christensen, G., Rabbitt, R., Miller, M.: Deformable templates using large deformation kinematics. IEEE Trans. Image Process. 5(10), 1435–1447 (2002)

    Article  Google Scholar 

  7. Diestel, J., Uhl, J.J. Jr.: Vector Measures. Mathematical Surveys, vol. 15. Am. Math. Soc., Providence (1977)

    MATH  Google Scholar 

  8. Dupuis, P., Grenander, U., Miller, M.I.: Variational Problems on Flows of Diffeomorphisms for Image Matching (1998)

    Google Scholar 

  9. Durrleman, S., Prastawa, M., Gerig, G., Joshi, S.: Optimal data-driven sparse parameterization of diffeomorphisms for population analysis. In: IPMI 2011, vol. 22, pp. 123–134 (2011)

    Google Scholar 

  10. Durrleman, S., Allassonnière, S., Joshi, S.: Sparse adaptive parameterization of variability in image ensembles. Int. J. Comput. Vis. 1–23 (2012). doi:10.1007/s11263-012-0556-1

  11. Fasshauer, G.E., Ye, Q.: Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators. Numer. Math. 119(3), 585–611 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grenander, U.: General Pattern Theory: A Mathematical Study of Regular Structures. Oxford University Press, London (1994)

    MATH  Google Scholar 

  13. Risser, L., Vialard, F., Wolz, R., Holm, D.D., Rueckert, D.: Simultaneous fine and coarse diffeomorphic registration: application to atrophy measurement in Alzheimer’s disease. In: MICCAI 2010 13(Pt 2), pp. 610–617 (2010)

    Google Scholar 

  14. Risser, L., Vialard, F.X., Wolz, R., Murgasova, M., Holm, D.D., Rueckert, D.: Simultaneous multi-scale registration using large deformation diffeomorphic metric mapping. IEEE Trans. Med. Imaging 30(10), 1746–1759 (2011)

    Article  Google Scholar 

  15. Sommer, S.: Accelerating Multi-Scale flows for LDDKBM diffeomorphic registration. In: GPUCV Workshop at ICCV 2011. IEEE Press, New York (2011)

    Google Scholar 

  16. Sommer, S., Lauze, F., Nielsen, M., Pennec, X.: Kernel bundle EPDiff: evolution equations for Multi-Scale diffeomorphic image registration. In: SSVM 2011. Springer, Berlin (2011)

    Google Scholar 

  17. Sommer, S., Nielsen, M., Lauze, F., Pennec, X.: A Multi-scale kernel bundle for LDDMM: towards sparse deformation description across space and scales. In: IPMI 2011. Springer, Berlin (2011)

    Google Scholar 

  18. Sommer, S., Nielsen, M., Pennec, X.: Sparsity and scale: compact representations of deformation for diffeomorphic registration. In: MMBIA at WACV 2012 (2012)

    Google Scholar 

  19. Sommer, S., Nielsen, M., Darkner, S., Pennec, X.: Higher-order momentum distributions and locally affine LDDMM registration. To appear in SIAM J. Imaging Sci.

  20. Trouvé, A.: An Infinite Dimensional Group Approach for Physics Based Models in Patterns Recognition (1995)

    Google Scholar 

  21. Vaillant, M., Miller, M., Younes, L., Trouvé, A.: Statistics on diffeomorphisms via tangent space representations. NeuroImage 23(Suppl. 1), S161–S169 (2004). doi:10.1016/j.neuroimage.2004.07.023

    Article  Google Scholar 

  22. Younes, L.: Shapes and Diffeomorphisms. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  23. Younes, L., Arrate, F., Miller, M.I.: Evolution equations in computational anatomy. NeuroImage 45(1, Suppl. 1), S40–S50 (2009)

    Article  Google Scholar 

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Authors and Affiliations

  1. The Image Group, Department of Computer Science, University of Copenhagen, Copenhagen, Denmark

    Stefan Sommer, François Lauze & Mads Nielsen

  2. BiomedIQ A/S, Copenhagen, Denmark

    Mads Nielsen

  3. Asclepios Project-Team, INRIA Sophia-Antipolis, Sophia Antipolis, France

    Xavier Pennec

Authors
  1. Stefan Sommer
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  2. François Lauze
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  3. Mads Nielsen
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  4. Xavier Pennec
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Correspondence to Stefan Sommer.

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Sommer, S., Lauze, F., Nielsen, M. et al. Sparse Multi-Scale Diffeomorphic Registration: The Kernel Bundle Framework. J Math Imaging Vis 46, 292–308 (2013). https://doi.org/10.1007/s10851-012-0409-0

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