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International Journal of Computer Vision

, Volume 127, Issue 1, pp 61–73 | Cite as

Fast Diffeomorphic Image Registration via Fourier-Approximated Lie Algebras

  • Miaomiao ZhangEmail author
  • P. Thomas Fletcher
Article

Abstract

This paper introduces Fourier-approximated Lie algebras for shooting (FLASH), a fast geodesic shooting algorithm for diffeomorphic image registration. We approximate the infinite-dimensional Lie algebra of smooth vector fields, i.e., the tangent space at the identity of the diffeomorphism group, with a low-dimensional, bandlimited space. We show that most of the computations for geodesic shooting can be carried out entirely in this low-dimensional space. Our algorithm results in dramatic savings in time and memory over traditional large-deformation diffeomorphic metric mapping algorithms, which require dense spatial discretizations of vector fields. To validate the effectiveness of FLASH, we run pairwise image registration on both 2D synthetic data and real 3D brain images and compare with the state-of-the-art geodesic shooting methods. Experimental results show that our algorithm dramatically reduces the computational cost and memory footprint of diffemorphic image registration with little or no loss of accuracy.

Keywords

Fourier-approximated Lie algebras Geodesic shooting Diffeomorphic image registration 

Notes

Acknowledgements

This work was supported by NIH Grant 5R01EB007688 and NSF CAREER Grant 1054057.

References

  1. Arnol’d, V. I. (1966). Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann Inst Fourier, 16, 319–361.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Arsigny, V., Commowick, O., Pennec, X., & Ayache, N. (2006). A log-Euclidean framework for statistics on diffeomorphisms. In Medical image computing and computer-assisted intervention—MICCAI 2006, Springer (pp. 924–931).Google Scholar
  3. Ashburner, J. (2007). A fast diffeomorphic image registration algorithm. Neuroimage, 38(1), 95–113.CrossRefGoogle Scholar
  4. Ashburner, J., & Friston, K. J. (2011). Diffeomorphic registration using geodesic shooting and Gauss–Newton optimisation. NeuroImage, 55(3), 954–967.CrossRefGoogle Scholar
  5. Beg, M., Miller, M., Trouvé, A., & Younes, L. (2005). Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal of Computer Vision, 61(2), 139–157.CrossRefGoogle Scholar
  6. Bullo, F. (1995). Invariant affine connections and controllability on Lie groups. Technical Report for Geometric Mechanics, California Institute of Technology.Google Scholar
  7. Christensen, G. E., Rabbitt, R. D., & Miller, M. I. (1996). Deformable templates using large deformation kinematics. IEEE Transactions on Image Processing, 5(10), 1435–1447.CrossRefGoogle Scholar
  8. Dupuis, P., Grenander, U., & Miller, M. I. (1998). Variational problems on flows of diffeomorphisms for image matching. Quarterly of Applied Mathematics, 56(3), 587.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Durrleman, S., Prastawa, M., Gerig, G., & Joshi, S. (2011). Optimal data-driven sparse parameterization of diffeomorphisms for population analysis. In Information processing in medical imaging, Springer (pp. 123–134).Google Scholar
  10. Hinkle, J., Fletcher, P. T., & Joshi, S. (2014). Intrinsic polynomials for regression on Riemannian manifolds. Journal of Mathematical Imaging and Vision, 50(1–2), 32–52.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hromatka, M., Zhang, M., Fleishman, G. M., Gutman, B., Jahanshad, N., Thompson, P., & Fletcher, P. T. (2015). A hierarchical Bayesian model for multi-site diffeomorphic image atlases. In Medical image computing and computer-assisted intervention—MICCAI 2015, Springer (pp. 372–379).Google Scholar
  12. Joshi, S., Davis, B., Jomier, M., & Gerig, G. (2004). Unbiased diffeomorphic atlas construction for computational anatomy. NeuroImage, 223(Supplement 1), 151–160.CrossRefGoogle Scholar
  13. Marcus, D. S., Wang, T. H., Parker, J., Csernansky, J. G., Morris, J. C., & Buckner, R. L. (2007). Open access series of imaging studies (OASIS): Cross-sectional mri data in young, middle aged, nondemented, and demented older adults. Journal of Cognitive Neuroscience, 19(9), 1498–1507.CrossRefGoogle Scholar
  14. Miller, M. I., Trouvé, A., & Younes, L. (2006). Geodesic shooting for computational anatomy. Journal of Mathematical Imaging and Vision, 24(2), 209–228.  https://doi.org/10.1007/s10851-005-3624-0.MathSciNetCrossRefGoogle Scholar
  15. Niethammer, M., Huang, Y., & Vialard, F. X. (2011). Geodesic regression for image time-series. In International conference on medical image computing and computer-assisted intervention, Springer (pp. 655–662).Google Scholar
  16. Singh, N., Hinkle, J., Joshi, S., & Fletcher, P. T. (2013). A vector momenta formulation of diffeomorphisms for improved geodesic regression and atlas construction. In International symposium on biomedial imaging (ISBI).Google Scholar
  17. Trouvé, A. (1998). Diffeomorphisms groups and pattern matching in image analysis. International Journal of Computer Vision, 28(3), 213–221.MathSciNetCrossRefGoogle Scholar
  18. Vaillant, M., Miller, M. I., Younes, L., & Trouvé, A. (2004). Statistics on diffeomorphisms via tangent space representations. NeuroImage, 23, S161–S169.CrossRefGoogle Scholar
  19. Vercauteren, T., Pennec, X., Perchant, A., & Ayache, N. (2009). Diffeomorphic demons: Efficient non-parametric image registration. NeuroImage, 45(1), S61–S72.CrossRefGoogle Scholar
  20. Vialard, F. X., Risser, L., Rueckert, D., & Cotter, C. J. (2012). Diffeomorphic 3D image registration via geodesic shooting using an efficient adjoint calculation. International Journal of Computer Vision, 97(2), 229–241.MathSciNetCrossRefzbMATHGoogle Scholar
  21. Younes, L., Arrate, F., & Miller, M. (2009). Evolutions equations in computational anatomy. NeuroImage, 45(1S1), 40–50.CrossRefGoogle Scholar
  22. Zhang, M., & Fletcher, P. T. (2014). Bayesian principal geodesic analysis in diffeomorphic image registration. In Medical image computing and computer-assisted intervention–MICCAI 2014, Springer (pp. 121–128).Google Scholar
  23. Zhang, M., & Fletcher, P. T. (2015). Finite-dimensional Lie algebras for fast diffeomorphic image registration. In Information processing in medical imaging.Google Scholar
  24. Zhang, M., Liao, R., Dalca, A. V., Turk, E. A., Luo, J., Grant, P. E., & Golland, P. (2017). Frequency diffeomorphisms for efficient image registration. In International conference on information processing in medical imaging, Springer (pp. 559–570).Google Scholar
  25. Zhang, M., Singh, N., & Fletcher, P. T. (2013). Bayesian estimation of regularization and atlas building in diffeomorphic image registration. J. C. Gee, S. Joshi, K. M. Pohl, W. M. Wells & L. Zöllei (Eds.), Information processing in medical imaging (pp. 37–48). Springer.Google Scholar
  26. Zhang, M., Wells III, W. M., & Golland, P. (2016). Low-dimensional statistics of anatomical variability via compact representation of image deformations. In International conference on medical image computing and computer-assisted intervention, Springer (pp. 166–173).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Lehigh UniversityBethlehemUSA
  2. 2.University of UtahSalt Lake CityUSA

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