Skip to main content
SpringerLink
Log in

Diffeomorphic 3D Image Registration via Geodesic Shooting Using an Efficient Adjoint Calculation

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

In the context of large deformations by diffeomorphisms, we propose a new diffeomorphic registration algorithm for 3D images that performs the optimization directly on the set of geodesic flows. The key contribution of this work is to provide an accurate estimation of the so-called initial momentum, which is a scalar function encoding the optimal deformation between two images through the Hamiltonian equations of geodesics. Since the initial momentum has proven to be a key tool for statistics on shape spaces, our algorithm enables more reliable statistical comparisons for 3D images.

Our proposed algorithm is a gradient descent on the initial momentum, where the gradient is calculated using standard methods from optimal control theory. To improve the numerical efficiency of the gradient computation, we have developed an integral formulation of the adjoint equations associated with the geodesic equations.

We then apply it successfully to the registration of 2D phantom images and 3D cerebral images. By comparing our algorithm to the standard approach of Beg et al. (Int. J. Comput. Vis. 61:139–157, 2005), we show that it provides a more reliable estimation of the initial momentum for the optimal path. In addition to promising statistical applications, we finally discuss different perspectives opened by this work, in particular in the new field of longitudinal analysis of biomedical images.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Allassonnière, S., Trouvé, A., & Younes, L. (2005). Geodesic shooting and diffeomorphic matching via textured meshes. In LNCS: Vol. 3757. Proc. of EMMCVPR. Berlin: Springer.

    Google Scholar 

  • Allassonnière, S., Kuhn, E., Trouvé, A., & Amit, Y. (2006). Generative model and consistent estimation algorithms for non-rigid deformable models. In ICASSP 2006 Proceedings (pp. 14–19).

    Google Scholar 

  • Allassonnière, S., Amit, Y., & Trouvé, A. (2007). Towards a coherent statistical framework for dense deformable template estimation. Journal of the Royal Statistical Society. Series B, 69(1), 3–29.

    MathSciNet  Google Scholar 

  • Ashburner, J. (2007). A fast diffeomorphic image registration algorithm. NeuroImage, 38, 95–113.

    Article  Google Scholar 

  • Avants, B. B., Epstein, C. L., Grossman, M., Gee, J. C. (2008). Symmetric diffeomorphic image registration with cross-correlation: Evaluating automated labeling of elderly and neurodegenerative brain. Medical Image Analysis, 12, 26–41.

    Article  Google Scholar 

  • Beg, M. F., Helm, P. A., McVeigh, E., Miller, M. I., & Winslow, R. L. (2004). Computational cardiac anatomy using MRI. Magnetic Resonance in Medecine, 52(5), 1167–1174.

    Article  Google Scholar 

  • Beg, M. F., Miller, M. I., Trouvé, A., & Younes, L. (2005). Computing large deformation metric mappings via geodesic flow of diffeomorphisms. International Journal of Computer Vision, 61, 139–157.

    Article  Google Scholar 

  • Cockburn, B., & Shu, C. W. (2001). Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing, 16(3), 173.

    Article  MathSciNet  MATH  Google Scholar 

  • Davis, B. C., Fletcher, P. T., Bullitt, E., & Joshi, S. (2010). Population shape regression from random design data. International Journal of Computer Vision, 90(2), 255–266.

    Article  Google Scholar 

  • Du, J., Younes, L., & Qiu, A. (2011). Whole brain diffeomorphic metric mapping via integration of sulcal and gyral curves, cortical surfaces, and images. NeuroImage, 56(1), 162–173.

    Article  Google Scholar 

  • Dupuis, P., Grenander, U., & Miller, M. (1998). Variational problems on flows of diffeomorphisms for image matching. Quarterly of Applied Mathematics

  • Durrleman, S., Pennec, X., Trouvé, A., Gerig, G., & Ayache, N. (2009). Spatiotemporal atlas estimation for developmental delay detection in longitudinal datasets. In MICCAI (Vol. 5761, pp. 297–304).

    Google Scholar 

  • Fletcher, P. T. (2004). Statistical variability in nonlinear spaces: application to shape analysis and dt-mri. PhD thesis, Department of Computer Science, University of North Carolina.

  • Fletcher, P. T., Lu, C., Pizer, M., & Joshi, S. (2004). Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions Medical Imaging, 995–1005.

  • Fletcher, P. T., Venkatasubramanian, S., & Joshi, S. (2008). Robust statistics on Riemannian manifolds via the geometric median. In Computer Vision and Pattern Recognition (pp. 1–8).

    Google Scholar 

  • Gratton, S., Lawless, A., & Nichols, N. (2007). Approximate Gauss-Newton methods for nonlinear least squares problems. SIAM Journal on Optimization, 18(1), 106–132.

    MathSciNet  MATH  Google Scholar 

  • Grenander, U. (1993). General pattern theory. Oxford: Oxford Science Publications.

    Google Scholar 

  • Grenander, U. & Miller, M. I. (1998). Computational anatomy: an emerging discipline. Quarterly of Applied Mathematics, LVI(4), 617–694.

    MathSciNet  Google Scholar 

  • Grenander, U., Srivastava, A., & Saini, S. (2006). Characterization of biological growth using iterated diffeomorphisms. In ISBI (pp. 1136–1139).

    Google Scholar 

  • Grenander, U., Srivastava, A., & Saini, S. (2007). A pattern-theoretic characterization of biological growth. IEEE Transactions on Medical Imaging, 26(5), 648–659.

    Article  Google Scholar 

  • Hart, G. L., Zach, C., & Niethammer, M. (2009). An optimal control approach for deformable registration. In Computer Vision and Pattern Recognition, Workshop (pp. 9–16).

    Chapter  Google Scholar 

  • Helm, P. A., Younes, L., Beg, M. F., Ennis, D. B., Leclercq, C., Faris, O. P., McVeigh, E., Kass, D., Miller, M. I., & Winslow, R. L. (2006). Evidence of structural remodeling in the dyssynchronous failing heart. Circulation Research, 98, 125–132.

    Article  Google Scholar 

  • Holm, D. D., Trouvé, A., & Younes, L., (2009, to appear). The Euler Poincaré theory of metamorphosis. Quarterly of Applied Mathematics.

  • Jiang, G. S., & Shu, C. W. (1996). Efficient implementation of weighted ENO schemes. Journal of Computational Physics, 126.

  • Joshi, S., & Miller, M. (2000). Landmark matching via large deformation diffeomorphisms. IEEE Transactions on Image Processing, 9(8), 1357–1370.

    Article  MathSciNet  MATH  Google Scholar 

  • Khan, A., & Beg, M. (2008). Representation of time-varying shapes in the large deformation diffeomorphic framework. In ISBI (pp. 1521–1524).

    Google Scholar 

  • Klein, A., Andersson, J., Ardekani, B. A., Ashburner, J., Avants, B., Chiang, M. C., Christensen, G. E., Collins, D. L., Gee, J., Hellier, P., Song, J. H., Jenkinson, M., Lepage, C., Rueckert, D., Thompson, P., Vercauteren, T., Woods, R., Mann, J., & Parseya, R. V. (2009). Evaluation of 14 nonlinear deformation algorithms applied to human brain MRI registration. NeuroImage, 46(3), 786–802.

    Article  Google Scholar 

  • Le Dimet, F. X., & Talagrand, O. (1986). Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus Series A, 38, 97–110.

    Article  Google Scholar 

  • Le Dimet, F. X., Ngodock, H. E., & Navon, I. M. (1997). Sensitivity analysis in variational data assimilation. Journal of the Meteorological Society Japan, 145–155.

  • Le Veque, R. J. (2002). Finite volume methods for hyperbolic problems. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Lee, J. M. (1997). Riemannian manifolds an introduction to curvature. Graduate texts in mathematics Berlin: Springer.

    MATH  Google Scholar 

  • Ma, J. Miller, M. I. Younes, L. (2010). A Bayesian generative model for surface template estimation. International Journal of Biomedical Imaging, 2010, 14.

    Article  Google Scholar 

  • Marsland, S., & McLachlan, R. I. (2007). A Hamiltonian particle method for diffeomorphic image registration. In Lecture Notes in Computer Science: Vol. 4548. Proceedings of Information Processing in Medical Images (pp. 396–407). Berlin: Springer.

    Chapter  Google Scholar 

  • Miller, M., & Younes, L. (2001). Group actions, homeomorphisms, and matching: a general framework. International Journal of Computer Vision, 41, 61–84.

    Article  MATH  Google Scholar 

  • Miller, M. I., Trouvé, A., & Younes, L. (2002). On the metrics and Euler-Lagrange equations of computational anatomy. Annual Review of Biomedical Engineering, 4, 375–405.

    Article  Google Scholar 

  • Miller, M. I., Trouvé, A., & Younes, L. (2006). Geodesic shooting for computational anatomy. Journal of Mathematical Imaging and Vision, 24(2), 209–228.

    Article  MathSciNet  Google Scholar 

  • Murgasova, M., Aljabar, P., Srinivasan, L., Counsell, S. J., Doria, V., Serag, A., Gousias, I. S., Boardman, J. P., Rutherford, M. A., Edwards, A. D., Hajnal, J. V., & Rueckert, D. (2011). A dynamic 4d probabilistic atlas of the developing brain. NeuroImage, 54(4), 2750–63.

    Article  Google Scholar 

  • Ngodock, H. E. (2005). Efficient implementation of covariance multiplication for data assimilation with the representer method. Ocean Modelling, 8(3), 237–251.

    Article  Google Scholar 

  • Niethammer, M., Hart, G., & Zach, C. (2009). An optimal control approach for the registration of image time-series. 2427–2434.

  • Pennec, X. (2009). Statistical computing on manifolds: From Riemannian geometry to computational anatomy. Emerging Trends in Visual Computing, 5416, 347–386.

    Article  Google Scholar 

  • Risser, L., Vialard, F. X., Wolz, R., Holm, D. D., & Rueckert, D., (2010). Simultaneous fine and coarse diffeomorphic registration: application to the atrophy measurement in Alzheimer’s disease. In International conference on medical image computing and computer assisted intervention (MICCAI). Lecture notes in computer science.

    Google Scholar 

  • Risser, L., Vialard, F., Wolz, R., Murgasova, M., Holm, D., & Rueckert, D. (2011). Simultaneous multiscale registration using large deformation diffeomorphic metric mapping IEEE Transactions on Medical Imaging, 99, 1. doi:10.1007/s11263-011-0481-8

    Google Scholar 

  • Sasaki, Y. (1970). Some basic formalism in numerical variational. Monthly Weather Review, 98, 875–883.

    Article  Google Scholar 

  • Shu, C. W. (2009). High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Review, 51(1), 82.

    Article  MathSciNet  MATH  Google Scholar 

  • Singh, N., Fletcher, P. T., Preston, J. S., Ha, L., King, R., Marron, J. S., Wiener, M., & Joshi, S. (2010). Multivariate statistical analysis of deformation momenta relating anatomical shape to neuropsychological measures. In T. Jiang, N. Navab, J. P. Pluim, & Viergever M. A. (Eds.), MICCAI (pp. 529–537). Berlin: Springer.

    Google Scholar 

  • Sled, J. G., Zijdenbos, A. P., & Evans, A. C. (1998). A nonparametric method for automatic correction of intensity nonuniformity in MRI data. IEEE Transactions on Medical Imaging, 17(1), 87–97.

    Article  Google Scholar 

  • Trouvé, A. (1998). Diffeomorphism groups and pattern matching in image analysis. International Journal of Computer Vision, 28(3), 213–221.

    Article  Google Scholar 

  • Trouvé, A., & Vialard, F. X. (2010, accepted for publication). Shape splines and stochastic shape evolutions: a second order point of view. Quarterly of Applied Mathematics.

  • Trouvé, A., & Younes, L. (2005). Local geometry of deformable templates. SIAM Journal of Mathematical Analysis.

  • Vaillant, M., Miller, M. I., Trouvé, A., Younes, L. (2004). Statistics on diffeomorphisms via tangent space representations. NeuroImage, 23(S1), S161–S169.

    Article  Google Scholar 

  • Vercauteren, T., Pennec, X., Perchant, A., & Ayache, N. (2009). Diffeomorphic demons: efficient non-parametric image registration. NeuroImage, 45(1), S61–S72.

    Article  Google Scholar 

  • Vialard, F. X. (2009). Hamiltonian approach to shape spaces in a diffeomorphic framework: from the discontinuous image matching problem to a stochastic growth model. PhD thesis, Ecole Normale Supérieure de Cachan.

  • Vialard, F. X., & Santambrogio, F. (2009). Extension to BV functions of the large deformation diffeomorphisms matching approach. Comptes Rendus Mathematique, 347(1-2), 27–32.

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, L., Beg, M. F., Ratnanather, J. T., Ceritoglu, C., Younes, L., Morris, J., Csernansky, J., & Miller, M. I. (2006). Large deformation diffeomorphism and momentum based hippocampal shape discrimination in dementia of the Alzheimer type. IEEE Transactions on Medical Imaging, 26, 462–470.

    Article  Google Scholar 

  • Younes, L. (2007). Jacobi fields in groups of diffeomorphisms and applications. Quarterly of Applied Mathematics, 65, 113–134.

    MathSciNet  MATH  Google Scholar 

  • Younes, L. (2010). Shapes and diffeomorphisms. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Younes, L., Arrate, F., & Miller, M. I. (2009). Evolutions equations in computational anatomy. NeuroImage, 45(1), S40–S50. Mathematics in Brain Imaging (Supplement 1)

    Article  Google Scholar 

  • Zeidler, E. (1995). Applied functional analysis: main principles and their applications. Applied Mathematical Sciences, 109.

Download references

Author information

Authors and Affiliations

  1. Institute for Mathematical Science, Imperial College London, 53 Prince’s Gate, SW7 2PG, London, UK

    François-Xavier Vialard

  2. Biomedical Image Analysis Group and Institute for Mathematical Sciences, Imperial College London, 53 Prince’s Gate, SW7 2PG, London, UK

    Laurent Risser

  3. Department of Computing, Imperial College London, 53 Prince’s Gate, SW7 2PG, London, UK

    Daniel Rueckert

  4. Department of Aeronautics, Imperial College London, 53 Prince’s Gate, SW7 2PG, London, UK

    Colin J. Cotter

Authors
  1. François-Xavier Vialard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Laurent Risser
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel Rueckert
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Colin J. Cotter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to François-Xavier Vialard.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vialard, FX., Risser, L., Rueckert, D. et al. Diffeomorphic 3D Image Registration via Geodesic Shooting Using an Efficient Adjoint Calculation. Int J Comput Vis 97, 229–241 (2012). https://doi.org/10.1007/s11263-011-0481-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11263-011-0481-8

Keywords

Search

Navigation