International Journal of Computer Vision

, Volume 97, Issue 2, pp 229–241 | Cite as

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

  • François-Xavier VialardEmail author
  • Laurent Risser
  • Daniel Rueckert
  • Colin J. Cotter


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.


Geodesic shooting Computational anatomy Adjoint equations Hamiltonian equations Large deformations via diffeomorphisms 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • François-Xavier Vialard
    • 1
    Email author
  • Laurent Risser
    • 2
  • Daniel Rueckert
    • 3
  • Colin J. Cotter
    • 4
  1. 1.Institute for Mathematical ScienceImperial College LondonLondonUK
  2. 2.Biomedical Image Analysis Group and Institute for Mathematical SciencesImperial College LondonLondonUK
  3. 3.Department of ComputingImperial College LondonLondonUK
  4. 4.Department of AeronauticsImperial College LondonLondonUK

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