A Hierarchical Geodesic Model for Diffeomorphic Longitudinal Shape Analysis

  • Nikhil Singh
  • Jacob Hinkle
  • Sarang Joshi
  • P. Thomas Fletcher
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7917)


Hierarchical linear models (HLMs) are a standard approach for analyzing data where individuals are measured repeatedly over time. However, such models are only applicable to longitudinal studies of Euclidean data. In this paper, we propose a novel hierarchical geodesic model (HGM), which generalizes HLMs to the manifold setting. Our proposed model explains the longitudinal trends in shapes represented as elements of the group of diffeomorphisms. The individual level geodesics represent the trajectory of shape changes within individuals. The group level geodesic represents the average trajectory of shape changes for the population. We derive the solution of HGMs on diffeomorphisms to estimate individual level geodesics, the group geodesic, and the residual geodesics. We demonstrate the effectiveness of HGMs for longitudinal analysis of synthetically generated shapes and 3D MRI brain scans.


Diffeomorphisms Longitudinal Hierarchical Model 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnol’d, V.I.: 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 (1966)zbMATHCrossRefGoogle Scholar
  2. 2.
    Beg, M., Miller, M., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. IJCV 61(2), 139–157 (2005)CrossRefGoogle Scholar
  3. 3.
    Durrleman, S., Pennec, X., Trouvé, A., Gerig, G., Ayache, N.: Spatiotemporal atlas estimation for developmental delay detection in longitudinal datasets. In: Yang, G.-Z., Hawkes, D., Rueckert, D., Noble, A., Taylor, C. (eds.) MICCAI 2009, Part I. LNCS, vol. 5761, pp. 297–304. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Fishbaugh, J., Prastawa, M., Durrleman, S., Piven, J., Gerig, G.: Analysis of longitudinal shape variability via subject specific growth modeling. In: Ayache, N., Delingette, H., Golland, P., Mori, K. (eds.) MICCAI 2012, Part I. LNCS, vol. 7510, pp. 731–738. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  5. 5.
    Fletcher, P.T.: Geodesic regression on Riemannian manifolds. In: MICCAI Workshop on Mathematical Foundations of Computational Anatomy, pp. 75–86 (2011)Google Scholar
  6. 6.
    Laird, N.M., Ware, J.H.: Random-effects models for longitudinal data. Biometrics 38(4), 963–974 (1982)zbMATHCrossRefGoogle Scholar
  7. 7.
    Lorenzi, M., Ayache, N., Frisoni, G.B., Pennec, X., The Alzheimer’s Disease Neuroimaging Initiative: Mapping the Effects of Aβ 1−42 Levels on the Longitudinal Changes in Healthy Aging: Hierarchical Modeling Based on Stationary Velocity Fields. In: Fichtinger, G., Martel, A., Peters, T. (eds.) MICCAI 2011, Part II. LNCS, vol. 6892, pp. 663–670. Springer, Heidelberg (2011)Google Scholar
  8. 8.
    Miller, M.I., Trouvé, A., Younes, L.: Geodesic shooting for computational anatomy. Journal of Mathematical Imaging and Vision 24(2), 209–228 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Muralidharan, P., Fletcher, P.: Sasaki metrics for analysis of longitudinal data on manifolds. In: IEEE Conference on CVPR, pp. 1027–1034 (June 2012)Google Scholar
  10. 10.
    Niethammer, M., Huang, Y., Vialard, F.-X.: Geodesic regression for image time-series. In: Fichtinger, G., Martel, A., Peters, T. (eds.) MICCAI 2011, Part II. LNCS, vol. 6892, pp. 655–662. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  11. 11.
    Singh, N., Hinkle, J., Joshi, S., Fletcher, P.T.: A vector momenta formulation of diffeomorphisms for improved geodesic regression and atlas construction. In: International Symposium on Biomedial Imaging (ISBI) (April 2013)Google Scholar
  12. 12.
    Younes, L., Arrate, F., Miller, M.I.: Evolution equations in computational anatomy. Neuroimage 45(1 suppl.), S40–S50 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Nikhil Singh
    • 1
  • Jacob Hinkle
    • 1
  • Sarang Joshi
    • 1
  • P. Thomas Fletcher
    • 1
  1. 1.Scientific Computing and Imaging InstituteUniversity of UtahSalt Lake CityUSA

Personalised recommendations