International Journal of Computer Vision

, Volume 117, Issue 1, pp 70–92 | Cite as

Hierarchical Geodesic Models in Diffeomorphisms

  • Nikhil Singh
  • Jacob Hinkle
  • Sarang Joshi
  • P. Thomas Fletcher


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. This paper develops the theory of hierarchical geodesic models (HGMs), which generalize HLMs to the manifold setting. Our proposed model quantifies longitudinal trends in shapes as a hierarchy of geodesics in the group of diffeomorphisms. First, individual-level geodesics represent the trajectory of shape changes within individuals. Second, a group-level geodesic represents the average trajectory of shape changes for the population. Our proposed HGM is applicable to longitudinal data from unbalanced designs, i.e., varying numbers of timepoints for individuals, which is typical in medical studies. We derive the solution of HGMs on diffeomorphisms to estimate individual-level geodesics, the group geodesic, and the residual diffeomorphisms. We also propose an efficient parallel algorithm that easily scales to solve HGMs on a large collection of 3D images of several individuals. Finally, we present an effective model selection procedure based on cross validation. We demonstrate the effectiveness of HGMs for longitudinal analysis of synthetically generated shapes and 3D MRI brain scans.


Longitudinal modeling Diffeomorphisms Mixed-effects modeling LDDMM 



This research is supported by NIH Grants U01NS082086, 5R01EB007688, U01 AG024904, R01 MH084795 and P41 RR023953, and NSF Grant 1054057. National Institutes of Health Grant U01 AG024904.


  1. Adams, J. F. (1969). Lectures on Lie groups. Chicago: University of Chicago Press.MATHGoogle Scholar
  2. Amit, Y., Grenander, U., & Piccioni, M. (1991). Structural image restoration through deformable templates. Journal of the American Statistical Association, 86(414), 376–387.CrossRefGoogle Scholar
  3. 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. Annales de l’institut Fourier, 16, 319–361.CrossRefMATHGoogle Scholar
  4. Burke, S. N., & Barnes, C. A. (2006). Neural plasticity in the ageing brain. Nature Reviews Neuroscience, 7(1), 30–40.CrossRefGoogle Scholar
  5. Chevalley, C. (1999). Theory of Lie groups: 1 (Vol. 1). Princeton: Princeton University Press.Google Scholar
  6. 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.CrossRefGoogle Scholar
  7. Durrleman, S., Pennec, X., Trouvé, A., Gerig, G., & Ayache, N. (2009). Spatiotemporal atlas estimation for developmental delay detection in longitudinal datasets. In MICCAI (pp. 297–304). Berlin: Springer.Google Scholar
  8. Fishbaugh, J., Prastawa, M., Durrleman, S., Piven, J., & Gerig, G. (2012). Analysis of longitudinal shape variability via subject specific growth modeling. MICCAI. Berlin: Springer.Google Scholar
  9. Fishbaugh, J., Prastawa, M., Gerig, G., & Durrleman, S. (2013). Geodesic image regression with a sparse parameterization of diffeomorphisms. In Geometric Science of Information (pp. 95–102). New York: Springer.Google Scholar
  10. Fitzmaurice, G. M., Laird, N. M., & Ware, J. H. (2012). Applied longitudinal analysis (Vol. 998). New Jersey: Wiley.Google Scholar
  11. Fletcher, P. T. (2013). Geodesic regression and the theory of least squares on riemannian manifolds. International Journal of Computer Vision, 105(2), 171–185.Google Scholar
  12. Fox, N. C., & Schott, J. M. (2004). Imaging cerebral atrophy: Normal ageing to alzheimerEijs disease. Lance, 363(9406), 392–394.CrossRefGoogle Scholar
  13. Grenander, U., & Miller, M. I. (1998). Computational anatomy: An emerging discipline. Quarterly of Applied Mathematics, LVI(4), 617–694.MathSciNetGoogle Scholar
  14. Hinkle, J., Muralidharan, P., Fletcher, P. T., & Joshi, S. (2012). Polynomial regression on riemannian manifolds. In Computer Vision–ECCV 2012 (pp. 1–14). New York: Springer.Google Scholar
  15. Hong, Y., Joshi, S., Sanchez, M., Styner, M., & Niethammer, M. (2012). Metamorphic geodesic regression. In N. Ayache, H. Delingette, P. Golland, & K. Mori (Eds.), Medical image computing and computer-assisted intervention âĂŞ MICCAI 2012. Lecture Notes in Computer Science (Vol. 7512, pp. 197–205). Berlin: Springer. doi: 10.1007/978-3-642-33454-2_25.
  16. Laird, N. M., & Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics, 38(4), 963–974.CrossRefMATHGoogle Scholar
  17. Lorenzi, M., Ayache, N., Frisoni, G. B., & Pennec, X. (2011). Mapping the effects of Ab142 levels on the longitudinal changes in healthy aging: Hierarchical modeling based on stationary velocity fields. In: MICCAI 2011. Heidelberg: Springer.Google Scholar
  18. Lorenzi, M., Pennec, X., Ayache, N., & Frisoni, G. (2012). Disentangling the normal aging from the pathological Alzheimer’s disease progression on cross-sectional structural MR images. MICCAI Workshop on Novel Imaging Biomarkers for Alzheimer’s Disease and Related Disorders (NIBAD’12) (pp. 145–154). France: Nice.Google Scholar
  19. Marcus, D. S., Fotenos, A. F., Csernansky, J. G., Morris, J. C., & Buckner, R. L. (2010). Open access series of imaging studies: Longitudinal mri data in nondemented and demented older adults. Journal of Cognitive Neuroscience, 22(12), 2677–2684.CrossRefGoogle Scholar
  20. Micheli, M., Michor, P. W., & Mumford, D. (2012). Sectional curvature in terms of the cometric, with applications to the riemannian manifolds of landmarks. SIAM Journal on Imaging Sciences, 5(1), 394–433.CrossRefMathSciNetMATHGoogle Scholar
  21. Miller, M., Banerjee, A., Christensen, G., Joshi, S., Khaneja, N., Grenander, U., et al. (1997). Statistical methods in computational anatomy. Statistical Methods in Medical Research, 6(3), 267–299.CrossRefGoogle Scholar
  22. Miller, M. I. (2004). Computational anatomy: Shape, growth, and atrophy comparison via diffeomorphisms. NeuroImage, 23, 19–33.CrossRefGoogle Scholar
  23. Miller, M. I., Trouvé, A., & Younes, L. (2006). Geodesic shooting for computational anatomy. Journal of Mathematical Imaging and Vision, 24(2), 209–228.Google Scholar
  24. Muralidharan, P., & Fletcher, P. (2012). Sasaki metrics for analysis of longitudinal data on manifolds. In: IEEE Conference on CVPR (pp. 1027–1034).Google Scholar
  25. Niethammer, M., Huang, Y., & Vialard, F. X. (2011). Geodesic regression for image time-series. In MICCAI 2011 (Vol. 6892, pp. 655–662). Berlin: Springer.Google Scholar
  26. Pinheiro, J. C., & Bates, D. M. (1995). Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics, 4(1), 12–35.Google Scholar
  27. Raz, N., & Rodrigue, K. M. (2006). Differential aging of the brain: Patterns, cognitive correlates and modifiers. Neuroscience & Biobehavioral Reviews, 30(6), 730–748.CrossRefGoogle Scholar
  28. Reuter, M., Rosas, H. D., & Fischl, B. (2010). Highly accurate inverse consistent registration: A robust approach. NeuroImage, 53(4), 1181–1196.CrossRefGoogle Scholar
  29. Reuter, M., Schmansky, N. J., Rosas, H. D., & Fischl, B. (2012). Within-subject template estimation for unbiased longitudinal image analysis. NeuroImage, 61(4), 1402–1418.CrossRefGoogle Scholar
  30. Singh, N., & Niethammer, M. (2014). Splines for diffeomorphic image regression. In: P. Golland, N. Hata, C. Barillot, J. Hornegger, & R. Howe (Eds.), Medical image computing and computer-assisted intervention âĂŞ MICCAI 2014. Lecture Notes in Computer Science (Vol. 8674, pp. 121–129). Springer. doi: 10.1007/978-3-319-10470-6_16.
  31. Singh, N., Hinkle, J., Joshi, S., & Fletcher, P. (2013a). A hierarchical geodesic model for diffeomorphic longitudinal shape analysis. In: J. Gee, S. Joshi, K. Pohl, W. Wells, & L. ZÃűllei (Eds.), Information processing in medical imaging. Lecture Notes in Computer Science (Vol. 7917, pp. 560–571). Berlin: Springer.Google Scholar
  32. Singh, N., Hinkle, J., Joshi, S., & Fletcher, P. (2013b). A vector momenta formulation of diffeomorphisms for improved geodesic regression and atlas construction. In: 2013 IEEE 10th International Symposium on Biomedical imaging (ISBI) (pp. 1219–1222). doi: 10.1109/ISBI.2013.6556700
  33. Singh, N., Hinkle, J., Joshi, S., & Fletcher, P. (2014). An efficient parallel algorithm for hierarchical geodesic models in diffeomorphisms. In: Proceedings of the 2014 IEEE International Symposium on Biomedical Imaging (ISBI).Google Scholar
  34. Sowell, E. R., Peterson, B. S., Thompson, P. M., Welcome, S. E., Henkenius, A. L., & Toga, A. W. (2003). Mapping cortical change across the human life span. Nature Neuroscience, 6, 309–315.CrossRefGoogle Scholar
  35. Thompson, D. W. (1942). On growth and form.Google Scholar
  36. Thompson, P. M., & Toga, A. W. (2002). A framework for computational anatomy. Computing and Visualization in Science, 5(1), 13–34.CrossRefMATHGoogle Scholar
  37. Winer, B. J. (1962). Statistical principles in experimental design. New York: McGraw-Hill Book Company.CrossRefGoogle Scholar
  38. Younes, L. (2010). Shapes and diffeomorphisms (Vol. 171). New York: Springer.MATHGoogle Scholar
  39. Younes, L., Qiu, A., Winslow, R. L., & Miller, M. I. (2008). Transport of relational structures in groups of diffeomorphisms. Journal of Mathematical Imaging and Vision, 32(1), 41–56.CrossRefMathSciNetGoogle Scholar
  40. Younes, L., Arrate, F., & Miller, M. I. (2009). Evolution equations in computational anatomy. NeuroImage, 45(1 Suppl), S40–S50.CrossRefGoogle Scholar
  41. Zhang, M., Singh, N., & Fletcher, P. (2013). Bayesian estimation of regularization and atlas building in diffeomorphic image registration. In: J. Gee, S. Joshi, K. Pohl, W. Wells, & L. ZÃűllei (Eds.), Information processing in medical imaging. Lecture Notes in Computer Science (Vol. 7917, pp. 37–48). Berlin: Springer. doi: 10.1007/978-3-642-38868-2_4

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Nikhil Singh
    • 1
  • Jacob Hinkle
    • 2
  • Sarang Joshi
    • 2
  • P. Thomas Fletcher
    • 2
  1. 1.University of North Carolina at Chapel HillChapel HillUSA
  2. 2.University of UtahSalt Lake CityUSA

Personalised recommendations