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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
Article

Abstract

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.

Keywords

Longitudinal modeling Diffeomorphisms Mixed-effects modeling LDDMM 

Notes

Acknowledgments

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.

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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

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