Discrete Geodesic Regression in Shape Space

  • Benjamin Berkels
  • P. Thomas Fletcher
  • Behrend Heeren
  • Martin Rumpf
  • Benedikt Wirth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8081)


A new approach for the effective computation of geodesic regression curves in shape spaces is presented. Here, one asks for a geodesic curve on the shape manifold that minimizes a sum of dissimilarity measures between given two- or three-dimensional input shapes and corresponding shapes along the regression curve. The proposed method is based on a variational time discretization of geodesics. Curves in shape space are represented as deformations of suitable reference shapes, which renders the computation of a discrete geodesic as a PDE constrained optimization for a family of deformations. The PDE constraint is deduced from the discretization of the covariant derivative of the velocity in the tangential direction along a geodesic. Finite elements are used for the spatial discretization, and a hierarchical minimization strategy together with a Lagrangian multiplier type gradient descent scheme is implemented. The method is applied to the analysis of root growth in botany and the morphological changes of brain structures due to aging.


Sugar Beet Regression Curve Rigid Body Motion Shape Space Geodesic Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Benjamin Berkels
    • 1
  • P. Thomas Fletcher
    • 2
  • Behrend Heeren
    • 1
  • Martin Rumpf
    • 1
  • Benedikt Wirth
    • 3
  1. 1.Institute for Numerical SimulationUniversität BonnGermany
  2. 2.Scientific Computing and Imaging InstituteUniversity of UtahUSA
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityUSA

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