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)

Abstract

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.

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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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