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Manifold Valued Statistics, Exact Principal Geodesic Analysis and the Effect of Linear Approximations

  • Stefan Sommer
  • François Lauze
  • Søren Hauberg
  • Mads Nielsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6316)

Abstract

Manifolds are widely used to model non-linearity arising in a range of computer vision applications. This paper treats statistics on manifolds and the loss of accuracy occurring when linearizing the manifold prior to performing statistical operations. Using recent advances in manifold computations, we present a comparison between the non-linear analog of Principal Component Analysis, Principal Geodesic Analysis, in its linearized form and its exact counterpart that uses true intrinsic distances. We give examples of datasets for which the linearized version provides good approximations and for which it does not. Indicators for the differences between the two versions are then developed and applied to two examples of manifold valued data: outlines of vertebrae from a study of vertebral fractures and spacial coordinates of human skeleton end-effectors acquired using a stereo camera and tracking software.

Keywords

Vertebral Fracture Tangent Space Hand Position Stereo Camera Human Skeleton 
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 2010

Authors and Affiliations

  • Stefan Sommer
    • 1
  • François Lauze
    • 1
  • Søren Hauberg
    • 1
  • Mads Nielsen
    • 1
    • 2
  1. 1.Dept. of Computer ScienceUniv. of CopenhagenDenmark
  2. 2.Nordic Bioscience ImagingHerlevDenmark

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