Continuous Orientation Representation for Arbitrary Dimensions – A Generalized Knutsson Mapping

  • Bernd Rieger
  • Lucas J. van Vliet
  • Piet W. Verbeek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6688)


In this paper we present a framework to construct a continuous orientation representation in arbitrary dimensions. Existing methods for 2D (doubling the angle) and 3D (Knutsson mapping) were found ad hoc. We show how they can be put in a general framework to derive suitable representations for filtering in spaces of arbitrary dimension. The dimensionality of the derived representation is shown to be minimal. Connections with the gradient structure tensor and Knutsson mapping are shown, like the fact that angle doubling works in each pair-cone of the Knutsson mapping. Finally, using projection operators we show how angles between vectors in the base space are related to vectors in the mapped spaces and in particular how to achieve preservation of isotropy.


orientation representation angle doubling projection operators 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Bernd Rieger
    • 1
  • Lucas J. van Vliet
    • 1
  • Piet W. Verbeek
    • 1
  1. 1.Quantitative Imaging GroupDelft University of TechnologyDelftThe Netherlands

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