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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)

Abstract

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.

Keywords

orientation representation angle doubling projection operators 

References

  1. 1.
    Granlund, G.H.: In search of a general picture processing operator. Computer Graphics and Image Processing 8, 155–173 (1978)CrossRefGoogle Scholar
  2. 2.
    Bigün, J., Granlund, G.H.: Optimal orientation detection of linear symmetry. In: Proceedings of the First IEEE International Conference on Computer Vision, London, pp. 433–438. IEEE Computer Society Press, Los Alamitos (1987)Google Scholar
  3. 3.
    Kass, M., Witkin, A.: Analyzing oriented patterns. Computer Vision, Graphics and Image Processing 37, 362–385 (1987)CrossRefGoogle Scholar
  4. 4.
    Knutsson, H.: Producing a continuous and distance preserving 5-d vector representation of 3-d orientation. In: IEEE Computer Society Workshop on Computer Architecture for Pattern Analysis and Image Database Management, Miami Beach, Florida, pp. 175–182 (1985)Google Scholar
  5. 5.
    Knutsson, H.: Representing local structure using tensors. In: The 6th Scandinavian Conference on Image Analysis, Oulu, Finland, pp. 244–251 (1989)Google Scholar
  6. 6.
    Rieger, B., van Vliet, L.J.: A systematic approach to nD orientation representation. Image and Vision Computing 22, 453–459 (2004)CrossRefGoogle Scholar
  7. 7.
    Granlund, G.H., Knutsson, H.: Signal processing for computer vision. Kluwer Academic Publishers, Boston (1995)CrossRefGoogle Scholar
  8. 8.
    Jähne, B.: Digital Image Processing, 4th edn. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  9. 9.
    van Ginkel, M., van de Weijer, J., Verbeek, P.W., van Vliet, L.J.: Curvature estimation from orientation fields. In: Ersboll, B.K., Johansen, P. (eds.) SCIA 1999, Proc. 11th Scandinavian Conference on Image Analysis, Kangerlussuaq, Greenland, pp. 545–551. Pattern Recognition Society of Denmark, Lyngby (1999)Google Scholar
  10. 10.
    Bigün, J., Granlund, G.H., Wiklund, J.: Multidimensional orientation estimation with applications to texture analysis and optical flow. IEEE Transactions on Pattern Analysis and Machine Intelligence 13, 775–790 (1991)CrossRefGoogle Scholar
  11. 11.
    Rieger, B., van Vliet, L.J.: Curvature of n-dimensional space curves in grey-value images. IEEE Transactions on Image Processing 11, 738–745 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Rieger, B., van Vliet, L.J., Verbeek, P.W.: Estimation of curvature based shape properties of surfaces in 3D grey-value images. In: Bigun, J., Gustavsson, T. (eds.) SCIA 2003. LNCS, vol. 2749, pp. 262–267. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    van Noorden, S., Caan, M.W.A., van der Graaf, M., van Vliet, L.J., Vos, F.M.: A comparison of the cingulum tract in ALS-B patients and controls using kernel matching. In: Jiang, T., Navab, N., Pluim, J.P.W., Viergever, M.A. (eds.) MICCAI 2010. LNCS, vol. 6362, pp. 249–256. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Westin, C.-F., Knutsson, H.: The Möbius strip parameterization for line extraction. In: Sandini, G. (ed.) ECCV 1992. LNCS, vol. 588, pp. 33–38. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  15. 15.
    Westin, C.F.: A Tensor Framework for Multidimensional Signal Processing. PhD thesis, Linköping University, Linköping, Sweden (1994)Google Scholar
  16. 16.
    Mühlich, M., Aach, T.: Analysis of multiple orientations. IEEE Transactions on Image Processing 18, 1424–1437 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Franken, E.M., Duits, R., Haar Romenij, B.M.: Nonlinear diffusion on the 2D euclidean motion group. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 461–472. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Faas, F.G.A., van Vliet, L.J.: 3D-orientation space; filters and sampling. In: Bigun, J., Gustavsson, T. (eds.) SCIA 2003. LNCS, vol. 2749, pp. 36–42. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  19. 19.
    Herberthsson, M., Brun, A., Knutsson, H.: Pairs of orientation in the plane. In: Proceedings of the SSBA Symposium on Image Analysis. SSBA (2006)Google Scholar
  20. 20.
    Stein, E., Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)zbMATHGoogle Scholar

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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