Energy Tensors: Quadratic, Phase Invariant Image Operators

  • Michael Felsberg
  • Erik Jonsson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3663)


In this paper we briefly review a not so well known quadratic, phase invariant image processing operator, the energy operator, and describe its tensor-valued generalization, the energy tensor. We present relations to the real-valued and the complex valued energy operators and discuss properties of the three operators. We then focus on the discrete implementation for estimating the tensor based on Teager’s algorithm and frame theory. The kernels of the real-valued and the tensor-valued operators are formally derived. In a simple experiment we compare the energy tensor to other operators for orientation estimation. The paper is concluded with a short outlook to future work.


Energy Operator Structure Tensor Dual Frame Energy Tensor Orientation Error 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Förstner, W., Gülch, E.: A fast operator for detection and precise location of distinct points, corners and centres of circular features. In: ISPRS Intercommission Workshop, Interlaken, pp. 149–155 (1987)Google Scholar
  2. 2.
    Bigün, J., Granlund, G.H.: Optimal orientation detection of linear symmetry. In: Proceedings of the IEEE First International Conference on Computer Vision, London, Great Britain, pp. 433–438 (1987)Google Scholar
  3. 3.
    Harris, C.G., Stephens, M.: A combined corner and edge detector. In: 4th Alvey Vision Conference, pp. 147–151 (1988)Google Scholar
  4. 4.
    Granlund, G.H., Knutsson, H.: Signal Processing for Computer Vision. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar
  5. 5.
    Jähne, B.: Digital Image Processing. Springer, Berlin (2002)zbMATHGoogle Scholar
  6. 6.
    Kovesi, P.: Image features from phase information. Videre: Journal of Computer Vision Research 1 (1999)Google Scholar
  7. 7.
    Kaiser, J.F.: On a simple algorithm to calculate the ’energy’ of a signal. In: Proc. IEEE Int’l. Conf. Acoust., Speech, Signal Processing, Albuquerque, New Mexico, pp. 381–384 (1990)Google Scholar
  8. 8.
    Potamianos, A., Maragos, P.: A comparison of the energy operator and the Hilbert transform approach to signal and speech demodulation. Signal Processing 37, 95–120 (1994)zbMATHCrossRefGoogle Scholar
  9. 9.
    Maragos, P., Bovik, A.C., Quartieri, J.F.: A multi-dimensional energy operator for image processing. In: SPIE Conference on Visual Communications and Image Processing, Boston, MA, pp. 177–186 (1992)Google Scholar
  10. 10.
    Larkin, K.G., Oldfield, M.A., Bone, D.J.: Demodulation and phase estimation of two-dimensional patterns. Australian patent AU 200110005 A1 (2001)Google Scholar
  11. 11.
    Felsberg, M., Granlund, G.: POI detection using channel clustering and the 2D energy tensor. In: 26. DAGM Symposium, Mustererkennung, Tübingen (2004)Google Scholar
  12. 12.
    Felsberg, M., Köthe, U.: Get: The connection between monogenic scale-space and gaussian derivatives. In: Proc. Scale Space Conference (2005)Google Scholar
  13. 13.
    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
  14. 14.
    Bovik, A.C., Maragos, P.: Conditions for positivity of an energy operator. IEEE Transactions on Signal Processing 42, 469–471 (1994)CrossRefGoogle Scholar
  15. 15.
    Weickert, J., Scharr, H.: A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance. Journal of Visual Communication and Image Representation, Special Issue On Partial Differential Equations In Image Processing, Computer Vision, And Computer Graphics, 103–118 (2002)Google Scholar
  16. 16.
    Knutsson, H., Andersson, M.: Robust N-dimensional orientation estimation using quadrature filters and tensor whitening. In: Proceedings of IEEE International Conference on Acoustics, Speech, & Signal Processing, Adelaide, Australia. IEEE, Los Alamitos (1994)Google Scholar
  17. 17.
    Burg, K., Haf, H., Wille, F.: Höhere Mathematik für Ingenieure, Band IV Vektoranalysis und Funktionentheorie. Teubner Stuttgart (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Michael Felsberg
    • 1
  • Erik Jonsson
    • 1
  1. 1.Computer Vision LaboratoryLinköping UniversityLinköpingSweden

Personalised recommendations