Advertisement

Advances in Applied Clifford Algebras

, Volume 23, Issue 4, pp 929–949 | Cite as

Hilbert and Riesz Transforms Using Atomic Function for Quaternionic Phase Computation

  • E. U. Moya-Sánchez
  • E. Bayro-Corrochano
Article
  • 221 Downloads

Abstract

Complex and hyper-complex valued filtering play a substantial role in signal processing, especially to obtain local features in the frequency and phase domain. In the case of 1D signals, the analytic signal is typically computed using the Hilbert transform. Such complex representation allows us to compute the phase and magnitude of the signal. For high-dimension signals, the Riesz transform and partial Hilbert transforms have been used as extensions of the analytic signal to compute the local phase and local orientation. A major goal of this work is to highlight the role of an atomic function (AF) up(x) as a kernel for the Hilbert and Riesz transforms. It is well known that the use of phase information has a big potential because it is invariant to illumination and rotations. In this regard, we show the advantages to carry out computations of local Riesz phase using the atomic function instead of the classical global-phase approach. In addition, we explain how the atomic function up(x), formulated in the quaternionic algebra framework, can be used as a building block to perform multiple analytical operations commonly used in image processing, such as low-pass filter, derivatives, local phase processing, steering quaternionic filters, multi-resolution analysis and symmetries detection.

Keywords

Quaternionic phase Hilbert Transform Riesz Transform 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. Doran and A. Lasenby, Geometric Algebra for Physicists. Cambridge University Press, 2003.Google Scholar
  2. 2.
    H. Pérez-Meana, V. F. Kravchenko, and V. I. Ponomaryov, Adaptive digital processing of multidimensional signals with applications. Moscow Fizmatlit, 2010.Google Scholar
  3. 3.
    V. F. Kravchenko, A. S. Gorshkov, and V. A. Rvachev, Estimation of the discrete derivative of a signal on the basis of atomic functions. Izmer. Tekhnika 1 (1992), pp. 8–10.Google Scholar
  4. 4.
    E. U. Moya-Sánchez and E. Bayro-Corrochano, Quaternion atomic function wavelet for applications in image processing. In Image Analysis, Computer Vision, and Applications, Proceedings of the fifteen Iberoamerican Conference on Pattern Recognition, CIARP’2010, LNCS 6419 Springer, pages 2010, pp. 346–353.Google Scholar
  5. 5.
    V. M. Kolodyazhny, and V. A. Rvachov, Atomic radial basis functions in numerical algorithms for solving boundary-value problems for the Laplace equation. Cybernetics and Systems Analysis 44 (2008), pp. 4–15.Google Scholar
  6. 6.
    G, Granlund and H, Kutsson, Signal Processing for Computer Vision. Lippincott Williams and Wilkins, 2002.Google Scholar
  7. 7.
    M. Felsberg, Low-level image processing with the structure multivector. PhD thesis, Christian-Albrecht, Kiel University, Kiel, Germany, 2002.Google Scholar
  8. 8.
    T. Bülow, Hypercomplex spectral signal representations for the processing and analysis of images. PhD thesis, Christian-Albrecht, Kiel University, Kiel, Germany, 1999.Google Scholar
  9. 9.
    J. Bernd, Digital Image Processing. Springer-Verlag, 1993.Google Scholar
  10. 10.
    J. Bigun, Vision with Direction. Springer, 2006.Google Scholar
  11. 11.
    V. M. Kolodyazhnya and V. A. Rvachov, Atomic functions: Generalization to the multivariable case and promising applications. Cybernetics and Systems Analysis, 46(6) 2007.Google Scholar
  12. 12.
    Oleg Kravchenko, Approximate Solutions of a Functional Differential Equation. From the Wolfram Demonstrations Project http://demonstrations.wolfram.com/ApproximateSolutionsOfAFunctionalDifferentialEquation/
  13. 13.
    E. Bayro Corrochano, Geometric Computing for Wavelet Transforms, Robot Vision, Learning, Control and Action. Springer, 2010.Google Scholar
  14. 14.
    E. U. Moya-Sánchez and E. Vázquez-Santacruz, A geometric bio-inspired model for recognition of low-level structures. In ICANN 2011, Part II, LNCS 6792, Springer 2011, pages pp. 429–436.Google Scholar
  15. 15.
    P. Kovesi, Invariant measures of image features from phase information. PhD thesis, University of Western Australia, 1996.Google Scholar
  16. 16.
    P. Kovesi, Image Features From Phase Congruency. A Journal of Computer Vision Research. MIT Press. Volume 1, Number 3, Summer 1999.Google Scholar
  17. 17.
    P. Kovesi, Symmetry and Asymmetry From Local Phase AI 97. Tenth Australian Joint Conference on Artificial Intelligence, 2 - 4 December 1997. Proceedings - Poster Papers. pp. 185-190.Google Scholar
  18. 18.
    W. Lennart, Local feature detection by higher order Riesz transforms on images. Master’s thesis, Christian-Albrecht, Kiel University, Kiel Germany, 2008.Google Scholar
  19. 19.
    E. Bayro Corrochano, The theory and use of the quaternion wavelet transform. Journal of Mathematical Imaging and Vision 24 (2006), pp. 19–35.Google Scholar
  20. 20.
    B. Svensson, A multidimensional filtering framework with applications to local structure analysis and image enhancement. PhD thesis, Linköping University, Linköping, Sweden, 2008.Google Scholar
  21. 21.
    V. M. Kolodyazhnya and V. A. Rvachev, Cybernetics and Systems Analysis 43 (2007).Google Scholar
  22. 22.
    S. Treil, S. Petermichl, and A. Volberg, hy the Riesz transforms are averages of the dyadic shifts?. Publ. Mat., 46(6), 2002.Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.CINVESTAV Campus GuadalajaraZapopanMexico

Personalised recommendations