Advertisement

Journal of Mathematical Imaging and Vision

, Volume 24, Issue 1, pp 19–35 | Cite as

The Theory and Use of the Quaternion Wavelet Transform

  • Eduardo Bayro-Corrochano
Article

Abstract

This paper presents the theory and practicalities of the quaternion wavelet transform (QWT). The major contribution of this work is that it generalizes the real and complex wavelet transforms and derives a quaternionic wavelet pyramid for multi-resolution analysis using the quaternionic phase concept. As a illustration we present an application of the discrete QWT for optical flow estimation. For the estimation of motion through different resolution levels we use a similarity distance evaluated by means of the quaternionic phase concept and a confidence mask. We show that this linear approach is amenable to be extended to a kind of quadratic interpolation.

Keywords:

image processing real and complex wavelets multi-resolution analysis wavelet pyramid quaternion wavelets quaternion wavelet pyramid disparity estimation optical flow 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Bayro-Corrochano, Geometric Computing for Perception Action Systems, Springer Verlag: Boston, 2001.Google Scholar
  2. 2.
    Ch. Bernard, 1997. “Discrete wavelet analysis for fast optic flow computation,” Applied and Computational Harmonic Analysis, Vol. 11, No. 1, pp. 32–63, 2001.Google Scholar
  3. 3.
    T. Bülow, Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images, PhD. thesis, University Christian Albrechts University of Kiel, 1999.Google Scholar
  4. 4.
    I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics: Philadelphia, 1992.Google Scholar
  5. 5.
    G.H. Granlund and H. Knutsson, Signal Processing for Computer Vision, Kluwer Academic Publishers: Dordrecht, The Netherlands, 1995.Google Scholar
  6. 6.
    D.J. Fleet and A.D. Jepson, “Computation of component image velocity from local phase information,” Int. Journal on Computer Vision, No. 5, pp. 77–104, 1990.Google Scholar
  7. 7.
    W.R. Hamilton, Elements of Quaternions, Longmans Green, London 1866. Chelsea, New York, 1969.Google Scholar
  8. 8.
    N. Kingsbury, “Image processing with complex wavelets,” Phil. Trans. R. Soc. Lond. A, Vol. 357, pp. 2543–2560, 1999.Google Scholar
  9. 9.
    J.-M. Lina, Complex Daubechies Wavelets: Filters Desing and Aplications, ISAAC Conference, Univ. of Delaware, June 1997.Google Scholar
  10. 10.
    J.F.A. Magarey and N.G. Kingsbury, “Motion estimation using a complex-valued wavelet transform,” IEEE Trans. Image Proc. Vol. 6, pp. 549–565, 1998.Google Scholar
  11. 11.
    G. Kaiser, A Friendly Guide to Wavelets, Birkhauser: Cambridge, USA, 1994.Google Scholar
  12. 12.
    S. Mallat, A Wavelet Tour of Signal Processing, 2nd edition, Academic Press: San Diego, CA, 1998.Google Scholar
  13. 13.
    S. Mallat, “A Theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Patt. Anal. and Mach. Intell., Vol. 11, No. 7, pp. 674–693, 1989.zbMATHGoogle Scholar
  14. 14.
    S. Mallat, A Wavelet Tour of Signal Processing, 2nd edition, Academic Press: San Diego, CA, 2001.Google Scholar
  15. 15.
    M. Mitrea, Clifford Waveletes, Singular Integrals and Hardy Spaces, Lecture Notes in Mathematics 1575, Spinger Verlag, 1994.Google Scholar
  16. 16.
    H.-P., Pan, “Uniform full information image matching complex conjugate wavelet pyramids,” XVIII ISPRS Congress, Viena, Vol. XXXI, July 1996.Google Scholar
  17. 17.
    L. Traversoni, “Image analysis using quaternion wavelet,” in Geometric Algebra in Science and Engineering Book, E. Bayro Corrochano and G. Sobczyk (Eds.), Springer Velag, 2001, Chap. 16.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Eduardo Bayro-Corrochano
    • 1
  1. 1.Computer Science Department, GEOVIS LaboratoryCentro de Investigación y de Estudios Avanzados, CINVESTAVGuadalajaraMexico

Personalised recommendations