QWT: Retrospective and New Applications

  • Yi Xu
  • Xiaokang Yang
  • Li Song
  • Leonardo TraversoniEmail author
  • Wei Lu


Quaternion wavelet transform (QWT) achieves much attention in recent years as a new image analysis tool. In most cases, it is an extension of the real wavelet transform and complex wavelet transform (CWT) by using the quaternion algebra and the 2D Hilbert transform of filter theory, where analytic signal representation is desirable to retrieve phase-magnitude description of intrinsically 2D geometric structures in a grayscale image. In the context of color image processing, however, it is adapted to analyze the image pattern and color information as a whole unit by mapping sequential color pixels to a quaternion-valued vector signal. This paper provides a retrospective of QWT and investigates its potential use in the domain of image registration, image fusion, and color image recognition. It is indicated that it is important for QWT to induce the mechanism of adaptive scale representation of geometric features, which is further clarified through two application instances of uncalibrated stereo matching and optical flow estimation. Moreover, quaternionic phase congruency model is defined based on analytic signal representation so as to operate as an invariant feature detector for image registration. To achieve better localization of edges and textures in image fusion task, we incorporate directional filter bank (DFB) into the quaternion wavelet decomposition scheme to greatly enhance the direction selectivity and anisotropy of QWT. Finally, the strong potential use of QWT in color image recognition is materialized in a chromatic face recognition system by establishing invariant color features. Extensive experimental results are presented to highlight the exciting properties of QWT.


Local Binary Pattern Stereo Match Quaternion Algebra Phase Congruency Contourlet Transform 
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.


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  1. 1.
    Bülow, T.: Hypercomplex spectral signal representations for the processing and analysis of images. Dissertation, Kiel: Institut für Informatik und Praktische Mathematik der Christian-Albrechts-Universitat zu Kiel (1999) Google Scholar
  2. 2.
    Corrochano, E.B.: The theory and use of the quaternion wavelet transform. J. Math. Imaging Vis. 24, 19–35 (2006) CrossRefGoogle Scholar
  3. 3.
    Chan, W.L., Choi, H., Baraniuk, R.: Quaternion wavelets for image analysis and processing. Proc. IEEE Int. Conf. Image Process. 5, 3057–3060 (2004) Google Scholar
  4. 4.
    Xu, Y., Zhou, J., Yang, X.: Quaternion wavelet phase based stereo matching for uncalibrated images. Pattern Recogn. Lett. 28(12), 1509–1522 (2007) CrossRefGoogle Scholar
  5. 5.
    Sangwine, S., Ell, T.: Hypercomplex Fourier transforms of color images. IEEE Trans. Image Process. 16(1), 22–35 (2007) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Jones, C., Abbott, A.: Color face recognition by hypercomplex Gabor analysis. In: 7th International Conference on Automatic Face and Gesture Recognition, pp. 126–131 (2006) Google Scholar
  7. 7.
    Lu, W., Xu, X.Y.Y., Song, L.: Local quaternionic Gabor binary patterns for color face recognition. In: International Conference on Acoustics, Speech, and Signal Processing, pp. 741–744 (2008) Google Scholar
  8. 8.
    Carré, P., Denis, P.: Quaternionic wavelet transform for colour images. Proc. SPIE 6383, 638 301/1–638 301/15 (2006). Invited paper Google Scholar
  9. 9.
    Hamilton, W.: Elements of Quaternions. Longman, Harlow (1866) Google Scholar
  10. 10.
    Felsberg, M.: Optimized fast algorithms for the quaternionic Fourier transform. In: Proc. 8th International Conference on Computer Analysis of Images and Patterns, vol. 1689, pp. 209–216 (1999) Google Scholar
  11. 11.
    Pei, S., Ding, J., Chang, J.: Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT. IEEE Trans. Signal Process. 49(11), 2783–2797 (2001) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Kingsbury, N.: Complex wavelets for shift invariant analysis and filtering of signals. Appl. Comput. Harmon. 10(3), 234–253 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    da Silva, E.A.B., Ghanbari, M.: On the performance of linear phase wavelet transform in low bitrate coding. IEEE Trans. Image Process. 5(5), 689–704 (1996) CrossRefGoogle Scholar
  14. 14.
    Gabor, D.: Theory of communication. J. IEE 93, 429–457 (1946) Google Scholar
  15. 15.
    Fleet, D., Jepson, A.: Stability of phase information. IEEE Trans. Pattern Anal. Mach. Intell. 15(12), 1253–1268 (1993) CrossRefGoogle Scholar
  16. 16.
    Lindeberg, T.: Feature detection with automatic scale selection. Int. J. Comput. Vis. 30(2), 79–116 (1998) CrossRefGoogle Scholar
  17. 17.
    Mikolajczyk, K., Tuytelaars, T.: A comparison of affine region detectors. Int. J. Comput. Vis. 65(1/2), 43–72 (2005) CrossRefGoogle Scholar
  18. 18.
    Kovesi, P.: Invariant feature measures of image features from phase information. Ph.D. Dissertation, University of Western Australia (1996) Google Scholar
  19. 19.
    Morrone, M., Owens, R.: Feature detection from local energy. Pattern Recogn. Lett. 6, 303–313 (1987) CrossRefGoogle Scholar
  20. 20.
    Mikolajczyk, K., Schmid, C.: A performance evaluation of local descriptors. IEEE Trans. Pattern Anal. Mach. Intell. 27(10), 1615–1630 (2005) CrossRefGoogle Scholar
  21. 21.
    Zhang, Z., Deriche, R., Faugeras, O., Luong, Q.: A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry. Artif. Intell. J. 78, 87–119 (1995) CrossRefGoogle Scholar
  22. 22.
    Do, M., Vetterli, M.: Contourlets. Beyond wavelets, pp. 1–27 (2001) Google Scholar
  23. 23.
    Lu, Z., Xu, Y., Yang, X., Song, L., Traversoni, L.: 2D quaternion Fourier transform: the spectrum properties and its application in color image registration. In: International Conference on Multimedia and Expo, pp. 1715–1718 (2007) Google Scholar
  24. 24.
    Ma, X., Xu, Y., Song, L., Yang, X., Burkhardt, H.: Color image watermarking using local quaternion Fourier spectral analysis. In: International Conference on Multimedia and Expo, pp. 233–236 (2008) Google Scholar
  25. 25.
    Ahonen, T., Hadid, A., Pietikainen, M.: Face description with local binary patterns: application to face recognition. IEEE Trans. Pattern Anal. Mach. Intell. 28(12), 2037–2041 (2006) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2010

Authors and Affiliations

  • Yi Xu
    • 1
  • Xiaokang Yang
    • 1
  • Li Song
    • 1
  • Leonardo Traversoni
    • 2
    Email author
  • Wei Lu
    • 1
  1. 1.Institute of Image Communication and Information Processing, Department of Electronic EngineeringShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Ciencias Básicas e IngenieriaUniv. Autonoma Met. (Iztapalapa)MexicoMexico

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