Journal of Computer Science and Technology

, Volume 21, Issue 1, pp 126–136 | Cite as

Quaternion Diffusion for Color Image Filtering

  • Zhong-Xuan LiuEmail author
  • Shi-Guo Lian
  • Zhen Ren


How to combine color and multiscale information is a fundamental question for computer vision, and quite a few color diffusion techniques have been presented. Most of these proposed techniques do not consider the direct interactions between color channel pairs. In this paper, a new method of color diffusion considering these effects is presented, which is based on quaternion diffusion (QD) equation. In addition to showing the solution to linear QD and its analysis, one form of nonlinear QD is discussed. Compared with other color diffusion techniques, considering the interactions between channel pairs, QD has the following advantages: 1) staircasing effect is avoided; 2) as diffusion tensor, the image derivative is regularized without requiring additional convolution; 3) less time is needed. Experimental results demonstrate the effectiveness of linear and nonlinear QD applied to natural color images for denoising by both visual and quantitative evaluations.


color image processing nonlinear filtering quaternion diffusion quaternion partial differential equation 


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  1. 1.
    Morrone M C, Denti V, Spinelli D. Different attentional resources modulated the gain mechanisms for color and luminance contrast. Vision Research, 2004, 44: 1389–1401.CrossRefGoogle Scholar
  2. 2.
    Teufel H J, Wehrhahn C. Chromatic induction in humans: how are the cone signals combined to provide opponent processing? Vision Research, 2004, 44: 2425–2435.CrossRefGoogle Scholar
  3. 3.
    Witkin A. Scale Space Filtering. In Proc. Int. Joint Conf. Artificial Intelligence, 1983, pp.1019–1023.Google Scholar
  4. 4.
    Florack L, Kuijper A. The topological structure of scale-space images. J. Mathematical Imaging and Vision, 2000, 12: 65–79.MathSciNetGoogle Scholar
  5. 5.
    Sapiro G. Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, 2001.Google Scholar
  6. 6.
    Lucchese L, Mitra S K. Color segmentation based on separate anisotropic diffusion of chromatic and achromatic channels. IEE Proc. Vision, Image and Signal Processing, 2001, 148(3): 141–150.Google Scholar
  7. 7.
    Sapiro G, Ringach D. Anisotropic diffusion of multivalued images with application to color filtering. IEEE Trans. Image Processing, 1996, 5(11): 1582–1586.Google Scholar
  8. 8.
    Weickert J. Coherence-enhancing diffusion of colour images. Image and Vision Computing, 1999, 17: 201–212.CrossRefGoogle Scholar
  9. 9.
    Tang B, Sapiro G, Caselles V. Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case. Int. J. Computer Vision, 2000, 36(2): 149–161.CrossRefGoogle Scholar
  10. 10.
    Yu Z Y, Bajaj C. Anisotropic vector diffusion in image smoothing. IEEE Proc. Image Processing, 2002, 1: 828–831.Google Scholar
  11. 11.
    Gerig G, Kübler O, Kikinis R, Jolesz F A. Nonlinear anisotropic filtering of MRI data. IEEE Trans. Medical Imaging, 1992, 11: 221–232.CrossRefGoogle Scholar
  12. 12.
    Sochen N, Kimmel R, Malladi R. A general framework for low level vision. IEEE Trans. Image Processing, 1998, 7(3): 310–318.MathSciNetGoogle Scholar
  13. 13.
    Boccignone G, Ferraro M, Caelli T. Generalized spatio-chromatic diffusion. IEEE Trans. Pattern Analysis and Machine Intelligence, 2002, 24(10): 1298–1309.CrossRefGoogle Scholar
  14. 14.
    Perona P, Malik J. Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Analysis and Machine Intelligence, 1990, 12(7): 629–639.CrossRefGoogle Scholar
  15. 15.
    Sangwine S J, Ell T A. Colour image filters based on hypercomplex convolution. In IEE Proc. Vision, Image and Signal Processing, 2000, 147(2): 89–93.Google Scholar
  16. 16.
    Moxey C E, Sangwine S J, Ell T A. Hypercomplex correlation techniques for vector images. IEEE Trans. Signal Processing, 2003, 51(7): 1941–1953.CrossRefMathSciNetGoogle Scholar
  17. 17.
    Soo C P, Chang J H, Ding J J. Quaternion matrix singular value decomposition and its application for color image processing. In Proc. IEEE International Conference on Image Processing'03, 2003, 1: 805–808.Google Scholar
  18. 18.
    Chan W L, Choi H, Baraniuk R. Quaternion wavelets for image analysis and processing. In Proc. IEEE International Conference on Image Processing, 2004, pp.2749–2752.Google Scholar
  19. 19.
    Felsberg M, Sommer G. The monogenic signal. IEEE Trans. Signal Processing, 2001, 49(12): 3136–3144.CrossRefMathSciNetGoogle Scholar
  20. 20.
    Felsberg M, Sommer G. The monogenic scale-space: A unifying approach to phase-based image processing in scale-space. J. Mathematical Imaging and Vision, 2004, 21: 5–26.MathSciNetGoogle Scholar
  21. 21.
    Gilboa G, Sochen N, Zeevi Y Y. Image enhancement and denoising by complex diffusion processes. IEEE Trans. Pattern Analysis and Machine Intelligence, 2004, 26(8): 1020–1036.CrossRefGoogle Scholar
  22. 22.
    Hamilton W R. Elements of Quaternions. London: Longmans, Green and Co., 1866.Google Scholar
  23. 23.
    Ebbinghaus H D, Hirzebruch F, Hermes H et al. Numbers. New York: Springer-Verlag, 1990.Google Scholar
  24. 24.
    Baylis W E (ed.). Clifford (Geometric) Algebras with Applications in Physics, Mathematics, and Engineering. Birkhäser-Boston, Inc., p.4.Google Scholar
  25. 25.
    Labunets V. Clifford algebras as unified language for image processing and pattern recognition. NATO Advanced Study Institute, Computational Noncommutative Algebra and Applications, July 6–19, 2003.
  26. 26.
    Chou J C K. Quaternion kinematic and dynamic differential equations. IEEE Trans. Robotics and Automation, 1992, 8(1): 53–64.CrossRefGoogle Scholar
  27. 27.
    Kähler U. Clifford analysis and the Navier-Stokes equations over unbounded domains. Advances in Applied Clifford Algebras, Special Issue on Clifford Analysis, 2001, 11(S2): 305–318.Google Scholar
  28. 28.
    Anastassiu H T, Atlamazoglou P E, Kaklamani D I. Application of bicomplex (quaternion) algebra to fundamental electromagnetics: A lower order alternative to the Helmholtz equation. IEEE Trans. Antennas and Propagation, 2003, 51(8): 2130–2136.CrossRefMathSciNetGoogle Scholar
  29. 29.
    Girard P R. Einstein's equations and Clifford algebra. Advances in Applied Clifford Algebras, 1999, 9(2): 225–230.zbMATHMathSciNetGoogle Scholar
  30. 30.
    Catte F, Lions P L, Morel J M, Coll T. Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal., 1992, 29(1): 182–192.MathSciNetGoogle Scholar
  31. 31.
    Petrovic A, Escoda O D, Vandergheynst P. Multiresolution segmentation of natural images: From linear to nonlinear scale-space representations. IEEE Trans. Image Processing, 2004, 13(8): 1104–1114.Google Scholar
  32. 32.
    Gupta S. Linear quaternion equation with application to spacecraft attitude propagation. IEEE Proc. Aerospace Conference, 1998, 1: 69–76.Google Scholar
  33. 33.
    Prudnikov A P, Brychkov Y A, Marichev O I. Integrals and Series. (English translation by N.M Queen), I, Gordon and Breach, 1986.Google Scholar
  34. 34.
    Bardos A J, Sangwine S J. Measuring noise in colour images. IEE Colloquium on Non-Linear Signal and Image Processing, May 1998, pp.8/1–8/4.Google Scholar
  35. 35.
    Trahanias P E, Venetsanopoulos A N. Vector directional filters: A new class of multi-channel image processing filters. IEEE Trans. Image Processing, 1993, 2(4): 528–534.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.France Telecom R &D BeijingBeijingP.R. China

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