Skip to main content
SpringerLink
Log in

Quaternion Diffusion for Color Image Filtering

  • Published:
Journal of Computer Science and Technology Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  3. Witkin A. Scale Space Filtering. In Proc. Int. Joint Conf. Artificial Intelligence, 1983, pp.1019–1023.

  4. Florack L, Kuijper A. The topological structure of scale-space images. J. Mathematical Imaging and Vision, 2000, 12: 65–79.

    MathSciNet  Google Scholar 

  5. Sapiro G. Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, 2001.

  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. 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. Weickert J. Coherence-enhancing diffusion of colour images. Image and Vision Computing, 1999, 17: 201–212.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  10. Yu Z Y, Bajaj C. Anisotropic vector diffusion in image smoothing. IEEE Proc. Image Processing, 2002, 1: 828–831.

    Google Scholar 

  11. Gerig G, Kübler O, Kikinis R, Jolesz F A. Nonlinear anisotropic filtering of MRI data. IEEE Trans. Medical Imaging, 1992, 11: 221–232.

    Article  Google Scholar 

  12. Sochen N, Kimmel R, Malladi R. A general framework for low level vision. IEEE Trans. Image Processing, 1998, 7(3): 310–318.

    MathSciNet  Google Scholar 

  13. Boccignone G, Ferraro M, Caelli T. Generalized spatio-chromatic diffusion. IEEE Trans. Pattern Analysis and Machine Intelligence, 2002, 24(10): 1298–1309.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

  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.

    Article  MathSciNet  Google Scholar 

  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.

  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.

  19. Felsberg M, Sommer G. The monogenic signal. IEEE Trans. Signal Processing, 2001, 49(12): 3136–3144.

    Article  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  22. Hamilton W R. Elements of Quaternions. London: Longmans, Green and Co., 1866.

    Google Scholar 

  23. Ebbinghaus H D, Hirzebruch F, Hermes H et al. Numbers. New York: Springer-Verlag, 1990.

    Google Scholar 

  24. Baylis W E (ed.). Clifford (Geometric) Algebras with Applications in Physics, Mathematics, and Engineering. Birkhäser-Boston, Inc., p.4.

  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. http://www.prometheus-inc.com/asi/algebra2003/lecturers.html

  26. Chou J C K. Quaternion kinematic and dynamic differential equations. IEEE Trans. Robotics and Automation, 1992, 8(1): 53–64.

    Article  Google Scholar 

  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.

  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.

    Article  MathSciNet  Google Scholar 

  29. Girard P R. Einstein's equations and Clifford algebra. Advances in Applied Clifford Algebras, 1999, 9(2): 225–230.

    MATH  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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. Gupta S. Linear quaternion equation with application to spacecraft attitude propagation. IEEE Proc. Aerospace Conference, 1998, 1: 69–76.

    Google Scholar 

  33. Prudnikov A P, Brychkov Y A, Marichev O I. Integrals and Series. (English translation by N.M Queen), I, Gordon and Breach, 1986.

  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.

  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 

Download references

Author information

Authors and Affiliations

  1. France Telecom R &D Beijing, Beijing, 100080, P.R. China

    Zhong-Xuan Liu, Shi-Guo Lian & Zhen Ren

Authors
  1. Zhong-Xuan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shi-Guo Lian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhen Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhong-Xuan Liu.

Additional information

Zhong-Xuan Liu was born in 1979. He received the Bachelor's degree in control theory from Shandong University in 2000 and Ph.D. degree in pattern recognition and image processing from Institute of Automation, Chinese Academy of Sciences in 2005. He is currently a researcher for video analysis and compression in Service Anticipation Multimedia Innovation (SAMI) Lab of France Telecom R&D Center (Beijing). He has published about 20 papers including 3 international journal papers (as the first author) and more than 10 international conference papers. His research interests include 2-D Empirical mode decomposition and nonlinear diffusion based image processing techniques, video watermarking, video analysis, and texture processing.

Shi-Guo Lian was born in 1979. He received the B.A.Sc. and Ph.D. degrees in information security from Nanjing University of Science & Technology in 2000 and 2005, respectively. He worked as a research assistant in the Department of Electronic Engineering of City University of Hong Kong from March 2004 to June 2004. He is now a researcher in Service Anticipation Multimedia Innovation (SAMI) Lab of France Telecom R&D Center (Beijing). His research interests include multimedia content protection, multimedia signal processing, chaos-based cipher and data authentication.

Zhen Ren received the M.S. degree in Southeast University, China and the Ph.D. degree from Grenoble National Institute of Technology, France. From Sep. 84 to Sep. 87, she was as computer engineer in Hydroelectricity Institute of Nankin of Hehai University, China; from Jan. 1997 to Mar. 2004, she has been worked in COSCO France S.A, Mitsubishi Electric Telecom France, French Embassy in China and Paycool Ltd; From Apr. 2004 to present, she has been in Multimedia and VAS Lab, France Telecom R&D Beijing. Her research interests include multimedia content protection and image/video processing.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, ZX., Lian, SG. & Ren, Z. Quaternion Diffusion for Color Image Filtering. J Comput Sci Technol 21, 126–136 (2006). https://doi.org/10.1007/s11390-006-0126-5

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11390-006-0126-5

Keywords

Search

Navigation