About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 162769, 10 pages
http://dx.doi.org/10.1155/2013/162769
Research Article

Convolution Theorems for Quaternion Fourier Transform: Properties and Applications

1Department of Mathematics, Hasanuddin University, Makassar 90245, Indonesia
2Division of Mathematical Sciences, Osaka Kyoiku University, Osaka 582-8582, Japan
3Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada K1N 6N5

Received 1 June 2013; Revised 1 September 2013; Accepted 7 September 2013

Academic Editor: Narcisa C. Apreutesei

Copyright © 2013 Mawardi Bahri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Ebling and G. Scheuermann, “Clifford Fourier transform on vector fields,” IEEE Transactionson Visualization and Computer Graphics, vol. 11, no. 4, pp. 469–479, 2005. View at Publisher · View at Google Scholar
  2. E. Hitzer and B. Mawardi, “Clifford Fourier transform on multivector fields and uncertainty principles for dimensions n=2(mod4) and n=3(mod4),” Advances in Applied Clifford Algebras, vol. 18, no. 3-4, pp. 715–736, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. B. Mawardi and E. Hitzer, “Clifford Fourier transformation and uncertainty principle for the Clifford geometric algebra Cl3,0,” Advances in Applied Clifford Algebras, vol. 16, no. 1, pp. 41–61, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. H. de Bie and N. de Schepper, “Fractional Fourier transform of hypercomplex signals,” Signal, Image and Video Processing, vol. 6, no. 3, pp. 381–388, 2012. View at Publisher · View at Google Scholar
  5. X. Guanlei, W. Xiaotong, and X. Xiagang, “Fractional quaternion Fourier transform, convolutionand correlation,” Signal Processing, vol. 88, no. 10, pp. 2511–2517, 2008. View at Publisher · View at Google Scholar
  6. M. Bahri, E. Hitzer, R. Ashino, and R. Vaillancourt, “Windowed Fourier transform of two-dimensional quaternionic signals,” Applied Mathematics and Computation, vol. 216, no. 8, pp. 2366–2379, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. Bahri, R. Ashino, and R. Vaillancourt, “Two-dimensional quaternion wavelet transform,” Applied Mathematics and Computation, vol. 218, no. 1, pp. 10–21, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. Bahri, “Quaternion algebra-valued wavelet transform,” Applied Mathematical Sciences, vol. 5, no. 71, pp. 3531–3540, 2011. View at Zentralblatt MATH · View at MathSciNet
  9. S. J. Sangwine, “Biquaternion (complexified quaternion) roots of -1,” Advances in Applied Clifford Algebras, vol. 16, no. 1, pp. 63–68, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. T. Bülow, Hypercomplex spectral signal representations for the processing and analysis of images [Ph.D. thesis], University of Kiel, Kiel, Germany, 1999.
  11. T. A. Ell, “Quaternion-Fourier transformations for analysis of two-dimensional lineartime-invariant partial differential systems,” in Proceedings of the 32nd IEEE Conference on Decision and Control, vol. 2, pp. 1830–1841, San Antonio, Tex, USA, December 1993.
  12. E. Hitzer, “Quaternion Fourier transform on quaternion fields and generalizations,” Advances in Applied Clifford Algebras, vol. 17, no. 3, pp. 497–517, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. Bahri, E. Hitzer, A. Hayashi, and R. Ashino, “An uncertainty principle for quaternion Fourier transform,” Computers & Mathematics with Applications, vol. 56, no. 9, pp. 2411–2417, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. D. Assefa, L. Mansinha, K. F. Tiampo, H. Rasmussen, and K. Abdella, “Local quaternion Fourier transform and color images texture analysis,” Signal Processing, vol. 90, no. 6, pp. 1825–1835, 2010. View at Publisher · View at Google Scholar
  15. R. Bujack, G. Scheuermann, and E. Hitzer, “A general geometric fourier transform convolution theorem,” Advances in Applied Clifford Algebras, vol. 23, no. 1, pp. 15–38, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. R. Bujack, G. Scheuermann, and E. Hitzer, “A general geometric Fourier transform,” in Proceedings of the 9th International Conference on Clifford Algebras and Their Applications in Mathematical Physics (ICCA '11), K. Guerlebeck, Ed., Weimar, Germany, July 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. E. Hitzer, “Directional uncertainty principle for quaternion Fourier transform,” Advances in Applied Clifford Algebras, vol. 20, no. 2, pp. 271–284, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. G. B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1999.
  19. T. A. Ell and S. J. Sangwine, “Hypercomplex Fourier transforms of color images,” IEEE Transactions on Image Processing, vol. 16, no. 1, pp. 22–35, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  20. S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, Calif, USA, 2nd edition, 1999. View at MathSciNet
  21. M. Petrou and C. Petrou, Image Processing: The Fundamentals, John Wiley & Sons, West Sussex, UK, 2nd edition, 2010.
  22. L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer, Berlin, Germany, 1983. View at Publisher · View at Google Scholar · View at MathSciNet