Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2017, Article ID 3795120, 11 pages
https://doi.org/10.1155/2017/3795120
Research Article

A Variation on Uncertainty Principle and Logarithmic Uncertainty Principle for Continuous Quaternion Wavelet Transforms

1Department of Mathematics, Hasanuddin University, Makassar 90245, Indonesia
2Division of Mathematical Sciences, Osaka Kyoiku University, Osaka 582-8582, Japan

Correspondence should be addressed to Ryuichi Ashino; pj.ca.ukioyk-akaso.cc@onihsa

Received 23 October 2016; Accepted 5 January 2017; Published 30 January 2017

Academic Editor: Lucas Jodar

Copyright © 2017 Mawardi Bahri and Ryuichi Ashino. 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. C. K. Chui, An Introduction to Wavelets, Academic Press, New York, NY, USA, 1992.
  2. K. Gröchenig, Foundation of Time-Frequency Analysis, Birkhäuser, Boston, Mass, USA, 2001.
  3. J. X. He and B. Yu, “Continuous wavelet transforms on the space L2 (R, H;dx),” Applied Mathematics Letters, vol. 17, no. 1, pp. 111–121, 2001. View at Publisher · View at Google Scholar
  4. R. S. Pathak, The Wavelet Transform, Atlantis Press, Amsterdam, The Netherlands, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  5. S. T. Ali and K. Thirulogasanthar, “The quaternionic affine group and related continuous wavelet transforms on complex and quaternionic Hilbert spaces,” Journal of Mathematical Physics, vol. 55, no. 6, 16 pages, 2014. View at Publisher · View at Google Scholar · View at Scopus
  6. M. Bahri, R. Ashino, and R. Vaillancourt, “Continuous quaternion Fourier and wavelet transforms,” International Journal of Wavelets, Multiresolution and Information Processing, vol. 12, no. 4, Article ID 1460003, 21 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  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 · View at Scopus
  8. E. Bayro-Corrochano, “The theory and use of the quaternion wavelet transform,” Journal of Mathematical Imaging and Vision, vol. 24, no. 1, pp. 19–35, 2006. View at Publisher · View at Google Scholar · View at Scopus
  9. Y. Xi, X. Yang, L. Song, L. Traversoni, and W. Lu, “QWT: retrospective and new application,” in Applied Geometric Algebras in Computer Science and Engineering, E. Bayro-Corrochano and G. Scheuermann, Eds., pp. 249–273, Springer, London, UK, 2010. View at Google Scholar
  10. S. Gai, G. Yang, and S. Zhang, “Multiscale texture classification using reduced quaternion wavelet transform,” International Journal of Electronics and Communications, vol. 67, no. 3, pp. 233–241, 2013. View at Publisher · View at Google Scholar · View at Scopus
  11. L. Akila and R. Roopkumar, “Ridgelet transform for quarternion-valued functions,” International Journal of Wavelets, Multiresolution and Information Processing, vol. 14, no. 1, 18 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  12. L. Akila and R. Roopkumar, “Multidimensional quaternionic Gabor transforms,” Advances in Applied Clifford Algebras, vol. 26, no. 3, pp. 985–1011, 2016. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. E. M. 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 · View at Scopus
  14. M. Bahri, E. S. Hitzer, A. Hayashi, and R. Ashino, “An uncertainty principle for quaternion Fourier transform,” Computers and Mathematics with Applications, vol. 56, no. 9, pp. 2398–2410, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. L.-P. Chen, K. I. Kou, and M.-S. Liu, “Pitt's inequality and the uncertainty principle associated with the quaternion Fourier transform,” Journal of Mathematical Analysis and Applications, vol. 423, no. 1, pp. 681–700, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. E. M. 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 · View at Scopus
  17. M. Bahri, “A modified uncertainty principle for two-sided quaternion Fourier transform,” Advances in Applied Clifford Algebras, vol. 26, no. 2, pp. 513–527, 2016. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. Y. Yang, P. Dang, and T. Qian, “Tighter uncertainty principles based on quaternion Fourier transform,” Advances in Applied Clifford Algebras, vol. 26, no. 1, pp. 479–497, 2016. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. T. Bülow, Hypercomplex spectral signal representations for the processing and analysis of images [Ph.D. thesis], University of Kiel, Kiel, Germany, 1999.
  20. J. P. Morais, S. Georgiev, and W. Sprößig, Real Quaternionic Calculus Handbook, Birkhäuser, New York, NY, USA, 2014. View at Publisher · View at Google Scholar
  21. K. I. Kou and J. Morais, “Asymptotic behaviour of the quaternion linear canonical transform and the Bochner-Minlos theorem,” Applied Mathematics and Computation, vol. 247, no. 15, pp. 675–688, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, New York, NY, USA, 2nd edition, 1950. View at MathSciNet