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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 645059, 9 pages
http://dx.doi.org/10.1155/2014/645059
Research Article

Image Watermarking in the Linear Canonical Transform Domain

School of Mathematics and Statistics of Beijing Institute of Technology, Beijing 100081, China

Received 26 December 2013; Revised 19 February 2014; Accepted 19 February 2014; Published 24 March 2014

Academic Editor: Juan J. Trujillo

Copyright © 2014 Bing-Zhao Li and Yu-Pu Shi. 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. I. J. Cox, J. Kilian, T. Leighton, and T. Shamoon, “Secure spread spectrum watermarking for images, audio and video,” in Proceedings of the IEEE International Conference on Image Processing (ICIP '96), vol. 3, pp. 243–246, Lausanne, Switzerland, September 1996. View at Scopus
  2. K. Eckhard, J. Rindfrey, and J. Zhao, “Copyright protection for multimedia data,” in Proceedings of the International Conference on Digital Media and Electronic Publishing, vol. 32, 1994.
  3. I. J. Cox, J. Kilian, F. T. Leighton, and T. Shamoon, “Secure spread spectrum watermarking for multimedia,” IEEE Transactions on Image Processing, vol. 6, no. 12, pp. 1673–1687, 1997. View at Publisher · View at Google Scholar · View at Scopus
  4. J. J. K. O'Ruanaidh, W. J. Dowling, and F. M. Boland, “Phase watermarking of digital images,” in Proceedings of the IEEE International Conference on Image Processing (ICIP '96), vol. 3, pp. 239–242, Lausanne, Switzerland, September 1996. View at Scopus
  5. V. Solachidis and I. Pitas, “Circularly symmetric watermark embedding in 2-D DFT domain,” IEEE Transactions on Image Processing, vol. 10, no. 11, pp. 1741–1753, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. S. Liu, B. M. Hennelly, and J. T. Sheridan, “Digital image watermarking spread-space technique based on double random phase encoding,” Optical Communications, vol. 300, pp. 162–177, 2013. View at Google Scholar
  7. M. Barni, F. Bartolini, V. Cappellini, and A. Piva, “A DCT-domain system for robust image watermarking,” Signal Processing, vol. 66, no. 3, pp. 357–372, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. A. Piva, M. Barni, F. Bartolini, and V. Cappellini, “DCT-based watermark recovering without resorting to the uncorrupted original image,” in Proceedings of the International Conference on Image Processing, vol. 1, pp. 520–523, October 1997. View at Scopus
  9. M. Kutter, “Watermarking world,” http://www.watermarkingworld.org/.
  10. C. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Transactions on Signal Processing, vol. 48, no. 5, pp. 1329–1337, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. H. M. Ozaktas, N. Erkaya, and M. A. Kutay, “Effect of fractional Fourier transformation on time-frequency distributions belonging to the Cohen class,” IEEE Signal Processing Letters, vol. 3, no. 2, pp. 40–41, 1996. View at Publisher · View at Google Scholar · View at Scopus
  12. H. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” Journal of the Optical Society of America A, vol. 10, no. 12, pp. 2522–2531, 1993. View at Google Scholar · View at Scopus
  13. S.-C. Pei, C.-C. Tseng, M.-H. Yeh, and J.-J. Shyu, “Discrete fractional hartley and fourier transforms,” IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 45, no. 6, pp. 665–675, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform With Applications in Optics and Signal Processing, Wiley, New York, NY, USA, 2000.
  15. R. Tao, B. Deng, and Y. Wang, Fractional Fourier Transform and Its Applications, University Press, Beijing, China, 2009.
  16. I. Djurovic, S. Stankovic, and I. Pitas, “Digital watermarking in the fractional Fourier transformation domain,” Journal of Network and Computer Applications, vol. 24, no. 2, pp. 167–173, 2001. View at Publisher · View at Google Scholar · View at Scopus
  17. M. A. Savelonas and S. Chountasis, “Noise-resistant watermarking in the fractional Fourier domain utilizing moment-based image representation,” Signal Processing, vol. 90, no. 8, pp. 2521–2528, 2010. View at Publisher · View at Google Scholar · View at Scopus
  18. A. Bultheel, “Digital watermarking of images in the fractional Fourier domain,” TW Report TW497, 2007. View at Google Scholar
  19. N. K. Nishchal, “Hierarchical encrypted image watermarking using fractional Fourier domain random phase encoding,” Optical Engineering, vol. 50, no. 9, Article ID 097003, 2011. View at Publisher · View at Google Scholar · View at Scopus
  20. T.-Z. Xu and B.-Z. Li, Linear Canonical Transform and Its Applications, Science Press, Beijing, China, 2013.
  21. S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Transactions on Signal Processing, vol. 48, no. 5, pp. 1338–1353, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  22. A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Transactions on Signal Processing, vol. 56, no. 6, pp. 2383–2394, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  23. F. S. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Processing Letters, vol. 16, no. 8, pp. 727–730, 2009. View at Google Scholar
  24. J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Processing, vol. 89, no. 4, pp. 641–648, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  25. B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Processing, vol. 87, no. 5, pp. 983–990, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. A. Stern, “Sampling of linear canonical transformed signals,” Signal Processing, vol. 86, no. 7, pp. 1421–1425, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  27. R. Tao, B.-Z. Li, Y. Wang, and G. K. Aggrey, “On sampling of band-limited signals associated with the linear canonical transform,” IEEE Transactions on Signal Processing, vol. 56, no. 11, pp. 5454–5464, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  28. B.-Z. Li and T.-Z. Xu, “Spectral analysis of sampled signals in the linear canonical transform domain,” Mathematical Problems in Engineering, vol. 2012, Article ID 536464, 19 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  29. B.-Z. Li and T.-Z. Xu, “Sampling in the linear canonical transform domain,” Mathematical Problems in Engineering, vol. 2012, Article ID 504580, 13 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  30. S.-C. Pei and J.-J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Transactions on Signal Processing, vol. 50, no. 1, pp. 11–26, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  31. D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Signal Processing Letters, vol. 16, no. 10, pp. 853–856, 2009. View at Publisher · View at Google Scholar · View at Scopus
  32. B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China. Series F. Information Sciences, vol. 49, no. 5, pp. 592–603, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  33. J. Zhao, R. Tao, Y.-L. Li, and Y. Wang, “Uncertainty principles for linear canonical transform,” IEEE Transactions on Signal Processing, vol. 57, no. 7, pp. 2856–2858, 2009. View at Publisher · View at Google Scholar · View at MathSciNet