International Journal of Theoretical Physics

, Volume 53, Issue 7, pp 2463–2484 | Cite as

Quantum Hilbert Image Scrambling

Article
First Online:
Received:
Accepted:

Abstract

Analogies between quantum image processing (QIP) and classical one indicate that quantum image scrambling (QIS), as important as quantum Fourier transform (QFT), quantum wavelet transform (QWT) and etc., should be proposed to promote QIP. Image scrambling technology is commonly used to transform a meaningful image into a disordered image by permutating the pixels into new positions. Although image scrambling on classical computers has been widely studied, we know much less about QIS. In this paper, the Hilbert image scrambling algorithm, which is commonly used in classical image processing, is carried out in quantum computer by giving the scrambling quantum circuits. First, a modified recursive generation algorithm of Hilbert scanning matrix is given. Then based on the flexible representation of quantum images, the Hilbert scrambling quantum circuits, which are recursive and progressively layered, is proposed. Theoretical analysis indicates that the network complexity scales squarely with the size of the circuit’s input n.

Keywords

Hilbert image scrambling Quantum circuit Quantum computation Quantum watermarking 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

This work is supported by the Beijing Municipal Education Commission Science and Technology Development Plan under Grants No. KM201310005021, KZ201210005007, the Fundamental Research Funds for the Central Universities under Grants No. 2012JBM041, and the Graduate Technology Fund of BJUT under Grants No. YKJ-2013-10282.

References

  1. 1.
    Vedral, V., Barenco, A., Ekert, A.: Quantum networks for elementary arithmetic operations. Phys. Rev. A 54(1), 147–153 (1996)ADSCrossRefMathSciNetGoogle Scholar
  2. 2.
    Feynman, R.: Quantum mechanical computers. Found. Phys. 16(6), 507–531 (1986)ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    Nielson, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)Google Scholar
  4. 4.
    Williams, C.: Explorations in quantum computing. In: Texts in Computer Science (2011)Google Scholar
  5. 5.
    Caraiman, S., Manta, V.: Image processing using quantum computing. In: International Conference on 16th System Theory, Control and Computing (ICSTCC), pp 1–6 (2012)Google Scholar
  6. 6.
    Beach, G., Lomont, C., Cohen, C.: Quantum image processing (QuIP). In: 32nd Applied Imagery Pattern Recognition Workshop, pp 39–44 (2003)Google Scholar
  7. 7.
    Venegas-Andraca, S.E., Ball, J.L.: Processing images in entangled quantum system. J. Quantum. Inf. Process. 9(1), 1–11 (2010)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Venegas-Andraca, S.E., Bose, S.: Storing, processing and retrieving an image using quantum mechanics. In: Proceedings of the SPIE Conference on Quantum Information and Computation, pp 137–147 (2003)Google Scholar
  9. 9.
    Latorre, J.I.: Image Compression and Entanglement. (2005). arXiv:quantph/0510031
  10. 10.
    Le, P.Q., Dong, F., Hirota, K.: A flexible representation of quantum images for polynomial preparation, image compression and processing operations. Quantum Inf. Process. 10(1), 63–84 (2011)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Zhang, Y., Lu, K., Gao, Y.-H., Wang, M.: NEQR: a novel enhanced quantum representation of digital images. Quantum Inf. Process. 12(12), 2833–2860 (2013)Google Scholar
  12. 12.
    Fijany, A., Williams, C.: Quantum Wavelet Transform: Fast Algorithm and Complete Circuits (1998). arXiv:quant-ph/9809004
  13. 13.
    Klappenecker, A., Roetteler, M.: Discrete cosine transforms on quantum computers. In: IEEER8-EURASIP Symposium on Image and Signal Processing and Analysis (ISPA01), pp. 464–468. Pula (2001)Google Scholar
  14. 14.
    Tseng, C., Hwang, T.: Quantum circuit design of 8 × 8 discrete cosine transforms using its fast computation flow graph. In: IEEE International Symposium on Circuits and Systems, pp 828–831 (2005)Google Scholar
  15. 15.
    Lomont, C.: Quantum Convolution and Quantum Correlation Algorithms are Physically Impossible (2003). arXiv:quant-ph/0309070
  16. 16.
    Zhang, W.-W., Gao, F., Liu, B., et al.: A watermark strategy for quantum images based on quantum Fourier transform. Quantum Inf. Process. 12(4), 793–803 (2013)Google Scholar
  17. 17.
    Weiwei, Z., Fei, G., Bing, L., et al.: A quantum watermark protocol. Int. J. Theor. Phys. 52, 504–513 (2013)Google Scholar
  18. 18.
    Iliyasu, A. M., Le, P. Q., Dong, F., Hirota, K.: Watermarking and authentication of quantum images based on restricted geometric transformations. Inf. Sci. 186, 126–149 (2012)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Hu, M.-Y., Tian, X.-L., Xia, S.-W., et al.: Image scrambling based on 3-D Hilbert curve. In: 3rd International Congress on Image and Signal Processing, vol. 1, pp. 147–149 (2010)Google Scholar
  20. 20.
    Guo, J.-M., Yang, Y., Wang, N.: Chaos-based gray image watermarking algorithm. In: International Conference on Uncertainty Reasoning and Knowledge Engineering, vol.1, pp. 158–160 (2011)Google Scholar
  21. 21.
    Lien, B.K.: Robust data hiding by Hilbert curve decomposition. In: IEEE Intelligent Information Hiding and Multimedia Signal Processing, pp 937–940 (2009)Google Scholar
  22. 22.
    Sun Y.-Y., Kong R.-Q., Wang X.-Y.: An image encryption algorithm utilizing Mandelbrot set. In: International Workshop on Chaos-Fractal Theory and Its Applications, pp 170–173 (2010)Google Scholar
  23. 23.
    Moreno, J., Otazu, X.: Image compression algorithm based on Hilbert scanning of embedded quadtrees: an introduction of the hi-set coder. In: IEEE International Conference on Multimedia and Expo, pp 1–6 (2011)Google Scholar
  24. 24.
    Lin, X.-H., Cai, L.-D.: Scrambling research of digital image based on Hilbert curve. Chinese J. Stereology Image Anal. 9(4), 224–227 (2004)Google Scholar
  25. 25.
    Shang, Z., Ren, H., Zhang, J.: A block location scrambling algorithm of digital image based on Arnold transformation. In: 9th International Conference for Young Computer Scientists, pp 2942–2947 (2008)Google Scholar
  26. 26.
    Zou, W.-G., Huang, J.-Y., Zhou C.-Y.: Digital image scrambling technology based on two dimension Fibonacci transformation and its periodicity. In: 3rd International Symposium on Information Science and Engineering, pp 415–418 (2010)Google Scholar
  27. 27.
    Wen M.-G., Huang S.-C., Han C.-C.: An information hiding scheme using magic squares. In: IEEE International Conference on Broadband, Wireless Computing, Communication and Applications, pp 556–560 (2010)Google Scholar
  28. 28.
    Wang, S., Xu, X.-S.: A new algorithm of Hilbert scanning matrix and its MATLAB program. J. Image Graph. 11(1), 119–122 (2006)Google Scholar
  29. 29.
    Le, P. Q., Iliyasu, A. M., Dong, F. Y., Hirota, K.: Fast geometric transformation on quantum images. IAENG Int. J. Appl. Math. 40(3), 113–123 (2010)MATHMathSciNetGoogle Scholar
  30. 30.
    Jiang, N., Wu, W.-Y., Wang, L.: The quantum realization of Arnold and Fibonacci image scrambling. Quantum Inf. Process. accepted.Google Scholar
  31. 31.
    Peano, G.: Sur une courbe, qui remplit toute une aire plane. Math. Ann. 36(1), 157–160 (1890)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Hilbert, D.: Ueber die stetige Abbildung einer Linie auf ein Flächenstück. Math. Ann. 38(3), 459–460 (1891)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Kamata, S., Eason, R.O., Bandou, Y.: A new algorithm for N-dimensional Hilbert scanning. IEEE Trans. Image Process. 8(7), 964–973 (1999)ADSCrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.College of ComputerBeijing University of TechnologyBeijingChina

Personalised recommendations