Skip to main content

Advertisement

Springer Nature Link
Log in

Hyper-complex number quaternion system-based method for image representation

  • Original Paper
  • Published:
Signal, Image and Video Processing Aims and scope Submit manuscript

Abstract

For centuries, the decimal number system always was dominant until the advent of digital computers, which brought the binary and other number systems into the spotlight. Except them, other numeral systems for image representation are also worth exploring. Quaternion, as a hyper-complex number mathematical tool, is widely concerned and used in image representation. However, the current mainstream quaternion representation is only applicable to color image representation, and its applicability is limited. Hence, the paper introduces a novel quaternion system-based for image representation, which is suitable not for color image but also for gray-scale image. The representation scheme is a numeral positional system which uses a quaternion as the base, and the method converts the traditional two-dimensional array representation of one image into a higher four-dimensional space. Then, some properties of the quaternion can be applied to image analysis and processing. Experiments show that the proposed representation method can retain the essence of the original image reversibly and losslessly. Furthermore, some potential image applications, such as image encryption,image scrambling and image secret sharing, are demonstrated that the proposed method has an practical research value in image security.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
€32.70 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Pseudo code 1
Pseudo code 2
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Data availibility

All of the data was obtained from the standard test images which were downloaded from https://www.imageprocessingplace.com.

References

  1. Abaunza, H., Castillo, P.: Quadrotor aggressive deployment, using a quaternion-based spherical chattering-free sliding-mode controller. Trans. Aerosp. Electron. Syst. 56(3), 1979–1991 (2020). https://doi.org/10.1109/TAES.2019.2937663

    Article  Google Scholar 

  2. Liu, Y., Zheng, Y., Lu, J., Cao, J., Rutkowski, L.: Constrained quaternion-variable convex optimization: a quaternion-valued recurrent neural network approach. IEEE Trans. Neural Netw. Learn. Syst. 31(3), 1022–1035 (2020). https://doi.org/10.1109/TNNLS.2019.2916597

    Article  MathSciNet  Google Scholar 

  3. Brignone, C., Mancini, G., Grassucci, E., Uncini, A., Comminiello, D.: Efficient sound event localization and detection in the quaternion domain. Trans. Circuits Syst. 69(5), 2453–2457 (2022). https://doi.org/10.1109/TCSII.2022.3160388

    Article  Google Scholar 

  4. Liu, L., Chen, C.L.P., Li, S.: Hallucinating color face image by learning graph representation in quaternion space. IEEE Trans. Cybern. 52(1), 265–277 (2022). https://doi.org/10.1109/TCYB.2020.2979320

    Article  Google Scholar 

  5. Brignone, C., Mancini, G., Grassucci, E., Uncini, A., Comminiello, D.: Efficient sound event localization and detection in the quaternion domain. IEEE Trans. Circuits Syst. II Express Briefs 69(5), 2453–2457 (2022). https://doi.org/10.1109/TCSII.2022.3160388

    Article  Google Scholar 

  6. Li, M., Yuan, X., Chen, H., Li, J.: Quaternion discrete Fourier transform-based color image watermarking method using quaternion QR decomposition. IEEE Access 8, 72308–72315 (2020). https://doi.org/10.1109/ACCESS.2020.2987914

    Article  Google Scholar 

  7. Vince, J.: Quaternions for Computer Graphics, pp. 73–88. Springer, London (2021). https://doi.org/10.1007/978-1-4471-7509-4

    Book  Google Scholar 

  8. Plantz, A.R., Berman, M.: Adoption of the octal number system. IEEE Trans. Comput. 20(5), 593–598 (1971). https://doi.org/10.1109/T-C.1971.223307

    Article  Google Scholar 

  9. Wadel, L.B.: Negative base number systems. IRE Trans. Electron. Comput. EC 6(2), 123 (1957). https://doi.org/10.1109/TEC.1957.5221586

    Article  Google Scholar 

  10. Dimitrov, V.S., Jullien, G.A.: Loading the bases: a new number representation with applications. IEEE Circuits Syst. Mag. 3(2), 6–23 (2003). https://doi.org/10.1109/MCAS.2003.1242832

    Article  Google Scholar 

  11. Bergman, G.: A number system with an irrational base. Math. Mag. 31(2), 98–110 (1957). https://doi.org/10.5948/UPO9780883859667.017

    Article  MathSciNet  Google Scholar 

  12. Knuth, D.E.: A imaginary number system. Commun. ACM 3(4), 245–247 (1960). https://doi.org/10.1145/367177.367233

    Article  MathSciNet  Google Scholar 

  13. Gilbert, W.J.: Complex numbers with three radix expansions. Can. J. Math. 34(06), 1335–1348 (1982). https://doi.org/10.4153/CJM-1982-093-4

    Article  MathSciNet  Google Scholar 

  14. Sun, W., Qi, D.X.: Complex number system based algorithm for engineering graph digital watermarking. J. Image Graph. 8, 626–630 (2003). https://doi.org/10.3969/j.issn.1006-8961.2003.z1.143

    Article  Google Scholar 

  15. Ahmed, N., Natarajan, T., Rao, K.R.: Discrete cosine transform. IEEE Trans. Comput. C 23(1), 90–93 (1974). https://doi.org/10.1109/T-C.1974.223784

    Article  MathSciNet  Google Scholar 

  16. Ahmed, N., Rao, K.R.: Orthogonal transforms for digital signal processing. In: ICASSP ’76. IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 1, pp. 136–140 (1976). https://doi.org/10.1109/ICASSP.1976.1170121

  17. Wang, S.J., Kuo, L.C., Jong, H.H., Wu, Z.H.: Representing images using points on image surfaces. IEEE Trans. Image Process. 14(8), 1043–1056 (2005). https://doi.org/10.1109/TIP.2005.851693

    Article  MathSciNet  Google Scholar 

  18. Zhou, H., Zheng, J., Lei, W.: Representing images using curvilinear feature driven subdivision surfaces. IEEE Trans. Image Process. 23(8), 3268–80 (2014). https://doi.org/10.1109/TIP.2014.2327807

    Article  MathSciNet  Google Scholar 

  19. Ell, T.A., Sangwine, S.J.: Hypercomplex Fourier transforms of color images. IEEE Trans. Image Process. 16(1), 22–35 (2007). https://doi.org/10.1109/TIP.2006.884955

  20. Wang, Y., Kou, K.I., Zou, C., Tang, Y.Y.: Robust sparse representation in quaternion space. IEEE Trans. Image Process. 30, 3637–3649 (2021). https://doi.org/10.1109/TIP.2021.3064193

    Article  MathSciNet  Google Scholar 

  21. Hamilton, R.W.: On quaternions: or on a new system of imaginaries in algebra. Philos. Mag. 30(203), 458–461 (1847). https://doi.org/10.1080/14786444708645898

    Article  Google Scholar 

  22. Zhang, F.: Quaternions and matrices of quaternions. Linear Akgebra Appl. 251(2), 21–57 (1997). https://doi.org/10.1016/0024-3795(95)00543-9

    Article  MathSciNet  Google Scholar 

  23. Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004). https://doi.org/10.1109/TIP.2003.819861

    Article  Google Scholar 

  24. Vía, J., Ramírez, D., Santamaría, I.: Properness and widely linear processing of quaternion random vectors. IEEE Trans. Inf. Theory 56(7), 3502–3515 (2010). https://doi.org/10.1109/TIT.2010.2048440

  25. Le Bihan, N., Mars, J.: Quaternion subspace method for vector-sensor wave separation. In: 2002 11th European Signal Processing Conference, pp. 1–4 (2002). https://doi.org/10.5281/ZENODO.37799

  26. Lan, T., Cai, Z.: A novel image representation method under a non-standard positional numeral system. IEEE Trans. Multimed. 23, 1301–1315 (2021). https://doi.org/10.1109/TMM.2020.2995258

    Article  Google Scholar 

  27. He, X., Chen, Y., Fan, Y., Kong, X., Cai, Z.: A reversible multimedia representation method and its applications in multimedia processing. IEEE Access 11, 62436–62448 (2023). https://doi.org/10.1109/ACCESS.2023.3288020

    Article  Google Scholar 

Download references

Funding

This work was supported in part by Guangzhou Science and Technology Project under Grant No. 202102080656, and in part by the Key Discipline Project of Guangzhou Xinhua University under Grant No. 2020XZD02, and in part by Guangdong key Discipline Scientific Research Capability Improvement Project with No. 2021ZDJS144.

Author information

Authors and Affiliations

  1. School of Information and Intelligent Engineering, Guangzhou Xinhua University, Dongguan, 523133, China

    Xiu He & Lina Zhang

  2. School of Biomedical Engineering, Guangdong Medical University, Dongguan, 523808, China

    Yuyun Chen

Authors
  1. Xiu He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuyun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lina Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Xiu He performed the data analyzes and wrote the manuscript. Yuyun Chen contributed to the conception of the study, analysis and manuscript preparation. Lina Zhang completed the revision and touch-up of the manuscript.

Corresponding author

Correspondence to Yuyun Chen.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file 1 (pdf 829 KB)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, X., Chen, Y. & Zhang, L. Hyper-complex number quaternion system-based method for image representation. SIViP 18, 7899–7907 (2024). https://doi.org/10.1007/s11760-024-03437-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11760-024-03437-1

Keywords

Search

Navigation