Skip to main content
SpringerLink
Log in

Uncertainty Principles for the Two-Sided Quaternion Linear Canonical Transform

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

The quaternion linear canonical transform (QLCT), as a generalized form of the quaternion Fourier transform, is a powerful analyzing tool in image and signal processing. In this paper, we propose three different forms of uncertainty principles for the two-sided QLCT, which include Hardy’s uncertainty principle, Beurling’s uncertainty principle and Donoho–Stark’s uncertainty principle. These consequences actually describe the quantitative relationships of the quaternion-valued signal in arbitrary two different QLCT domains, which have many applications in signal recovery and color image analysis. In addition, in order to analyze the non-stationary signal and time-varying system, we present Lieb’s uncertainty principle for the two-sided short-time quaternion linear canonical transform (SQLCT) based on the Hausdorff–Young inequality. By adding the nonzero quaternion-valued window function, the two-sided SQLCT has a great significant application in the study of signal local frequency spectrum. Finally, we also give a lower bound for the essential support of the two-sided SQLCT.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Bahri, A modified uncertainty principle for two-sided quaternion Fourier transform. Adv. Appl. Clifford Algebras 26(2), 513–527 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Bahri, R. Ashino, A simplified proof of uncertainty principle for quaternion linear canonical transform. Abstr. Appl. Anal. 2016(6), 1–11 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Bahri, E. Hitzer, A. Hayashi, R. Ashino, An uncertainty principle for quaternion Fourier transform. Comput. Math. Appl. 56(9), 2398–2410 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Bahri, Resnawati, S. Musdalifah, A version of uncertainty principle for quaternion linear canonical transform. Abstr. Appl. Anal. Article ID 8732457 (2018). https://doi.org/10.1155/2018/8732457

  5. P. Bas, N.L. Bihan, J.M. Chassery, Color image watermarking using quaternion Fourier transform, in ICASSP, Hong-kong, hal-00166555, pp. 521–524 (2003). https://hal.archives-ouvertes.fr/hal-00166555

  6. E. Bayro-Corrochano, N. Trujillo, M. Naranjo, Quaternion Fourier descriptors for preprocessing and recognition of spoken words using images of spatiotemporal representations. J. Math. Imaging Vis. 28(2), 179–190 (2007)

    Article  MathSciNet  Google Scholar 

  7. A. Bonami, B. Demange, P. Jaming, Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transform. Rev. Mat. Iberoam. 19, 23–55 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. K. Brahim, E. Tefjeni, Uncertainty principle for the two-sided quaternion windowed Fourier transform. J. Pseudo-Differ. Oper. Appl. (2019). https://doi.org/10.1007/s11868-019-00283-5

  9. K. Brahim, E. Tefjeni, Uncertainty principle for the two-sided quaternion windowed Fourier transform. Integral Transf. Spec. Funct. 30(5), 362–382 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. L.P. Chen, K.I. Kou, M.S. Liu, Pitt’s inequality and the uncertainty principle associated with the quaternion Fourier transform. J. Math. Anal. Appl. 423(1), 681–700 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. D.L. Donoho, P.B. Stark, Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49(3), 906–931 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. T.A. Ell, Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems. in Proceeding of the 32nd Conference on Decision and Control, San Antonio, Texas, vol. 2, pp. 1830–1841 (1993)

  13. L. Grafakos, Classical Fourier Analysis (Springer, New York, 2014)

    MATH  Google Scholar 

  14. A.M. Grigoryan, J. Jenkinson, S.S. Agaian, Quaternion Fourier transform based alpha-rooting method for color image measurement and enhancement. Signal Process. 109, 269–289 (2015)

    Article  Google Scholar 

  15. A.M. Grigoryan, A. John, S.S. Agaian, Alpha-rooting color image enhancement method by two-side 2-D quaternion discrete Fourier transform followed by spatial transformation. IJACEEE (2018). https://doi.org/10.5121/ijaceee.2018.6101

  16. Y.E. Haoui, S. Fahlaoui, The uncertainty principle for the two-sided quaternion Fourier transform. Mediterr. J. Math. 14(6), 221 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Y.E. Haoui, S. Fahlaoui, Beurling’s theorem for the quaternion Fourier transform. J. Pseudo Differ. Oper. Appl (2019). https://doi.org/10.1007/s11868-019-00281-7

  18. G.H. Hardy, A theorem concerning Fourier transforms. J. Lond. Math. Soc. 8(3), 227–231 (1933)

    Article  MathSciNet  MATH  Google Scholar 

  19. W. Heisenberg, Uber den anschaulichen Inhalt der quanten theoretischen Kinematik und Mechanik. Z. Phys. 43, 172–198 (1927)

    Article  MATH  Google Scholar 

  20. E. Hitzer, Quaternion Fourier transform on quaternion fields and generalizations. Adv. Appl. Clifford Algebras 17(3), 497–517 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. L. Hörmander, A uniqueness theorem of Beurling for Fourier transform pairs. Ark. Math. 2, 237–240 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. H. Huo, Uncertainty principles for the offset linear canonical transform. Circuits Sys. Signal Process. 38, 395–406 (2019)

    Article  MathSciNet  Google Scholar 

  23. H. Huo, W. Sun, L. Xiao, Uncertainty principles associated with the offset linear canonical transform. Math. Meth. Appl. Sci. 42, 466–474 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. K.I. Kou, J. Morais, Asymptotic behaviour of the quaternion linear canonical transform and the Bochner–Minlos theorem. Appl. Math. Comput. 247, 675–688 (2014)

    MathSciNet  MATH  Google Scholar 

  25. K.I. Kou, J.Y. Ou, J. Morais, J. Morais, On uncertainty principle for quaternionic linear canonical transform. Abstr. Appl. Anal. 2013(1), 94–121 (2013)

    MathSciNet  MATH  Google Scholar 

  26. P. Lian, Uncertainty principle for the quaternion Fourier transform. J. Math. Anal. Appl. 467, 1258–1269 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Moshinsky, C. Quesne, Linear canonical transformations and their unitary representations. J. Math. Phys. 12(8), 1772–1880 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  28. S.C. Pei, J.J. Ding, J.H. Chang, Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT. IEEE Trans. Signal Process. 49(11), 2783–2797 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. J. Wang, C. Zhou, S. Wang, Color image adaptive watermarking algorithm using fractional quaternion Fourier transform. J. Electr. Inform. Technol. 38(11), 2832–2839 (2016)

    Google Scholar 

  30. G.L. Xu, X.T. Wang, X.G. Xu, Fractional quaternion Fourier transform, convolution and correlation. Signal Process. 88(10), 2511–2517 (2008)

    Article  MATH  Google Scholar 

  31. Y. Yang, P. Dang, T. Qian, Tighter uncertainty principles based on quaternion Fourier transform. Adv. Appl. Clifford Algebras 26(1), 479–497 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Y.N. Zhang, B.Z. Li, Generalized uncertainty principles for the two-sided quaternion linear canonical transform, in Proceedings of the IEEE International Conference on Acoustics Speech and Signal Processing, ICASSP, pp. 4594–4598 (2018). https://doi.org/10.1109/ICASSP.2018.8461536

  33. Y.N. Zhang, B.Z. Li, Novel uncertainty principles for two-sided quaternion linear canonical transform. Adv. Appl. Clifford Algebras 28(1), 15 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions. This paper was in part supported by the Natural Science Foundation of China Grant No. 11371050.

Author information

Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China

    Xiaoyu Zhu & Shenzhou Zheng

Authors
  1. Xiaoyu Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shenzhou Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shenzhou Zheng.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, X., Zheng, S. Uncertainty Principles for the Two-Sided Quaternion Linear Canonical Transform. Circuits Syst Signal Process 39, 4436–4458 (2020). https://doi.org/10.1007/s00034-020-01376-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-020-01376-z

Keywords

Mathematics Subject Classification

Search

Navigation