Advances in Applied Clifford Algebras

, Volume 26, Issue 1, pp 479–497 | Cite as

Tighter Uncertainty Principles Based on Quaternion Fourier Transform



The quaternion Fourier transform (QFT) and its properties are reviewed in this paper. Under the polar coordinate form for quaternion-valued signals, we strengthen the stronger uncertainty principles in terms of covariance for quaternion-valued signals based on the right-sided quaternion Fourier transform in both the directional and the spatial cases. We also obtain the conditions that give rise to the equal relations of two uncertainty principles. Examples are given to verify the results.

Mathematics Subject Classification

46F10 30G35 94A12 


Uncertainty principle Quaternion Fourier transform Covariance 


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  1. 1.
    Aytur O., Ozaktas H.M.: Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transform. Opt. Commun. 120, 166–170 (1995)CrossRefADSGoogle Scholar
  2. 2.
    Bas, P., Le Bihan, N., Chassery, J.M.: Color image watermarking using quaternion Fourier transform. In: Proceedings of the IEEE International Conference on Acoustics Speech and Signal and Signal Processing, ICASSP, Hong-kong, pp. 521–524 (2003)Google Scholar
  3. 3.
    Bayro-Corrochano E., Trujillo N., Naranjo M.: Quaternion Fourier descriptors for preprocessing and recognition of spoken words using images of spatiotemporal representations. J. Math. Imaging Vis. 28(2), 179–190 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bahri M., Hitzer E., Hayashi A., Ashino R.: An uncertainty principle for quaternion Fourier transform. Comput. Math. Appl. 56, 2398–2410 (2008)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bernstein, S., Bouchot, J.-L., Reinhardt, M., Heise, B.: Generalized analytic signals in image processing: comparison, theory and applications. In: Hitzer, E., Sangwine, S.J. (eds.) Quaternion and Clifford Fourier Transforms and Wavelets. Trends in Mathematics, pp. 221–246. Birkhäuser, Basel (2013)Google Scholar
  6. 6.
    Bülow, T: (1999) Hypercomplex spectral signal representations for the processing and analysis of images. Ph.D. Thesis, Institut für Informatik und Praktische Mathematik, University of Kiel, GermanyGoogle Scholar
  7. 7.
    Bülow, T., Felsberg, M., Sommer, G.: Non-commutative hypercomplex Fourier transforms of multidimensional signals. Geometric computing with Clifford algebras. Springer Berlin Heidelberg, pp. 187–207 (2001)Google Scholar
  8. 8.
    Cohen L.: Time-Frequency Analysis: Theory and Applications. Prentice Hall Inc., Upper Saddle River (1995)Google Scholar
  9. 9.
    Da, ZX.: Modern signal processing. Tsinghua University Press, Beijing, 2nd edn, pp. 362 (2002)Google Scholar
  10. 10.
    Dang P., Deng G.T., Qian T.: A sharper uncertainty principle. J. Funct. Anal. 265, 2239–2266 (2013)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Dang P., Deng G.T., Qian T.: A tighter uncertainty principle for linear canonical transform in terms of phase derivative. IEEE Trans. Signal Process. 61, 5153–5164 (2013)MathSciNetCrossRefADSGoogle Scholar
  12. 12.
    Dembo A., Cover T.M.: Information theoretic inequalities. IEEE Trans. Inform. Theory 37(6), 1501–1508 (1991)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Ell, T.A.: Quaternion-Fourier transfotms for analysis of two-dimensional linear time-invariant partial differential systems. In: Proceeding of the 32nd Conference on Decision and Control, San Antonio, Texas, pp. 1830–1841 (1993)Google Scholar
  14. 14.
    Felsberg, M.: Low-Level image processing with the structure multivector. Ph.D. Thesis, Institut für Informatik und Praktische Mathematik, University of Kiel, Germany (2002)Google Scholar
  15. 15.
    Felsberg M., Sommer G.: The monogenic signal. IEEE Trans. Signal Process. 49(12), 3136–3144 (2001)MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Georgiev, S., Morais, J.: Bochner’s Theorems in the framework of quaternion analysis. Quaternion and Clifford Fourier transforms and wavelets, pp. 85–104. Springer, Basel (2013)Google Scholar
  17. 17.
    Georgiev, S., Morais, J., Kou, KI., Sprössig, W.: Bochner–Minlos Theorem and Quaternion Fourier Transform. Quaternion and Clifford Fourier Transforms and Wavelets, pp. 105–120. Springer, Basel (2013)Google Scholar
  18. 18.
    Hardy, G., Littlewood, JE., Polya, G.: Inequalities, 2nd edn. Press of University of Cambridge (1951)Google Scholar
  19. 19.
    Heinig H., Smith M.: Extensions of the Heisenberg-Weyl inequality. Int. J. Math. Math. Sci. 9, 185–192 (1986)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Hitzer E.M.S.: Quaternion Fourier transform on quaternion fields and generalizations. Adv. Appl. Clifford Algebr. 17(3), 497–517 (2007)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Hitzer E.M.S.: Directional uncertainty principle for quaternion Fourier transform. Adv. Appl. Clifford Algebr. 20, 271–284 (2010)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Iwo, B.B.: Entropic uncertainty relations in quantum mechanics. In: Accardi, L., Von Waldenfels, W. (eds.) Quantum probability and applications II, Lecture Notes in Mathematics 1136, pp. 90–103. Springer, Berlin (1985)Google Scholar
  23. 23.
    Iwo B.B.: Formulation of the uncertainty relations in terms of the Rényi entropies. Phys. Rev. A 74, 052101 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Iwo, B.B.: Rényi entropy and the uncertainty relations. In: Adenier, G., Fuchs, C.A., Yu, A. (eds.) Foundations of probability and physics, Khrennikov, Aip Conf. Proc. 889, pp. 52–62. American Institute of Physics, Melville (2007)Google Scholar
  25. 25.
    Kou, K.I., Ou, J.-Y., Morais, J.: On uncertainty principle for quaternionic linear canonical transform. Abstract Appl. Anal. 2013, 725952 (2013). doi: 10.1155/2013/725952
  26. 26.
    Loughlin P.J., Cohen L.: The uncertainty principle: global, local, or both?. IEEE Trans. Signal Porcess. 52(5), 1218–1227 (2004)MathSciNetCrossRefADSGoogle Scholar
  27. 27.
    Majernik V., Eva M., Shpyrko S.: Uncertainty relations expressed by Shannon-like entropies. CEJP 3, 393–420 (2003)ADSGoogle Scholar
  28. 28.
    Maassen, H.: A discrete entropic uncertainty relation, Quantum probability and applications. Lecture Notes in Mathematics, pp. 263–266. Springer, Berlin/Heidelberg (1990)Google Scholar
  29. 29.
    Maassen H., Uffink J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60(12), 1103–1106 (1988)MathSciNetCrossRefADSGoogle Scholar
  30. 30.
    Mustard D.: Uncertainty principle invariant under fractional Fourier transform. J. Aust. Math. Soc. Ser. B 33, 180–191 (1991)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Nicewarner, K.E., Sanderson, A.C.: A General Representation for Orientational Uncertainty Using Random Unit Quaternions. In: Proceedings of IEEE International Conference on Robotics and Automation, pp. 1161–1168 (1994)Google Scholar
  32. 32.
    Ozaktas H.M., Aytur O.: Fractional Fourier domains. Signal Process. 46, 119–124 (1995)MATHCrossRefGoogle Scholar
  33. 33.
    Pei S.C., Ding J.J., Chang J.H.: Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT. IEEE Trans. Signal Process. 49(11), 2783–2797 (2001)MathSciNetCrossRefADSGoogle Scholar
  34. 34.
    Rényi, A.: On measures of information and entropy. In: Fourth Berkeley Symposium on Mathematical Statistics and Probability, pp. 547–561 (1961)Google Scholar
  35. 35.
    Sangwine S.J., Ell T.A.: Hypercomplex Fourier transgorms of color images. IEEE Trans. Image Process. 16(1), 22–35 (2007)MathSciNetMATHCrossRefADSGoogle Scholar
  36. 36.
    Shinde S., Gadre V.M.: An uncertainty principle for real signals in the fractional Fourier transform domain. IEEE Trans. Signal Process. 49(11), 2545–2548 (2001)MathSciNetCrossRefADSGoogle Scholar
  37. 37.
    Sudbery A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85, 199–225 (1979)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Stern A.: Sampling of compact signals in offset linear canonical transform domains. Signal Image Video Process. 1(4), 259–367 (2007)CrossRefGoogle Scholar
  39. 39.
    Stern A.: Uncertainty principles in linear canonical transform domains and some of their implications in optics. J. Opt. Soc. Am. A 25(3), 647–652 (2008)CrossRefADSGoogle Scholar
  40. 40.
    Wódkiewicz K.: Operational approach to phase-space measurements in quantum mechanics. Phys. Rev. Lett. 52(13), 1064–1067 (1984)MathSciNetCrossRefADSGoogle Scholar
  41. 41.
    Xu G.L., Wang X.T., Xu X.G.: Three uncertainty relations for real signals associated with linear canonical transform. IET Signal Process. 3(1), 85–92 (2009)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Yang, Y., Kou, K.I.: Novel uncertainty principles associated with 2D Quaternion Fourier transforms. Integral Transform. Spec. Funct. (to appear)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.School of Mathematics and Computational ScienceSun Yat-Sen UniversityGuangzhouChina
  2. 2.Faculty of Information TechnologyMacau University of Science and TechnologyMacaoChina
  3. 3.Department of MathematicsUniversity of MacauMacaoChina

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