The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations

  • Eckhard Hitzer
  • Stephen J. Sangwine
Part of the Trends in Mathematics book series (TM)


The two-sided quaternionic Fourier transformation (QFT) was introduced in [2] for the analysis of 2D linear time-invariant partial-differential systems. In further theoretical investigations [4, 5] a special split of quaternions was introduced, then called ±split. In the current chapter we analyze this split further, interpret it geometrically as an orthogonal 2D planes split (OPS), and generalize it to a freely steerable split of H into two orthogonal 2D analysis planes. The new general form of the OPS split allows us to find new geometric interpretations for the action of the QFT on the signal. The second major result of this work is a variety of new steerable forms of the QFT, their geometric interpretation, and for each form, OPS split theorems, which allow fast and efficient numerical implementation with standard FFT software.


Quaternion signals orthogonal 2D planes split quaternion Fourier transformations steerable transforms geometric interpretation fast implementations. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    H.S.M. Coxeter. Quaternions and reflections. The American Mathematical Monthly, 53(3):136–146, Mar. 1946.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    T.A. Ell. Quaternion-Fourier transforms for analysis of 2-dimensional linear timeinvariant partial-differential systems. In Proceedings of the 32nd Conference on Decision and Control, pages 1830–1841, San Antonio, Texas, USA, 15–17 December 1993. IEEE Control Systems Society.Google Scholar
  3. [3]
    T.A. Ell and S.J. Sangwine. Hypercomplex Fourier transforms of color images. IEEE Transactions on Image Processing, 16(1):22–35, Jan. 2007.MathSciNetCrossRefGoogle Scholar
  4. [4]
    E. Hitzer. Quaternion Fourier transform on quaternion fields and generalizations. Advances in Applied Clifford Algebras, 17(3):497–517, May 2007.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    E. Hitzer. Directional uncertainty principle for quaternion Fourier transforms. Advances in Applied Clifford Algebras, 20(2):271–284, 2010.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    E. Hitzer. OPS-QFTs: A new type of quaternion Fourier transform based on the orthogonal planes split with one or two general pure quaternions. In International Conference on Numerical Analysis and Applied Mathematics, volume 1389 of AIP Conference Proceedings, pages 280–283, Halkidiki, Greece, 19–25 September 2011. American Institute of Physics.Google Scholar
  7. [7]
    E. Hitzer. Two-sided Clifford Fourier transform with two square roots of 1in Cℓ p,q. In Proceedings of the 5th Conference on Applied Geometric Algebras in Computer Science and Engineering (AGACSE 2012), La Rochelle, France, 2–4 July 2012.Google Scholar
  8. [8]
    S.J. Sangwine. Fourier transforms of colour images using quaternion, or hypercomplex, numbers. Electronics Letters, 32(21):1979–1980, 10 Oct. 1996.Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.College of Liberal Arts, Department of Material ScienceInternational Christian UniversityTokyoJapan
  2. 2.School of Computer Science and Electronic EngineeringUniversity of EssexColchesterUK

Personalised recommendations