Skip to main content
Springer Nature Link
Log in

Two-Sided Clifford Fourier Transform with Two Square Roots of −1 in Cl(p, q)

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

We generalize quaternion and Clifford Fourier transforms to general two-sided Clifford Fourier transforms (CFT), and study their properties (from linearity to convolution). Two general multivector square roots \({\in}\) Cl(p, q) of −1 are used to split multivector signals, and to construct the left and right CFT kernel factors.

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
$34.99 /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 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

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. R. Abłamowicz, Computations with Clifford and Grassmann Algebras. Adv. Appl. Clifford Algebras 19 No. 3–4 (2009), 499–545.

  2. Abłamowicz R., Fauser B., Podlaski K., Rembieliński J: Idempotents of Clifford Algebras. Czechoslovak Journal of Physics. 53(11), 949–954 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  3. R. Abłamowicz and B. Fauser, CLIFFORD with Bigebra – A Maple Package for Computations with Clifford and Grassmann Algebras. http://math.tntech.edu/rafal/ (\({\copyright}\) 1996–2012).

  4. T. Batard, M. Berthier, C. Saint-Jean, Clifford-Fourier Transform for Color Image Processing. In E. Bayro-Corrochano and G. Scheuermann (eds.), Geometric Algebra Computing for Engineering and Computer Science Springer Verlag, Berlin, 2010, 135–161.

  5. T. Batard, M. Berthier, The Spinor Representation of Images. In: E. Hitzer, S.J. Sangwine (eds.), Quaternion and Clifford Fourier Transforms and Wavelets, Trends in Mathematics (TIM) 27, Birkhäuser, Basel, 2013, 177–195.

  6. H. De Bie, Clifford algebras, Fourier transforms and quantum mechanics. Math. Methods Appl. Sci. 35 (2012), 2198–2228.

    Google Scholar 

  7. F. Brackx, N. De Schepper, F. Sommen, The Fourier transform in Clifford analysis. Adv. Imag. Elect. Phys. 156 (2008), 55–203.

  8. T. Bülow, Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images. Ph.D. Thesis, University of Kiel, 1999.

  9. T. Bülow, M. Felsberg and G. Sommer, Non-commutative Hypercomplex Fourier Transforms of Multidimensional Signals. In G. Sommer (ed.), Geometric Computing with Clifford Algebras: Theoretical Foundations and Applications in Computer Vision and Robotics. Springer-Verlag, Berlin, 2001, 187–207.

  10. R. Bujack, G. Scheuermann, E. Hitzer, A General Geometric Fourier Transform. In: E. Hitzer, S.J. Sangwine (eds.), Quaternion and Clifford Fourier Transforms andWavelets, Trends in Mathematics (TIM) 27, Birkhäuser, Basel, 2013, 155–176.

  11. R. Bujack, G. Scheuermann, E. Hitzer, A General Geometric Fourier Transform Convolution Theorem. Adv. Appl. Clifford Algebras 23 (1) (2013), 15–38, DOI:10.1007/s00006-012-0338-4.

  12. R. Bujack, H. De Bie, N. De Schepper and G. Scheuermann, Convolution products for hypercomplex Fourier transforms. Journal of Mathematical Imaging and Vision, published online 08 March 2013, 1–19, DOI:10.1007/s10851-013-0430-y.

  13. T. A. Ell, Quaternion-Fourier Transforms for Analysis of Two-Dimensional Linear Time-Invariant Partial Differential Systems. In Proc. of the 32nd Conf. on Decision and Control, IEEE (1993), 1830–1841.

  14. T. A. Ell, S. J. Sangwine, Hypercomplex Fourier Transforms of Color Images. IEEE Transactions on Image Processing, 16(1), (2007) 22–35.

  15. Falcao M.I., Malonek H.R.: Generalized Exponentials through Appell sets in \({\mathbb{R}^{n+1}}\) and Bessel functions. AIP Conference Proceedings. 936, 738–741 (2007)

    Article  ADS  Google Scholar 

  16. E. Hitzer, B. Mawardi, Clifford Fourier Transform on Multivector Fields and Uncertainty Principles for Dimensions n = 2 (mod 4) and n = 3 (mod 4). Adv. Appl. Clifford Algebras 18 (3-4) (2008), 715–736.

  17. Hitzer E., Quaternion Fourier Transformation on Quaternion Fields and Generalizations. Adv. in App. Cliff. Alg. 17 (2007), 497–517.

    Article  MATH  MathSciNet  Google Scholar 

  18. Hitzer E., Directional Uncertainty Principle for Quaternion Fourier Transforms. Adv. in App. Cliff. Alg. 20(2) (2010), 271–284.

    Article  MATH  MathSciNet  Google Scholar 

  19. E. Hitzer, R. Abłamowicz, Geometric Roots of −1 in Clifford Algebras Cl(p, q) with p + q \({\leq 4}\). Adv. Appl. Clifford Algebras 21(1) (2011), 121–144, DOI:10.1007/s00006-010-0240-x.

  20. E. Hitzer, OPS-QFTs: A New Type of Quaternion Fourier Transforms Based on the Orthogonal Planes Split with One or Two General Pure Quaternions. Numerical Analysis and Applied Mathematics ICNAAM 2011, AIP Conf. Proc. 1389 (2011), 280–283; DOI:10.1063/1.3636721.

  21. E. Hitzer, J. Helmstetter, R. Abłamowitz, Square roots of −1 in real Clifford algebras. In E. Hitzer, S.J. Sangwine (eds.), Quaternion and Clifford Fourier Transforms and Wavelets, Trends in Mathematics (TIM) 27, Birkhäuser, Basel, 2013, 123–153. Preprints: http://arxiv.org/abs/1204.4576, http://www.tntech.edu/files/math/reports/TR_2012_3.pdf.

  22. E. Hitzer, J. Helmstetter, and R. Abłamowicz, Maple worksheets created with CLIFFORD for a verification of results in [21]. http://math.tntech.edu/rafal/publications.html (\({\copyright}\) 2012).

  23. E. Hitzer, T. Nitta, Y. Kuroe, Applications of Clifford’s Geometric Algebra. Adv. Appl. Clifford Alg. 23 (2013), 377–404, DOI:10.1007/s00006-013-0378-4. Preprint: http://arxiv.org/abs/1305.5663.

  24. E. Hitzer, S. J. Sangwine, The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations. In E. Hitzer, S.J. Sangwine (eds.), Quaternion and Clifford Fourier Transforms and Wavelets, Trends in Mathematics (TIM) 27, Birkhäuser, Basel, 2013, 15–39.

  25. Lounesto P.: Clifford Algebras and Spinors. CUP, Cambridge (UK) (2001)

    Book  MATH  Google Scholar 

  26. Waterloo Maple Incorporated, Maple, a general purpose computer algebra system. Waterloo, http://www.maplesoft.com (\({\copyright}\) 2012).

  27. Sangwine S. J., Biquaternion (Complexified Quaternion) Roots of −1. Adv. Appl. Clifford Algebras 16(1) (2006), 63–68.

    Article  MATH  MathSciNet  Google Scholar 

  28. Sangwine S. J., Fourier transforms of colour images using quaternion, or hypercomplex, numbers, Electronics Letters, 32(21) (1996), 1979–1980.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Liberal Arts, Department of Material Science, International Christian University, Tokyo, 181-8585, Japan

    Eckhard Hitzer

Authors
  1. Eckhard Hitzer
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Eckhard Hitzer.

Additional information

In memory of our dear friend Rev. Olaug Hansen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hitzer, E. Two-Sided Clifford Fourier Transform with Two Square Roots of −1 in Cl(p, q). Adv. Appl. Clifford Algebras 24, 313–332 (2014). https://doi.org/10.1007/s00006-014-0441-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00006-014-0441-9

Keywords