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Optimized Fast Algorithms for the Quaternionic Fourier Transform

  • Michael Felsberg
  • Gerald Sommer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1689)

Abstract

In this article, we deal with fast algorithms for the quaternionic Fourier transform (QFT). Our aim is to give a guideline for choosing algorithms in practical cases. Hence, we are not only interested in the theoretic complexity but in the real execution time of the implementation of an algorithm. This includes floating point multiplications, additions, index computations and the memory accesses. We mainly consider two cases: the QFT of a real signal and the QFT of a quaternionic signal. For both cases it follows that the row-column method yields very fast algorithms. Additionally, these algorithms are easy to implement since one can fall back on standard algorithms for the fast Fourier transform and the fast Hartley transform. The latter is the optimal choice for real signals since there is no redundancy in the transform. We take advantage of the fact that each complete transform can be converted into another complete transform. In the case of the complex Fourier transform, the Hartley transform, and the QFT, the conversions are of low complexity. Hence, the QFT of a real signal is optimally calculated using the Hartley transform.

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References

  1. 1.
    R. N. Bracewell. The Fourier transform and its applications. McGraw Hill, 1986.Google Scholar
  2. 2.
    T. Bülow and G. Sommer. Algebraically Extended Representation of Multi-Dimensional Signals. In Proceedings of the 10th Scandinavian Conference on Image Analysis, pages 559–566, 1997.Google Scholar
  3. 3.
    T. Bülow and G. Sommer. Multi-Dimensional Signal Processing Using an Algebraically Extended Signal Representation. In G. Sommer and J.J. Koenderink, editors, Int'l Workshop on Algebraic Frames for the Perception-Action Cycle, AFPAC'97, Kiel, volume 1315 of LNCS, pages 148–163. Springer, 1997.CrossRefGoogle Scholar
  4. 4.
    V. M. Chernov. Discrete orthogonal transforms with data representation in composition algebras. In Proceedings of the 9th Scandinavian Conference on Image Analysis, pages 357–364, 1995.Google Scholar
  5. 5.
    Digital Equipment Corporation. Digital Semiconductor Alpha 21164PC Microprocessor Data Sheet3, 1997.Google Scholar
  6. 6.
    T. A. Ell. Hypercomplex Spectral Transformations. PhD thesis, University of Minnesota, 1992.Google Scholar
  7. 7.
    M. Felsberg. Signal Processing Using Frequency Domain Methods in Clifford Algebra4. Master’s thesis, Christian-Albrechts-University of Kiel, 1998.Google Scholar
  8. 8.
    M. Felsberg et al. Fast Algorithms of Hypercomplex Fourier Transforms. In G. Sommer, editor, Geometric Computing with Clifford Algebra, Springer Series in Information Sciences. Springer, Berlin, 1999. to appear.Google Scholar
  9. 9.
    B. Jähne. Digitale Bildverarbeitung. Springer, Berlin, 1997.Google Scholar
  10. 10.
    W. Press et al. Numerical Recipes in C. Cambridge University Press, 1994.Google Scholar
  11. 11.
    Silicon Graphics, Inc. MIPS RISC Technology R10000 Microprocessor Technical Brief, 1998.Google Scholar
  12. 12.
    Sun Microsystems, Inc. UltraSPARC-II Data Sheet6, 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Michael Felsberg
    • 1
  • Gerald Sommer
    • 1
  1. 1.Christian-Albrechts-University of KielInstitute of Computer Science and Applied Mathematics Cognitive SystemsKielGermany

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