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Two-Dimensional Clifford Windowed Fourier Transform

  • Mawardi BahriEmail author
  • Eckhard M. S. Hitzer
  • Sriwulan Adji
Chapter

Abstract

Recently several generalizations to higher dimension of the classical Fourier transform (FT) using Clifford geometric algebra have been introduced, including the two-dimensional (2D) Clifford–Fourier transform (CFT). Based on the 2D CFT, we establish the two-dimensional Clifford windowed Fourier transform (CWFT). Using the spectral representation of the CFT, we derive several important properties such as shift, modulation, a reproducing kernel, isometry, and an orthogonality relation. Finally, we discuss examples of the CWFT and compare the CFT and CWFT.

Keywords

Fringe Pattern Window Function Clifford Algebra Geometric Algebra Scalar Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Brackx, F., De Schepper, N., Sommen, F.: The two-dimensional Clifford–Fourier transform. J. Math. Imaging Vis. 26(1), 5–18 (2006) zbMATHCrossRefGoogle Scholar
  2. 2.
    Bülow, T.: Hypercomplex spectral signal representations for the processing and analysis of images. Ph.D. thesis, University of Kiel, Germany (1999) Google Scholar
  3. 3.
    Bülow, T., Felsberg, M., Sommer, G.: Non-commutative hypercomplex Fourier transforms of multidimensional signals. In: Sommer, G. (ed.) Geom. Comp. with Cliff. Alg., Theor. Found. and Appl. in Comp. Vision and Robotics, pp. 187–207. Springer, Berlin (2001) Google Scholar
  4. 4.
    Gröchenig, K., Zimmermann, G.: Hardy’s Theorem and the short-time Fourier transform of Schwartz functions. J. Lond. Math. Soc. 2(63), 205–214 (2001) CrossRefGoogle Scholar
  5. 5.
    Hitzer, E., Mawardi, B.: Clifford Fourier transform on multivector fields and uncertainty principle for dimensions n=2 (mod 4) and n=3 (mod 4). Adv. Appl. Clifford Algebr. 18(3–4), 715–736 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kemao, Q.: Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications, and implementations. Opt. Laser Eng. 45, 304–317 (2007) CrossRefGoogle Scholar
  7. 7.
    Weisz, F.: Multiplier theorems for the short-time Fourier transform. Integr. Equ. Oper. Theory 60(1), 133–149 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Zhong, J., Zeng, H.: Multiscale windowed Fourier transform for phase extraction of fringe pattern. Appl. Opt. 46(14), 2670–2675 (2007) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2010

Authors and Affiliations

  • Mawardi Bahri
    • 1
    Email author
  • Eckhard M. S. Hitzer
    • 2
  • Sriwulan Adji
    • 1
  1. 1.School of Mathematical SciencesUniversiti Sains MalaysiaPenangMalaysia
  2. 2.Department of Applied PhysicsUniversity of FukuiFukuiJapan

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