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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Brackx, F., De Schepper, N., Sommen, F.: The two-dimensional Clifford–Fourier transform. J. Math. Imaging Vis. 26(1), 5–18 (2006)
Bülow, T.: Hypercomplex spectral signal representations for the processing and analysis of images. Ph.D. thesis, University of Kiel, Germany (1999)
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)
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)
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)
Kemao, Q.: Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications, and implementations. Opt. Laser Eng. 45, 304–317 (2007)
Weisz, F.: Multiplier theorems for the short-time Fourier transform. Integr. Equ. Oper. Theory 60(1), 133–149 (2008)
Zhong, J., Zeng, H.: Multiscale windowed Fourier transform for phase extraction of fringe pattern. Appl. Opt. 46(14), 2670–2675 (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag London
About this chapter
Cite this chapter
Bahri, M., Hitzer, E.M.S., Adji, S. (2010). Two-Dimensional Clifford Windowed Fourier Transform. In: Bayro-Corrochano, E., Scheuermann, G. (eds) Geometric Algebra Computing. Springer, London. https://doi.org/10.1007/978-1-84996-108-0_5
Download citation
DOI: https://doi.org/10.1007/978-1-84996-108-0_5
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-84996-107-3
Online ISBN: 978-1-84996-108-0
eBook Packages: Computer ScienceComputer Science (R0)