Abstract
The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straightforward definition of a general geometric Fourier transform covering most versions in the literature. We show which constraints are additionally necessary to obtain certain features such as linearity or a shift theorem. As a result, we provide guidelines for the target-oriented design of yet unconsidered transforms that fulfill requirements in a specific application context. Furthermore, the standard theorems do not need to be shown in a slightly different form every time a new geometric Fourier transform is developed since they are proved here once and for all.
Mathematics Subject Classification (2010). Primary 15A66, 11E88; secondary 42A38, 30G35.
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
T. Batard, M. Berthier, and C. Saint-Jean. Clifford Fourier transform for color image processing. In Bayro-Corrochano and Scheuermann [2], pages 135–162.
E.J. Bayro-Corrochano and G. Scheuermann, editors. Geometric Algebra Computing in Engineering and Computer Science. Springer, London, 2010.
F. Brackx, N. De Schepper, and F. Sommen. The Clifford–Fourier transform. Journal of Fourier Analysis and Applications, 11(6):669–681, 2005.
F. Brackx, N. De Schepper, and F. Sommen. The two-dimensional Clifford–Fourier transform. Journal of Mathematical Imaging and Vision, 26(1):5–18, 2006.
F. Brackx, N. De Schepper, and F. Sommen. The Clifford–Fourier integral kernel in even dimensional Euclidean space. Journal of Mathematical Analysis and Applications, 365(2):718–728, 2010.
F. Brackx, N. De Schepper, and F. Sommen. The Cylindrical Fourier Transform. In Bayro-Corrochano and Scheuermann [2], pages 107–119.
T. Bülow. Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images. PhD thesis, University of Kiel, Germany, Institut für Informatik und Praktische Mathematik, Aug. 1999.
W.K. Clifford. Applications of Grassmann’s extensive algebra. American Journal of Mathematics, 1(4):350–358, 1878.
H. De Bie and F. Sommen. Vector and bivector Fourier transforms in Clifford analysis. In K. Guerlebeck and C. Koenke, editors, 18th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, page 11, 2009.
J. Ebling. Visualization and Analysis of Flow Fields using Clifford Convolution. PhD thesis, University of Leipzig, Germany, 2006.
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.
T.A. Ell and S.J. Sangwine. Hypercomplex Fourier transforms of color images. IEEE Transactions on Image Processing, 16(1):22–35, Jan. 2007.
M. Felsberg. Low-Level Image Processing with the Structure Multivector. PhD thesis, Christian-Albrechts-Universität, Institut für Informatik und Praktische Mathematik, Kiel, 2002.
D. Hestenes and G. Sobczyk. Clifford Algebra to Geometric Calculus. D. Reidel Publishing Group, Dordrecht, Netherlands, 1984.
E. Hitzer. Quaternion Fourier transform on quaternion fields and generalizations. Advances in Applied Clifford Algebras, 17(3):497–517, May 2007.
E. Hitzer and R. Abłamowicz. Geometric roots of −1in Clifford algebras Cℓ p,q with p + q ≤ 4. Advances in Applied Clifford Algebras, 21(1):121–144, 2010. Published online 13 July 2010.
E. Hitzer, J. Helmstetter, and R. Abłamowicz. Square roots of −1in real Clifford algebras. In K. Gürlebeck, editor, 9th International Conference on Clifford Algebras and their Applications, Weimar, Germany, 15–20 July 2011. 12 pp.
E.M.S. Hitzer and B. Mawardi. Clifford Fourier transform on multivector fields and uncertainty principles for dimensions n = 2(mod 4) and n = 3(mod4). Advances in Applied Clifford Algebras, 18(3-4):715–736, 2008.
B. Jancewicz. Trivector Fourier transformation and electromagnetic field. Journal of Mathematical Physics, 31(8):1847–1852, 1990.
S.J. Sangwine and T.A. Ell. The discrete Fourier transform of a colour image. In J.M. Blackledge and M.J. Turner, editors, Image Processing II Mathematical Methods, Algorithms and Applications, pages 430–441, Chichester, 2000. Horwood Publishing for Institute of Mathematics and its Applications. Proceedings Second IMA Conference on Image Processing, De Montfort University, Leicester, UK, September 1998.
F. Sommen. Hypercomplex Fourier and Laplace transforms I. Illinois Journal of Mathematics, 26(2):332–352, 1982.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this chapter
Cite this chapter
Bujack, R., Scheuermann, G., Hitzer, E. (2013). A General Geometric Fourier Transform. In: Hitzer, E., Sangwine, S. (eds) Quaternion and Clifford Fourier Transforms and Wavelets. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0603-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0603-9_8
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0602-2
Online ISBN: 978-3-0348-0603-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)