The Two-Dimensional Clifford-Fourier Transform

Authors:
Fred Brackx;Nele Schepper;Frank Sommen
Affiliations:
Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Gent, Belgium B-9000;Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Gent, Belgium B-9000;Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Gent, Belgium B-9000
Venue:
Journal of Mathematical Imaging and Vision
Year:
2006

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Abstract

Recently several generalizations to higher dimension of the Fourier transform using Clifford algebra have been introduced, including the Clifford-Fourier transform by the authors, defined as an operator exponential with a Clifford algebra-valued kernel.In this paper an overview is given of all these generalizations and an in depth study of the two-dimensional Clifford-Fourier transform of the authors is presented. In this special two-dimensional case a closed form for the integral kernel may be obtained, leading to further properties, both in the L 1 and in the L 2 context. Furthermore, based on this Clifford-Fourier transform Clifford-Gabor filters are introduced.