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Convolution Products for Hypercomplex Fourier Transforms

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Abstract

Hypercomplex Fourier transforms are increasingly used in signal processing for the analysis of higher-dimensional signals such as color images. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. The present paper develops and studies two conceptually new ways to define convolution products for such transforms. As a by-product, convolution theorems are obtained that will enable the development and fast implementation of new filters for quaternionic signals and systems, as well as for their higher dimensional counterparts.

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Acknowledgements

The authors would like to thank Todd Ell and Stephen Sangwine for email communication about the qFT, as well as Eckhard Hitzer for communication about the GFT and proofreading Sect. 4.

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Authors and Affiliations

  1. Institut für Informatik, Universität Leipzig, Augustuplatz 10, 04109, Leipzig, Germany

    Roxana Bujack & Gerik Scheuermann

  2. Department of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent University, Galglaan 2, 9000, Ghent, Belgium

    Hendrik De Bie & Nele De Schepper

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  1. Roxana Bujack
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  2. Hendrik De Bie
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Correspondence to Hendrik De Bie.

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Bujack, R., De Bie, H., De Schepper, N. et al. Convolution Products for Hypercomplex Fourier Transforms. J Math Imaging Vis 48, 606–624 (2014). https://doi.org/10.1007/s10851-013-0430-y

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