Convolution Products for Hypercomplex Fourier Transforms

Authors:
Roxana Bujack;Hendrik Bie;Nele Schepper;Gerik Scheuermann
Affiliations:
Institut für Informatik, Universität Leipzig, Leipzig, Germany 04109;Department of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent University, Ghent, Belgium 9000;Department of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent University, Ghent, Belgium 9000;Institut für Informatik, Universität Leipzig, Leipzig, Germany 04109
Venue:
Journal of Mathematical Imaging and Vision
Year:
2014

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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.