Discrete octonion Fourier transform and the analysis of discrete 3-D data

Abstract

AbstractThe purpose of this article is to further develop the theory of octonion Fourier transformations (OFT), but from a different perspective than before. It follows the earlier work by Błaszczyk and Snopek, where they proved a few essential properties of the OFT of real-valued functions of three continuous variables. The research described in this article applies to discrete transformations, i.e. discrete-space octonion Fourier transform (DSOFT) and discrete octonion Fourier transform (DOFT). The described results combine the theory of Fourier transform with the analysis of solutions for difference equations, using for this purpose previous research on algebra of quadruple-complex numbers. This hypercomplex generalization of the discrete Fourier transformation provides an excellent tool for the analysis of 3-D discrete linear time-invariant (LTI) systems and 3-D discrete data.