Convolution Products for Hypercomplex Fourier Transforms

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. MSC 2010 : 30G35; 42B10; 44A35; 94A12

Some Properties of the Spinor Fourier Transform

“…This is rather surprising, as in all the other examples of hypercomplex Fourier transforms (see e.g. [4]), the transform is only diagonalized by one basis and not by both.…”

“…Examples of Fourier transforms for which eigenfunctions are used to construct them, can be found in e.g. [1,2,3,4,11,12,13]. The main issue that hinders further development of applications is the lack of a suitable convolution theorem for such hypercomplex Fourier transforms.…”

“…In this work, a color image is considered to be a vector-valued function. As the link with group representations is often important when dealing with Fourier transforms, the spinor Fourier transform was constructed using group morphisms from R 2 to Spin (4). In fact, this transform can also be compared to the general geometric Fourier transform introduced by Bujack, Scheuermann and Hitzer in [5].…”

Connecting spatial and frequency domains for the quaternion Fourier transform

“…We may now formulate the following theorem, see [6], which expresses the Mustard convolution as a finite linear combination of classical convolutions: f * g + f * g (0,1) + f * g (1,0) + f * g (1,1)…”

“…The key idea to solve this problem is to find a suitable decomposition of the two signals to be convolved, so that the action of the qFT simplifies to a sum of products. This will be tackled using a mathematical tool we call Mustard convolution, introduced in [6] as follows:…”