Convolution products for hypercomplex Fourier transforms

TL;DR

This paper introduces two new convolution products for hypercomplex Fourier transforms, enabling advanced filter design for higher-dimensional signals like color images, and provides convolution theorems to facilitate their implementation.

Contribution

It develops and analyzes two novel convolution definitions for hypercomplex Fourier transforms, addressing a key gap in filter design for quaternionic and higher-dimensional signals.

Findings

New convolution theorems for hypercomplex Fourier transforms

Facilitation of fast filter implementation for quaternionic signals

Enhanced analysis tools for higher-dimensional signal processing

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.