Connecting spatial and frequency domains for the quaternion Fourier transform

TL;DR

This paper explores the relationship between classical and Mustard convolutions in quaternion Fourier transforms, providing a formula to express classical convolution as a combination of Mustard convolutions, aiding quaternion image processing.

Contribution

It derives a finite linear combination expression linking classical and Mustard convolutions in quaternion Fourier analysis, enhancing understanding of their interplay.

Findings

Derived a formula expressing classical convolution as a combination of Mustard convolutions.

Provided insights into the spectral properties of quaternion convolutions.

Facilitated future development in quaternion image processing techniques.

Abstract

The quaternion Fourier transform (qFT) is an important tool in multi-dimensional data analysis, in particular for the study of color images. An important problem when applying the qFT is the mismatch between the spatial and frequency domains: the convolution of two quaternion signals does not map to the pointwise product of their qFT images. The recently defined `Mustard' convolution behaves nicely in the frequency domain, but complicates the corresponding spatial domain analysis. The present paper analyses in detail the correspondence between classical convolution and the new Mustard convolution. In particular, an expression is derived that allows one to write classical convolution as a finite linear combination of suitable Mustard convolutions. This result is expected to play a major role in the further development of quaternion image processing, as it yields a formula for the qFT spectrum of the classical convolution.