The discrete quaternion Fourier transform

TLDR: The discrete quaternion Fourier transform (DQFT) as discussed by the authors is a generalization of the two-dimensional complex DFT to the case of images with more than two (but not more than four) image components.

Abstract: This paper presents a two-dimensional discrete Fourier transform (DFT), applicable to colour images (and multispectral images generally). It is based on quaternion numbers, which may be thought of as a generalisation of the complex numbers with one real part and three imaginary parts. The transform is referred to the discrete quaternion Fourier transform (DQFT). The DQFT is a useful generalisation of the two-dimensional complex DFT to the case of images with more than two (but not more than four) image components. Natural colour images are a significant example of such images and their three components, for example, red, green and blue, may now be transformed as a whole. There are some important further theoretical steps needed, however, to make generalisation of monochrome frequency domain techniques in image processing widely applicable to colour and other multispectral images.