Hypercomplex Fourier transforms of color images

Hypercomplex Fourier transforms based on quaternions have been proposed by several groups for use in image processing, particularly of color images. So far, however, there has not been a coherent explanation of what the spectral domain coefficients produced by a hypercomplex Fourier transform represent and this paper attempts to present such an explanation for the first time making use of the polar form of a quaternion and a separation of a quaternion spectral coefficient into components parallel and perpendicular to the hypercomplex exponentials in the transform (the basis functions)