Spatial and spectral quaternionic approaches for colour images

Hypercomplex or quaternions numbers have been used recently for both greyscale and colour image processing. Fast, numerous hypercomplex 2D Fourier transforms were presented as a generalization of the complex 2D Fourier transform to this new hypercomplex space. Thus, the major problem was to put an interpretation of what information the Fourier coefficients could provide. In this paper, we first define the conditions on the spectrum coefficients needed to reconstruct a colour image without loss of information through the inverse quaternionic Fourier transform process. The result is used to interpret the quaternionic spectrum coefficients of this specific colour Fourier transform. Secondly, with this apprehension of the quaternion numbers and the corresponding colour spectrum space, we define spatial and frequential strategies to filter colour images.