Quaternionic wavelet transform for colour images

Quaternionic Wavelet Transform (QWT) already exist but it dealt with greyscale images. In this paper we propose a quaternionic wavelet transform aimed to colour image processing. To encode colour information in our new transformation, a pixel in the spatial domain is represented by a quaternion as described by Sangwine. First, we propose to use the discrete quaternionic Fourier transform to study the spectral information of the colour image. It is well known that the frequency space of a real signal is a complex hermitian signal, we then studied the digital spectral domain of the quaternionic Fourier transform in order to analyze symmetry properties. This study gives us one characterization of the colour Fourier domain. Second we use the quaternion formalism to define a wavelet transform for colour images. We propose to generalize the filter bank construction to quaternionic formalism. Especially, we describe conditions on quaternionic filters to obtain a perfect reconstruction. We build a first colour quaternionic filter bank: the colour Shannon Wavelet. This family of functions are based on a windowing process in the quaternionic Fourier space.