Hypercomplex polynomial wavelet-filter bank transform for color image

Abstract

Starting from a link with the discrete polynomial decomposition, we propose an extension of the non-uniform filter bank framework to vector-valued signals to carry out a non marginal color wavelet transform. Recently quaternions or Clifford algebras have been used to perform color image processing. This paper uses the embedding of color information into the imaginary part of a Quaternion. From this coding strategy, discrete Hypercomplex 2D polynomial transform is presented as generalizing the real polynomial transform to the Hypercomplex domain. We illustrate how this numerical approach can be used to define Wavelet filter bank transform for color image, with piecewise Hypercomplex polynomial. From this, the connection between discrete polynomial representation and multiscale, filter bank algorithm already defined for Grayscale image is described and the proposed color representation is discussed. This new hypercomplex polynomial-wavelet transform has an extra parameter: an imaginary unit quaternion corresponding to a privileged color direction. This new color representation satisfies the conditions of orthogonality, perfect reconstruction and linear phase filter. Moreover, a link between hypercomplex polynomial projections and discrete multiscale differentiators is studied. We also give a practical study of this new Hypercomplex transform in the contexts of image denoising, which validates the interest of our construction.