Quaternionic Analysis of Colour Images

The aim of this paper is to present the basics of a mathematical theory of a new Fourier transform of colour images from the point of view of Clifford analysis. Quaternionic-valued signals have long been used as a model for colour images and Fourier transforms for such signals have been studied in [3], [?], [6], [?], [7]. In this paper we use a version of the Fourier kernel markedly from those in these works, which arises naturally in Clifford analysis from a consideration of the so-called Clifford-Hermite functions. The kernel we use first appeared in [1], [2]. Properties of the associated Fourier transform are developed including appropriate convolution theorems. Fourier series for images which are periodic with respect to the canonical lattice Z in R are investigated and applied to the problem of determining which signals have the property that their Z-shifts are orthogonal with respect to the natural inner product – information that is crucial in the construction of quaternionic-valued wavelets on the plane. Finally we investigate quaternionic-valued quadrature mirror filters and solutions of the two-dimensional quaternionic dilation equation.