Quaternionic Transform

Analogous to complex numbers, quaternions can be represented by a magnitude and three phases with

$$\begin{eqnarray} q=\mid\mid q \mid\mid e^{i\phi} e^{j\theta} e^{k\psi}.\end{eqnarray}$$ (10)

Ell (1992) introduced the quaternionic Fourier transform (QFT) for two-dimensional signals,

$$F^q(\textbf{u})= \int_{R^2} e^{-i 2 \pi u x} f(\textbf{x}) e^{-j 2 \pi v y} d^2\textbf{x},$$ (11)

where $\textbf{x}=(x,y)^T$ and $\textbf{u}=(u,v)^T$ $\in$ R² and f is a two-dimensional quaternion signal. Because two-dimensional signals can be decomposed into even and odd components along either the x- or y-axis, f can be written

$$\begin{eqnarray} f~=~f_{ee}+f_{oe}+f_{eo}+f_{oo},\nonumber\end{eqnarray}$$

with, for example, f_oe denoting the part of f that is odd with respect to x and even with respect to y. The QFT can now be decomposed to

$$\begin{eqnarray} F^q(\textbf{u})= ~\int_{R^2}cos(2\pi u x)cos(2\pi v y)f(\textbf... ...xt{sin}(2\pi u x)\text{sin}(2\pi v y) f(\textbf{x})d^2\textbf{x}.\end{eqnarray}$$ (12)

The QFT is an invertible transform and most standard Fourier theorems hold for QFTs with minimal variation. These theorems will not be rederived here, as complete proofs of the QFT extension of Rayleigh's, the shift, the modulation, the derivative, and the convolution theorem exist elsewhere Bülow (1999); Ell (1992).