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Analogous to complex numbers, quaternions can be represented by
a magnitude and three phases with
 
(10) 
Ell (1992) introduced the quaternionic Fourier transform (QFT) for
twodimensional signals,
 
(11) 
where and R^{2} and
f is a twodimensional quaternion signal. Because
twodimensional signals can be decomposed into even and odd components along
either the x or yaxis, f can be written
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
 

 
 
 (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).
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Stanford Exploration Project
4/5/2006