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The three different types of signal phase, as described by
Bülow (1999) are global,
instantaneous, and local. Global phase is the angular
phase of the complex Fourier transform of a signal. It gives a realvalued number for each point in the frequency domain that is indicative of
the relative position of the frequency components. Instead of giving
the phase of a certain frequency component, instantaneous and local
phase give the phase at a certain position in a real signal.
The instantaneous and local phases, however, do this in
different ways. The instantaneous phase is the angular
phase of the complex value at each signal position. The instantaneous
phase
has the drawback that while it provides local information, that information depends on the entire
signal. This causes the instantaneous phase at any point to be
effected by changes at any other point in the image, regardless of
separation distance. To overcome this problem
quadrature filters are employed. The local phase is defined as angular
phase of the quadrature filter response at a particular position of
the signal. In this paper, in order to obtain a local quaternion
phase, quaterernonic Gabor filters are used. While quaternionic
Gabor filters are not exact quaternonic quadrature filters, they are a good
approximation to them.
Gabor filters are linear timeinvariant (LTI) filters that exhibit
many useful properties, which have led to their use in a wide range of signal
processing applications. The impulse response of a twodimensional Gabor filter is
 
(13) 
where is the Gaussian with aspect ratio ,
 
(14) 
Analogous to equation 13, a quaternionic Gabor filter is defined such that the impulse response to the filter is,
 

 (15) 
where is defined as in equation 14.
The quaternionic Gabor filter can be split into its even and odd symmetries, just as
the QFT and the original image. In that case the filter h can be written as
 
(16) 
Note that h^{q}_{ee}, h^{q}_{oe}, h^{q}_{eo}, and h^{q}_{oo} are realvalued
functions (see Figure ).
hs
Figure 1 The h_{ee}, h_{oe}, h_{eo}, and
h_{oo} symmetries of a quaternionic Gabor filter
Next: Quaternionic Disparity Estimation
Up: Witten: Quaternionbased Signal Processing
Previous: Quaternionic Transform
Stanford Exploration Project
4/5/2006