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Hypercomplex numbers are multi-dimensional numbers which have more
than one complex plane. The most common type of hypercomplex numbers have
3 complex dimensions and one real one. These were introduced by
Hamilton (1866) and he termed them quaternions. The most common
application of quaternions has been towards Maxwell's equations.
Recently, however, quaternions have been applied to signal processing,
most notably pattern recognition. They have been useful for color
image analysis where previous techniques have failed. This is because
each complex quaternion axis can be associated with an RGB axis
Sanwine and Ell (2000) which allows for color edge detection. In addition, they
can be used for image segmentation, finding structure based not only
upon color, but repeating patters. This has proven useful for finding
defects in textiles Bülow and Sommer (2001). Another application is image
disparity, which is used by Bülow (1999) to show how a pattern
changes between two frames of a movie.
Image disparity offers many geophysical applications, as will be
shown, because only organized structures, patterns, are detected by
image disparity. Thus noise will have minimal effect. Since it is a phase based
technique, even low amplitude signal still retain enough information
to be viable for image disparity estimation. The applications that
will be discussed here are time lapse analysis and edge detection.

To this end, the basics of hypercomplex mathematics will be shown,
as will the quaternionic Fourier transform. Then the Gabor filter
must be defined and extended to the quaternionic case. This will
allow for the image disparity estimation to be calculated based upon
multiple complex phases. Examples and applications to time lapse
analysis and boundary detection with synthetics and real data are then
shown.

** Next:** Hypercomplex Mathematics
** Up:** Witten: Quaternion-based Signal Processing
** Previous:** Witten: Quaternion-based Signal Processing
Stanford Exploration Project

4/5/2006