Introduction

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.