Personal Research Interests
I first came across quaternions while working for Martin Marietta Aerospace in 1983. I was employed as a controls engineer and in the course of my work came upon the use of quaternions to describe vehicle attitude, or orientation in space. Quaternions are particularly well suited in describing 3-dimensional rotations. If you are interested in a through presentation of the application of quaternions as rotation operators, see either Jack Kuipers' book Quaternions and Rotation Sequences or, for the more mathematically inclined, Simon Altmann's book Rotations, Quaternions, and Double Groups.
Four years later, while working as a graduate research assistant at the University of Minnesota, I was doing image processing research using Wigner's spatial/spatial-frequency transformation in an attempt to mimic the Gestalt grouping properties of the human visual system. The hypothesis of this research was that the human visual system tended to visually group objects based on the spatial distribution of energy across different frequency bands.
When the time came for me to decide on a doctorial thesis topic, I blended the areas of spatial/spatial-frequency representations, systems & controls theory, and quaternion analytic functions. I reasoned that if complex analytic functions could provide a powerful mathematical basis for the study of linear time-invariant systems of ordinary differential equations, then why couldn't quaternion analytic functions provide the same utility for systems of partial differential equations? The result of this study was published in my thesis entitled Hypercomplex Spectral Transforms. For more detailed information in this area see the section on Spectral Transforms.
Six years later, Dr. Sangwine, currently with the University of Essex in England, read my work. He contacted me asking a simple question, "Could the same techniques I had laid out in my thesis be used to design color image processing filters?" He believed it could and so did I. We have been in collaboration ever since and are slowly build up the theory of hypercomplex color image analysis. Numerous advances in this area have been published over the years, as evidenced by our publication list. For more detailed information on this topic see the section on Color Image Processing.