Describing the orientation or “pose” of a rigid object in three-dimensional space is a fundamental technique in computer graphics, computer aided design (CAD) and computer vision. Quaternions are an encoding for orientation that are often used in these fields because of certain inherent algebraic advantages. One open problem in this area has been to develop continuous, rational quaternion curves to parameterize orientation.
A technique for creating rational quaternion splines will be presented as well as applications of these splines to animation and to the specification of a class of surfaces known as “swept surfaces.”
The inverse of the problem is also considered. Given empirical data such as a 3-D medical image, it is shown how to fit a particular variety of swept surface, Rational Discrete Generalized Cylinders, to this data using physically-motivated active surface techniques.
Finally, a technique for determining axial direction and topology of lung vasculature from CT images will be demonstrated. This approach unifies elements of image morphology and the differential geometry of surfaces to locate axes and junctions in branching networks of vessels.