Multi-Scale Description of Space Curves and Three-Dimensional Objects

Farzin Mokhtarian
Publishing date
November 1987

This paper addresses the problem of representing the shape of three-dimensional or space curves. This problem is important since space curves can be used to model the shape of many three-dimensional objects effectively and economically. A number of shape representation methods that operate on two-dimensional objects and can be extended to apply to space curves are reviewed briefly and their shortcomings discussed.

Next, the concepts of curvature and torsion of a space curve are explained. The curvature and torsion functions of a space curve specify it uniquely up to rotation and translation. Arc-length parametrization followed by Gaussian convolution is used to compute curvature and torsion on a space curve at varying levels of detail. Larger values of the scale parameter of the Gaussian bring out more basic features of the curve. Information about the curvature and torsion of the curve over a continuum of scales are combined to produce the curvature and torsion scale space images of the curve. These images are essentially invariant under rotation, uniform scaling and translation of the curve and are used as a representation for it. Using this representation, a space curve can be successfully matched to another one of similar shape.

The application of this technique to a common three-dimensional object is demonstrated. Finally, the proposed representation is evaluated according to several criteria that any shape representation method should ideally satisfy. It is shown that the curvature and torsion scale space representation satisfies those criteria better than other possible candidate methods.