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

Abstract

The authors address 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. Arc-length parametrization followed by Gaussian convolution is used to compute curvature and torsion on a space curve at varying levels of detail. Information of both 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. The application of this technique to a common three-dimensional object is demonstrated. The proposed representation is then evaluated according to several criteria that any shape representation method should ideally satisfy.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>