Convergence Properties of Curvature and Torsion Scale Space Representations

Multi-scale, curvature-based shape representation techniques for planar curves and multi-scale, torsion-based shape representation techniques for space curves have been proposed to the computer vision community by Mokhtarian & Mackworth [1986], Mackworth & Mokhtarian [1988] and Mokhtarian [1988]. These representations are referred to as the regular, renormalized and resampled curvature and torsion scale space images and are computed by combining information about the curvature or torsion of the input curve at a continuum of detail levels. Arc length parametric representations of planar or space curves are convolved with Gaussian functions of varying standard deviation to compute evolved versions of those curves. The process of generating evolved versions of a curve as the standard deviation of the Gaussian function goes from 0 to $\infty$ is referred to as the evolution of that curve. When evolved versions of the curve are computed through an iterative process in which the curve is reparametrized by arc length in each iteration, the process is referred to as arc length evolution. This paper contains a number of important results on the convergence properties of curvature and torsion scale space representations. It has been shown that every closed planar curve will eventually become simple and convex during evolution and arc length evolution and will remain in that state. This result is very important and shows that curvature scale space images are well-behaved in the sense that we can always expect to find a scale level at which the number of curvature zero-crossing points goes to zero and know that new curvature zerocrossing points will not be created beyond that scale level which can be considered to be the high end of the curvature scale space image. It has also been shown that every closed space curve will eventually tend to a closed planar curve during evolution and arc length evolution and that every closed space curve will eventually enter a state in which new torsion zero-crossing points will not be created during evolution and arc length evolution and will remain in that state. Furthermore, the proofs are not difficult to comprehend. They can be understood by readers without an extensive knowledge of mathematics.