Arc Length Evolution and the Resampled Scale Space Representations

The curvature scale space representations of planar curves are computed by combining information about the curvature of those curves at multiple levels of detail. Similarly, curvature and torsion scale space representations of space curves are computed by combining information about the curvature and torsion of those curves at varying levels of detail. Curvature and torsion scale space representations satisfy a number of criteria such as efficiency, invariance, detail, sensitivity, robustness and uniqueness [Mokhtarian & Mackworth 1986] which makes them suitable for recognizing a noisy curve at any scale or orientation. The renormalized curvature and torsion scale space representations [Mackworth & Mokhtarian 1988] are more suitable for recognition of curves with non-uniform noise added to them but can only be computed for closed curves. The resampled curvature and torsion scale space representations introduced in this paper are shown to be more suitable than the renormalized curvature and torsion scale space representations for recognition of curves with non-uniform noise added to them. Furthermore, these representations can also be computed for open curves. A number of properties of the representation are also investigated and described. An important new property presented in this paper is that no new curvature zero-crossing points can be created in the resampled curvature scale space representation of simple planar curves.