The Renormalized Curvature Scale Space and the Evolution Properties of Planar Curves

The Curvature Scale Space Image of a planar curve is computed by convolving a path-based parametric representation of the curve with a Gaussian function of variance $\sigma^{2}$, extracting the zeroes of curvature of the convolved curves and combining them in a scale space representation of the curve. For any given curve $\Gamma$, the process of generating the ordered sequence of curves \{ \( \Gamma_{\sigma} \mid \sigma \geq 0\) \} is known as the evolution of $\Gamma$.

It is shown that the normalized arc length parameter of a curve is, in general, not the normalized arc length parameter of a convolved version of that curve. A new method of computing the curvature scale space image reparametrizes each convolved curve by its normalized arc length parameter. Zeroes of curvature are then expressed in that new parametrization. The result is the Renormalized Curvature Scale Space Image and is more suitable for matching curves similar in shape.

Scaling properties of planar curves and the curvature scale space image are also investigated. It is shown that no new curvature zero-crossings are created at the higher scales of the curvature scale space image of a planar curve in $C_{2}$ if the curve remains in $C_{2}$ during evolution. Several positive and negative results are presented on the preservation of various properties of planar curves under the evolution process. Among these results is the fact that every polynomially represented planar curve in $C_{2}$ intersects itself just before forming a cusp point during evolution.