# The ICICS/CS Reading Room

## UBC CS TR-87-37 Summary

- No on-line copy of this technical report is available.

- The Renormalized Curvature Scale Space and the Evolution Properties of Planar Curves, November 1987 Alan K. Mackworth and Farzin Mokhtarian
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.

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