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 σ², extracting the zeros of curvature of the convolved curves and combining them in a scale space representation of the curve. For any given curve Γ, the process of generating the ordered sequence of curves {Γ_σ|σ⩾0} is the evolution of Γ. It is shown that the normalized arc length parameter of a curve is, in general, not the normalized arch length parameter of a convolved version of that curve. A novel method of computing the curvature scale space image reparametrizes each convolved curve by its normalized arc length parameter. Zeros 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 C₁ if the curve remains in C₁ during evolution. Several results are presented on the preservation of various properties of planar curves under the evolution process