A Theory of Multi-Scale Curvature-Based Shape Representation for Planar Curves

This paper presents a multi-scale, curvature-based shape representation technique for planar curves which satisfies several criteria, considered necessary for any shape representation method, better than other shape representation techniques. As a result, the representation is suitable for tasks which call for recognition of a noisy curve of arbitrary shape at an arbitrary scale or orientation. The method rests on the concept of describing a curve at varying levels of detail using features that are invariant with respect to transformations which do not change the shape of the curve. Three different ways of computing the representation are described in this paper. These three methods result in three different representations: the curvature scale space image, the renormalized curvature scale space image, and the resampled curvature scale space image. The process of describing a curve at increasing levels of abstraction is referred to as the evolution of that curve. Several evolution properties of planar curves are described in this paper. Some of these properties show that evolution is a physically plausible operation and characterize possible behaviours of planar curves during evolution. Some show that the representations proposed in this paper in fact satisfy some of the required criteria. Others impose constraints on the location of a planar curve as it evolves. Together, these evolution properties provide a theoretical foundation for the representation methods introduced in this paper.