Robust Shape Directions

This chapter deals with shape affine normalization. This method associates with all shapes deduced from each other by an affine distortion a single normalized shape. A crucial ingredient for normalization is the computation of a small affine covariant set of robust straight lines associated with a shape. The set of all tangent lines to a shape has this covariance property, but it is too large. A very successful idea is to use bitangent lines, that is, lines tangent to a shape at two different points. If the shape has a finite number of inflexion points it also has a finite number of bitangent lines. In Sect. 3.3 a well-established curve affine invariant smoothing algorithm will be briefly described. This smoothing permits a drastic reduction of the number of bitangent lines. Yet, not all shapes can be encoded by using bitangents. Convex shapes have no bitangents and simple shapes have only a few. This explains why shape recognition algorithms compute other robust straight lines associated with the shape. Flat parts of curves are informally defined as intervals of the curve along which the direction of the tangent line does not vary too much. For instance, large enough polygons show as many reliable flat parts as sides. This chapter will present a simple parameterless definition of flat parts, based again on the Helmholtz principle.