Generic evolutions of edges on families of diffused greyvalue surfaces

Abstract

In a previous paper we have listed some stable features of lines of maximal slope (“edges”) of a single generic greyvalue surface. In 1-parameter families of Gaussian blurred greyvalue surfaces the stable features, such asA ₃-curvature extrema and ordinary nodes of the edge line, occur for open intervals of the parameter axis, but at isolated “bad” parameter values degenerate features, such asA ₄-curvature extrema, isolated points and rhamphoid cusps, can occur. An efficient edge-based “scale-space” description of a 1-parameter family of greyvalue surfaces would consist of a set of “representative images”, where each element of this set represents exactly one open interval in the complement of the “bad” parameter values. It transpires that the number of image regions in the complement of the edge lines can increaselocally as the standard deviation of the Gaussian kernel increases (globally, i.e. when the standard deviation tends to infinity, one, of course, expects a single image region). This somewhat surprising behaviour is not restricted to certain “pathological” 1-parameter families of greyvalue surfaces but, on the contrary, arises ingeneric 1-parameter families. It also turns out that the same local edge features occur in generic solutions of certain non-linear (greyvalue) diffusion equations.