Properties of ridges and cores for two-dimensional images

Pizer and Eberly introduced the "core" as the analogue of the medial axis for greyscale images. For two-dimensional images, it is obtained as the "ridge" of a "medial function" defined on 2 + 1-dimensional scale space. The medial function is defined using Gaussian blurring and measures the extent to which a point is in the center of the object measured at a scale. Numerical calculations indicate the core has properties quite different from the medial axis. In this paper we give the generic properties of ridges and cores for two-dimensional images and explain the discrepancy between core and medial axis properties. We place cores in a larger "relative critical set structure", which coherently relates disjoint pieces of core. We also give the generic transitions which occur for sequences of images varying with a parameter such as time. The genericity implies the stability of the full structure in any compact viewing area of scale space under sufficiently small L-2 perturbations of the image intensity function. We indicate consequences for finding cores and also for adding "markings" to completely determine the structure of the medial function.