On the Local Form and Transitions of Symmetry Sets, Medial Axes, and Shocks

摘要

In this paper we explore the local geometry of the medial axis (MA) and shocks (SH), and their structural changes under deformations, by viewing these symmetries as subsets of the symmetry set (SS) and present two results. First, we establish that the local form of the medial axis must generically be one of three cases, which we denote by the A notation explained below (here, it merely serves as a reference to sections of the paper): endpoints (A 3), interior points (A 1 2), and junctions (A 1 3). The local form of shocks is then derived from a sub-classification of these points into six types. Second, we address the (classical) instabilities of the MA, i.e., abrupt changes in the representation arising from slight changes in shape, as when a new branch appears with slight protrusion. The identification of these ‘transitions’ is clearly crucial in robust object recognition. We show that for the medial axis only two such instabilities are generically possible: (i) when four branches come together (A 1 4), and (ii) when a new branch grows out of an existing one (A 1 A 3). Similarly, there are six cases of shock instabilities, derived as sub-classifications of the MA instabilities. We give an explicit example of a dent forming in an ellipse where many of the transitions described in the paper can be seen to appear.