Abstract

The instabilities of the medial axis of a shape under de- formations have long been recognized as a ma jor obstacle to its use in recognition and other applications. These instabilities, or transitions, oc- cur when the structure of the medial axis graph changes abruptly under deformations of shape. The recent classification of these transitions in 2D for the medial axis and for the shock graph, was a key factor both in the development of an ob ject recognition system and an approach to perceptual organization. This paper classiffies generic transitions of the 3D medial axis, by examining the order of contact of spheres with the surface, leading to an enumeration of possible transitions, which are then examined on a case by case basis. Some cases are ruled out as never occurring in any family of deformations, while others are shown to be non-generic in a one-parameter family of deformations. Finally, the re- maining cases are shown to be viable by developing a specific example for each. We relate these transitions to a classiffication by Bogaevsky of sin- gularities of the viscosity solutions of the Hamilton-Jacobi equation. We believe that the classiffication of these transitions is vital to the successful regularization of the medial axis and its use in real applications.