Transitions of the 3D Medial Axis under a One-Parameter Family of Deformations

The instabilities of the medial axis of a shape under deformations have long been recognized as a major obstacle to its use in recognition and other applications. These instabilities, or transitions, occur 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 in the development of an object recognition system where the classified instabilities were utilized to represent deformation paths. The classification of generic transitions of the 3D medial axis could likewise potentially lead to a similar representation in 3D. In this paper, these transitions are classified 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 nongeneric in a one-parameter family of deformations. Finally, the remaining cases are shown to be viable by developing a specific example for each. Our work is inspired by that of Bogaevsky, who obtained the transitions as part of an investigation of viscosity solutions of Hamilton-Jacobi equations. Our contribution is to give a more down-to-earth approach, bringing this work to the attention of the computer vision community, and to provide explicit constructions for the various transitions using simple surfaces. We believe that the classification of these transitions is vital to the successful regularization of the medial axis in its use in real applications.