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

This work is inspired by that of Bogaevsky, who obtained the transitions as part of an investigation of viscosity solutions of Hamilton-Jacobi equations by examining the order of contact of spheres with the surface, leading to an enumeration of possible transitions.

Abstract

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 both in the development of an object recognition system and an approach to perceptual organization. This paper classifies 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 remaining cases are shown to be viable by developing a specific example for each. We relate these transitions to a classification by Bogaevsky of singularities of the viscosity solutions of the Hamilton-Jacobi equation. We believe that the classification of these transitions is vital to the successful regularization of the medial axis and its use in real applications.