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