Medial Axis Computation and Evolution

Abstract

In an effort to articulate models for the intuitive representation and manipulation of 2D and 3D forms, Blum invented the notion of a medial axis. His insight was to consider a disk as a basic geometric primitive and to use it to describe the reflective symmetries of an object. This representation became very popular in a variety of fields including computer vision, computer aided design, graphics, and medical image analysis. In this chapter we review the generic local structure of the medial axis due to Giblin and Kimia. We then provide an overview of algorithms to compute this representation; these algorithms are based on an integral measure of the average outward flux of a vector field defined as the gradient of the Euclidean distance function to the object’s boundary. Finally we examine the sensitivity of medial loci to boundary perturbations by modeling this as a skeletal evolution. We consider the common case where the maximal inscribed disk at a medial axis point has first-order tangency to the object boundary at two bitangent points. We derive an expression for the (local) velocity of a medial axis point as induced by motions of these bitangent points. It turns out that the medial axis computation and evolution are both closely connected by the object angle which measures the degree of parallelism (locally) between the boundaries at the two bitangent points. Our analysis provides some justification for the use of methods that consider measures proportional to the object angle at a medial axis point to indicate their stability under boundary deformation.