The global medial structure of regions in R^3

TL;DR

This paper classifies the geometric complexity of compact regions in R^3 using the Blum medial axis, providing algorithms and structures to analyze their topology and invariants.

Contribution

It introduces a structure theorem for the Blum medial axis, decomposing it into irreducible components with a two-level graph representation, and links this to topological invariants.

Findings

Provides an algorithm for decomposing the medial axis into irreducible components.

Defines a two-level extended graph structure for medial components.

Characterizes contractible regions using the extended graph data.

Abstract

For compact regions Omega in R^3 with generic smooth boundary B, we consider geometric properties of Omega which lie midway between their topology and geometry and can be summarized by the term "geometric complexity". The "geometric complexity" of Omega is captured by its Blum medial axis M, which is a Whitney stratified set whose local structure at each point is given by specific standard local types. We classify the geometric complexity by giving a structure theorem for the Blum medial axis M. We do so by first giving an algorithm for decomposing M using the local types into "irreducible components" and then representing each medial component as obtained by attaching surfaces with boundaries to 4--valent graphs. The two stages are described by a two level extended graph structure. The top level describes a simplified form of the attaching of the irreducible medial components to each other, and the second level extended graph structure for each irreducible component specifies how to construct the component. We further use the data associated to the extended graph structures to compute topological invariants of Omega such as the homology and fundamental group in terms of the singular invariants of M defined using the local standard types and the extended graph structures. Using the classification, we characterize contractible regions in terms of the extended graph structures and the associated data.