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.
Deep Dive into The global medial structure of regions in R^3.
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
We consider a compact region Ω ⊂ R 3 with generic smooth boundary B . We are interested in geometric properties of Ω and B which lie midway between their topology and geometry and can be summarized by the term "geometric complexity". Our goal in this paper is first to give a structure theorem for the "geometric complexity" of such regions. Second, we directly relate this structure to the topology of the region and deduce how the topology places restrictions on the geometric complexity and how the structure capturing the geometric complexity determines the topology of the region.
To explain what we mean by geometric complexity, we first consider R 2 . If B is a simple closed curve as in Figure 1 (a), then by the Jordan Curve and Schoenflies Theorems, Ω in Figure 1 (b) is topologically a 2-disk, and hence contractible. Hence, the region in Figure 1 (b) is topologically simple; however, it has considerable “geometric complexity”. It is this geometric complexity which is important for understanding shape features for both 2 and 3 dimensional objects in a number of areas such as computer and medical imaging, biology, etc. This complexity is not captured by traditional geometrical invariants such as the local curvature of B nor by global geometric invariants given by integrals over B or Ω.
Nor do traditional results such as the Riemann mapping theorem, which provides the existence of a conformal diffeomorphism of D 2 with Ω, provide a comparison of the geometric complexity of Ω with D 2 . For 2-dimensional contractible regions, a theorem of Grayson [14], building on the combined work of Gage and Hamilton [10,11], shows that under curvature flow the boundary B evolves so the fingers of the region shrink and the region ultimately simplifies and shrinks to a convex region which contracts to a “round point”. Throughout this evolution, the boundary of the region remains smooth. The order of shrinking and disappearing of subregions provides a model of the geometric complexity of the region. Unfortunately, this approach already fails in R 3 , where the corresponding mean curvature flow may develop singularities, as in the case of the “dumbell” surface found by Grayson (also see eg Sethian [18]).
The structure theorem for the “geometric complexity” which we give is based on the global structure of the “Blum medial axis” M . The medial axis is a singular space which encodes both the topology and geometry of the region (see Figure 2). It is defined in all dimensions and has multiple descriptions including locus of centers of spheres in Ω which are tangent to B at two or more points (or have a degenerate tangency) as in Blum and Nagel [2], the shock set for the eikonal/ “grassfire” flow as in Kimia, Tannenbaum and Zucker [15], which is also a geometric flow on Ω, and the Maxwell set for the family of distance to the boundary functions as in Mather [17]. It has alternately been called the central set by Yomdin [20], and has as an analogue the cut-locus for regions without conjugate points in Riemannian manifolds.
The multiple descriptions allow its local structure to be explicitly determined for regions in R n+1 with generic smooth boundaries: M is an n-dimensional Whitney stratified set (by Mather [17]) which is a strong deformation retract of Ω. The local structure of M is given by a specific list of local models, by Blum and Nagel for n = 1, by Yomdin [20] for n ≤ 3 (where it is called the central set) and Mather [17] for n ≤ 6, and is given a precise singularity theoretic geometrical description by Giblin [12] for n = 2. Results of Buchner [5] show the cut locus has analogous properties for regions without conjugate points (and our structure theorem extends to this more general case). Furthermore, by results in [7,8,9,6], we can derive the local, relative, and global geometry of both Ω and B in terms of geometric properties defined on M . For example, for generic compact regions in R 2 , M is a 1-dimensional singular space whose singular points either have Y-shaped branching or are end points. This defines a natural graph structure with vertices for the branch and end points and edges representing the curve segments joining these points (see Figure 3). This graph structure encodes the geometric complexity of the region. Furthermore, Ω is contractible if and only if the graph is a tree. Then the tree structure can be used to contract the region (also for computer imaging a tree structure is a desirable feature, as trees can be searched in polynomial time). A computer scientist Mads Nielsen has asked whether in the case of contractible Ω ⊂ R 3 , the Blum medial axis still has a “tree structure”. We answer this question as part of the structure theorem for the medial axis M for generic regions.
While the global structure of the medial axis as a Whitney stratified set with specific local singular structure does capture the geometric complexity of Ω, it is insufficient in We give a brief overview of the form of the struc
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