doi:10.1016/S0010-4485(03)00055-1 
Copyright © 2004 Elsevier Ltd. All rights reserved.
Linear one-sided stability of MAT for weakly injective 3D domain
Max Planck Institute for Computer Science, Stuhlsatzenhausweg 85, D-66123, Saarbrücken, Germany
Available online 17 April 2003.
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Abstract
Despite its usefulness in many applications, the medial axis transform (MAT) is very sensitive to the change of the boundary in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. However, it is known that MATs of 2D domains are stable if we view this phenomenon with the one-sided Hausdorff distance. This result depends on the fact that MATs are stable if the differences between them are measured with the recently introduced hyperbolic Hausdorff distance. In this paper, we extend the result for the one-sided stability of the MAT to a class of 3D domains called weakly injective, which contains many important 3D shapes typically appearing in solid modeling. Especially, the weakly injective 3D domains can have sharp features like corners or edges. In fact, by using the stability of the MAT under the hyperbolic Hausdorff distance, we obtain an explicit bound for the one-sided Hausdorff distance of the MAT of a weakly injective 3D domain with respect to that of a perturbed domain, which is linear with respect to the domain perturbation. We discuss some consequences of this result concerning the computation and the approximation of the MAT of 3D objects.
Author Keywords: Medial axis transform; Skeleton; Stability; Hausdorff distance; Hyperbolic Hausdorff distance; Weakly injective domain
Fig. 1. Instability of MAT: a small perturbation (under the Hausdorff distance) of the domain on the upper left to the one on the lower left leads to a drastic change in their corresponding MATs (upper right and lower right, resp.). But the one-sided Hausdorff distance of the original MAT (upper right) with respect to the perturbed one (lower right) still remains small.
Fig. 2. Local geometry of 2D MA and MAT around a generic point: MA and MAT are
C1 curves around a generic point.
Fig. 3. Three types of 1-prong points for 2D MAT.
Fig. 4. MA (MAT) can be decomposed into finitely many patches called
elements. Each patch has trivial topology, and according to its dimension, it is either called a
surface element or a
line element. Note that such a decomposition is not unique.
Fig. 5. Two types of the elements in MA: (a) a surface element (left), and (b) a line element (right). Both can be extended slightly at their boundaries in the
C1 manner (so that the resulting extensions are locally
C1 manifolds at the boundaries), with possible exceptions at some boundary points of a surface element (c.f. the top corner point in (a)).
Fig. 6. Domain decomposition for 3D domain. Corresponding to their MAT decomposition, 3D domains can also be decomposed into simpler ones.
Fig. 7. Two kinds of generic points in MAT(Ω). (Up) Generic point in a surface element. The maximal ball has exactly two contact points with the domain boundary. (Below) Generic point in a line element. The contact points form a circle, which, together with the MA point, makes a cone.
Fig. 8. Types of 1-prong points in 3D MAT. In addition to these, there are other types which are more 3D specific.
Fig. 9. Examples of 3D domains: (left) weakly injective and (right) not weakly injective.
Fig. 10. One-sided Hausdorff distance measures how approximately a set
A is contained in a set
B. Here,
A is contained in the ε-neighborhood of
B, which is equivalent to
Fig. 11. Two-sided Hausdorff distance measures how similar two sets
A and
B are. Here, each of
A and
B is contained in each other's ε-neighborhood, which is equivalent to
Fig. 12. The hyperbolic distance
dh(
P1|
P2) from
P1=(
p1,
r1) to
P2=(
p2,
r2) is the same with the one-sided Hausdorff distance
Fig. 14. The graph of the coefficient function (a) the logarithmic graph of
g on the whole interval (0,π/2], (b) the (normal) graph of
g on the interval [π/6,π/2].
Fig. 15. Example showing the tightness of the bound in
Theorem 1.
Fig. 16. Hexahedron: θ
Ω=π/4,
g(θ
Ω)=28.089243…
Fig. 17. Cylinder: θ
Ω=π/4,
g(θ
Ω)=28.089243…
Fig. 18. Regular tetrahedron: θ
Ω=54.73561…°,
g(θ
Ω)=19.3923…
Fig. 19. Torus: θ
Ω=π/2,
g(θ
Ω)=9.
Fig. 20. Spherical ball and Ω
λ: θ
Ω=θ
Ωλ=π/2,
g(θ
Ω)=
g(θ
Ωλ)=9. The spherical ball Ω can be regarded as the limit of Ω
λ as λ→0.
Fig. 21. (Top left) the polyhedral domain Ω
n approximating the unit spherical ball centered at the origin, and (Top right) its MA. The vertices of ∂Ω
n are distributed regularly on the unit sphere with the angle α
n=π/2
n. Actually, the origin is slightly outside of the MA pieces (except the ones on the
xy-plane). (Bottom) The angle θ
Ωn is realized by a point
A on an MA piece between two triangular boundary pieces around the north pole
P.
Fig. 22. Pruning strategy. (a) The original normal domain Ω′ with its MA. (b) The approximating weakly injective domain Ω with its MA. (c) The Hausdorff distance between Ω and Ω′. Here, (d) Comparison of MA(Ω) and MA(Ω′). Note that MA(Ω) captures an essential part of MA(Ω′), while simplifying MA(Ω′).
Fig. A1. Two spheres
Br1,m(
p1,m) and
Br2,m(
p2,m). Since Ω is weakly injective, there exist two different contact points
q1,m and
q2,m for each
P1,m.
Fig. A2. Two spheres
Br1,m(
p1,m) and
Br2,m(
p2,m) projected onto the
xy-plane.
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