Description of Wrap

Our presentation of the algorithm differs slightly from that of Edelsbrunner and this is due to our description’s reliance on integral lines of the flow map φQinduced by Q which correspond

to limit curves studied by Edelsbrunner only with opposite orientation.

As usual, a target surface Σ in Rn _{is known through a finite ε-sample P ⊂ Σ. We define a}

set Q of weighted points consisting of the Voronoi vertices in Vor P with every vertex q ∈ Q given the weight wq = hP(q), i.e. the square of the distance between q and its closest points

in P . Moreover, we denote by Q+ _{the set Q with addition of a symbolic point of infinite}

weight at infinity.

As we saw in Section 1.4.4, with this definition of Q+_{, we can write}

Vor P = Del Q+ and Del P = Vor Q+.

Notice the above equalities still hold when Q+ is replaced with Q if we restricted ourselves to the convex hulls of the sets whose Delaunay complex is being considered. In other words, if we clip Voronoi faces of Vor P by intersecting them with conv Q, Vor P = Del Q and likewise when restricted to conv P , Del P = Vor Q and in this sense the point-sets P (unweighted) and Q (weighted), as well as their Voronoi and Delaunay complexes are dual to each other. An immediate consequence of this duality is that the critical points of hQ are exactly the

Figure 6.1: Precedence relation between Delaunay simplices is shown by arrows. An arrow crossing an edge signifies that the flow on that edge is transversal while arrows ending on an edge show that the flow on the edge is tangential.

same as those of hP. We call those simplices of Del P that contain a critical point, i.e. those

that intersect their dual faces in Vor P , critical simplices (or centered in Edelsbrunner’s terminology).

Recall that the flow orbit φQ(x) of any point x ∈ Rn is a piece-wise linear curve that does

not turn in the relative interior of a Voronoi face of Vor Q. Thus, the intersection of a flow orbit of φQ and a full-dimensional simplex of Del P = Vor Q is a line segment (if not empty).

With lower dimensional simplices there is a second possibility, namely, a flow line can cross the simplex and thus intersect it in a single point. In such a case, the simplex is called transversal (or equivocal according to Edelsbrunner) relative to the flow map φQ. Of course,

the flow line can just as well intersect a non-full-dimensional simplex in a line segment in which case we say that the flow is tangential on the simplex in question or simply call the simplex tangential with respect to φQ (Edelsbrunner calls such simplices confident). Notice

that critical simplices are in fact tangential simplices that contain a critical point.

We say a simplex τ precedes a simplex σ and denote it by τ ≺ σ if τ and σ are incident simplices, i.e. τ is either a face or a coface of σ, and some flow line of φQ enters the relative

interiors of σ immediately after leaving the relative interior of τ . More formally, when τ is a coface of σ, τ ≺ σ if there exists a point x and and a time t0 > 0 and a real number

Algorithm (original)Wrap(sample point-set P ) 1 Let ∆ ⊆ D be the set of critical simplices. 2 _{Let O = {τ ∈ D : ω E τ and ∀σ ∈ ∆ : σ 5 τ }.} 3 Return I = D \ O.

Figure 6.2: The original Wrap algorithm.

when τ is a proper face of σ, τ ≺ σ if for some point x there exist time t0 > 0 and real

α > t0 such that φQ(t0, x) ∈ rel int τ and φQ(t, x) ∈ rel int σ for t0 < t < α. We define the

relation “4” as the reflexive transitive closure of “≺”, namely, τ 4 σ if there is a sequence τ = τ0 ≺ · · · ≺ τk = σ with k ≥ 0. Figure 6.1 shows the precedence relation between

Delaunay simplices.

Remark _{Edelsbrunner’s definition of the precedence relation, which we denote by “C” is} slightly different from ours, in that τ C σ if τ ≺ σ, and in addition, the flow on one of τ or σ is transversal. This definition thus invalidates τ C σ in the case where τ ≺ σ but the flow is tangential in τ and reaches the face σ of τ and continues tangentially on σ. Note that no other case is possible; flow cannot cross two incident simplices transversally and cannot move tangentially from a face to a coface. The reflexive transitive closure of “C” is denoted by “E”.

Some subtlety is associated with having a point (a minimum) at infinity in Q+_{. In practice}

this point has no role in the computation of the Delaunay triangulation of Q. However, because of it, Rn_{\ conv Q become the Voronoi cell in Vor Q}+ _{associated to the critical point}

(minimum) at infinity. Consequently, the Wrap algorithm must treat Rn _{\ conv P as a}

special abstract critical simplex ω that contains this critical point. Since this critical point is infinitely far away, every simplex τ of Del P that is contained in the boundary of conv P is considered preceded by ω.

With these preliminaries covered, Edelsbrunner’s Wrap algorithm can now be stated as shown in Figure 6.2.

As stated, the output of Wrap is not guaranteed to agree topologically with the sampled surface Σ. In fact, Edelsbrunner proves that the produced output I is the boundary of a contractible volume. However, Edelsbrunner also suggests methods for extending the

Figure 6.3: An two-dimensional example of the execution of Wrap. The removed Delaunay simplices are shown in red.

algorithm in order to allow production of non-contractible output. For example, he suggests to consider, in addition to the simplices that are preceded by ω, those that are preceded by other “significant critical simplices”. The results in rest of this chapter are witness to the accuracy of this insightful intuition; the other “significant critical simplices” turn out to be critical simplices associated to outer medial axis critical points of hQ.

In Figure 6.5, we present a modified version of Wrap which can capture the topology of Σ or rather the bounded volume S enclosed by it. As mentioned above, the modification rests primarily on separation of critical points which allows us in particular to filter out the so-called surface critical points which are in essence (at least from a topological standpoint) the artifacts of discretization of the surface. Essentially, our algorithm amends Wrap by adding to ω all other outer medial axis critical simplices.

The rest of this chapter is dedicated to proving that this modified version of Wrap produces an output that is geometrically close to S and has the same homotopy type as S provided that the input P to the algorithm is an ε-sample of Σ for a sufficiently small value of ε. In the rest of this chapter Wrap refers to this modified version.

−→

Figure 6.4: Extension of the sample points for simulation of the a critical point at infinty. Left: Delaunay triangulation of a set of points. Right: the point-set on the left is enclosed in a large enough ball and the boundary of the ball is sampled. The Delaunay triangulation of the original point-set is a subcomplex of the Delaunay triangulation of the extended one.