EuroCG 2011, Morschach, Switzerland, March 28–30, 2011

Boundary of a non-uniform point cloud for reconstruction

Nicolas Chevallier∗ Yvan Maillot†

Abstract

This paper deals with the problem of shape recon-struction. We define a filtration of the Delaunay com-plex of a point cloud. This filtration allows to se-lect points in a point cloud that should be boundarypoints. Theoretical guarantees are given when thepoint cloud samples a region with smooth boundary.A simple and efficient algorithm computing the filtra-tion is described.

1 Introduction

We will focus on the reconstruction of an open setΩ in Rd with smooth boundary, from a set of pointsnon-uniformly distributed inside Ω and not just lyingon its boundary.

With our hypothesis, Ω, the closure of Ω, is a man-ifold with boundary. In the past decade, many algo-rithms have been proposed for the reconstruction ofmanifolds [2, 3, 4]. Some of them give guarantees,but only for surfaces without boundary. Recently, in[7], Dey et al. have been the firsts to ensure the re-construction of surfaces with boundary. Nevertheless,their algorithm is limited to 2-dimensional surfaces inR3, from a sufficiently dense sample. Besides, theiralgorithm faces problems when the sample is not uni-form and no theoretical guarantees are provided inthis case.

Given a point cloud S sampling a bounded open setΩ in Rd with smooth boundary, our work proposes:

• To define a filtration of the Delaunay complex ofS. This filtration takes into account in a verynatural way that the local density of the cloudmay change according to the local complexity ofΩ. This filtration depending on a positive realnumber α is called Locally-Density-Adaptative-α-complex, LDA-α-complex for short.

• To select a subset in the point cloud S, calledLDA-α-boundary, that presumably constitutes asample of the boundary F of Ω.

• Thanks to a result of Chazal and Lieutier [6] toprove that the LDA-α-boundary carries topolog-ical and geometric properties of F .

∗UHA, LMIA/SD, Mulhouse, France,nicolas.chevallier@uha.fr†UHA, LMIA/MAGE, Mulhouse, France,

yvan.maillot@uha.fr

• To give an efficient algorithm for computingLDA-α-complex and LDA-α-boundary.

2 Mathematical and geometric concepts

Notations. d(x, y) = ‖x− y‖ denotes the Eu-clidean distance between to points x, y in Rd.B(x, r) = y ∈ Rd, d(x, y) ≤ r denotes the closed

ball of center x and radius r ∈ [0,∞[.Let A be a subset of Rd, AC denotes its comple-

mentary in Rd, A its closure, Ao its interior, ∂A itsboundary, conv(A) its convex hull, and diamA its di-ameter, i.e. diamA = supd(x, y) : x, y ∈ A.

Delaunay complex. Let S be a finite point set inRd. An empty ball is a ball containing no point of Sin its interior.

The Delaunay complex of S, Del(S), is the set ofconvex polytopes K = conv(S ∩ B) where B is anempty ball.

Maximal ball. A ball B of Rd is called a maximalball if B is an empty ball such that the dimension ofthe convex polytope conv(B ∩ S) is d.

Restricted Delaunay complex. Let X be apoint set in Rd. A convex polytope K in the De-launay complex Del(S) is in the Delaunay complex ofS restricted to X, denoted by Del|X(S), if there existsan empty ball B = B(x, r) such that x ∈ X and thevertices of K are on the boundary of B. This meansthat K is a face of conv(S ∩ B). The set Del|X(S)of all these convex polytopes K is a sub-complex ofDel(S).

We assume in the following that S is a finite pointset in Rd which is contained in no proper affine sub-space of Rd.

3 Idea of sampling and reconstruction

We aim at reconstructing a bounded open set Ω withsmooth boundary from a sample set S included in Ωand not only on its boundary. Moreover, the sampledoes not have to be uniformly distributed, i.e., itslocal density may be different in places. Nevertheless,the information about the shape to reconstruct has tobe carried by the sample. The sample has to be densewhere a big amount of details is required and a suddenvariation of local density indicates the presence of ahole or a hollow, as shown on figure 1 close to theeyes of the lizard or on its fingers. Since variationsof the local density are allowed, the sample may be

This is an extended abstract of a presentation given at EuroCG 2011. It has been made public for the benefit of the community and should be considered a preprint rather than a formally reviewed paper. Thus, this work is expected to appear in a conference with formal proceedings and/or in a journal.

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less dense where the shape is simple. However, thedensity must change gradually, as shown on figure 1on the back of the lizard, to avoid the formation ofnon-existent holes.

Figure 1: A shape (left), a possible sample (right)

The problem comes down to finding an efficient wayto measure the variation of local density of the sampleset S.

4 Formalization of the idea

The Delaunay complexcan be efficiently used tomeasure the density varia-tion of the sample set S.For instance, in the figureopposite, the maximal ballB is much larger than someof its neighbours. Thismeans there is a wide area with no point of S whilethe sampling around is dense. This area is proba-bly a hole and the polytope t included in B must beeliminated.

4.1 Eliminating polytopes

Polytopes will be eliminated by eliminator balls. Aneliminator ball is an empty ball large enough com-pared to the maximal balls around.

Definition 1 Let α be a real number in ]0,∞[. Anempty ball B = B(c, r) is α-eliminator if for eachpoint p of S lying on ∂B, there exists a maximal ballB′ = B′(c′, r′) with p lying on ∂B′ such that r ≥ αr′.

Here are some interesting observations:

• There exists an α∞ such that ∀α ≥ α∞ there isno α-eliminator ball since the radius of a maximalball is strictly positive and the radius of an emptyball is finite.

• There exists an α0 such that ∀α ≤ α0 all emptyballs containing a 1-polytope are α-eliminatorsince the radius of an empty ball containing a1-polytope is strictly positive and the radius of amaximal ball is finite.

• A maximal ball B = B(c, r) is 1-eliminator sincethe ball B′(c′, r′) with p lying on ∂B′ such thatr ≥ r′ is B itself.

Definition 2 A polytope K of Del(S) is α-eliminated if each empty ball B such that K ⊆conv(S ∩ ∂B) is α-eliminator.

The following observations are deduced from theprevious ones: No polytope is α∞-eliminated. All thepolytopes except the 0-polytopes are α0-eliminated.All d-polytopes are 1-eliminated.

4.2 LDA-α-complex

Definition 3 The LDA-α-complex is the set of allpolytopes K of Del(S) that are not α-eliminated.

We can collect some easy facts: The LDA-α-complex of S is a sub-complex of Del(S). The LDA-α0-complex of S is S. The LDA-α∞-complex of S isDel(S). The LDA-1-complex of S is a sub-complex ofDel(S) with no d-polytope. If α1 ≤ α2 then LDA-α1-complex ⊆ LDA-α2-complex.

The LDA-α-complex is indeed very close to theconformal-α-shape [5] with α−p equals to 0 and α+

p

equals to the radius of the smallest maximal ball con-taining p.

4.3 LDA-α-boundary

Definition 4 The LDA-α-boundary is the union ofthe sets of the vertices of all the α-eliminated poly-topes and of the set of all the vertices of the convexhull of S.

5 Algorithm

We present a very simple and efficient algorithm inthe full version of this paper. A short description isgiven in what follows.

5.1 Description

There are four main steps. First, Del(S) is computed.The second step computes for each v ∈ S, the radiusm(v) of the smallest maximal ball containing v .Thethird step computes for each polytope K in Del(S),the radius e(K) of the smallest empty ball contain-ing K. The fourth step determines the polytopes inDel(S) that are in the LDA-α-complex.

The fourth step selects the polytopes of Del(S) indecreasing order of dimensions. First, for each d-polytope K, if the inequality e(K) < αm(v) holds forat least one vertex v of K, then K is in the LDA-α-complex. Moreover, the LDA-α-boundary is de-termined at the same time since the elements of theLDA-α-boundary are the vertices of the d-polytopesthat are not selected (together with the vertices of theconvex hull of S). Next, once the polytopes of dimen-sions ≥ k have been selected, a (k− 1)-polytope K ofDel(S) is in the LDA-α-complex if either one of the

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EuroCG 2011, Morschach, Switzerland, March 28–30, 2011

A sample Its LDA-2 Its LDA-2

of a lizard -complexe -boundary

Figure 2: The lizard

Three balls with LDA- with

sampled (a) α-shapes (b) α-shapes (c)

Figure 3: The balls

k-polytope containing K is in the LDA-α-complex orif e(K) < αm(v) for at least one vertex v of K.

In the full version of this paper we prove that thetime-complexity of the algorithm is O(|Del(S)|) plusthe time required to compute the Delaunay complexDel(S).

5.2 Little examples

Thanks to CGAL [1], this algorithm has been im-plemented in 2D and 3D. Two examples are shownin this subsection. Figure 2 shows the sample of alizard, its LDA-2-complex, and its LDA-2-boundary.The density of this sample is strong only where it isneeded. Figure 3 (a) shows 3D-balls that have beenrandomly sampled according to non-constant proba-bility densities. The reconstructions work with LDA-α-complexes while it is well known that α-shapes mayfail in that case.

6 Well distributed subsets

Given a bounded open set Ω with smooth boundaryand a finite point set S in Ω, we will give some con-ditions about S ensuring that with a right choice ofα, the LDA-α-boundary of S carries the topologicalinformations about ∂Ω.

We first define (ε, δ)-sample with a general lipschitzfunction. We will specify such a function later.

Definition 5 Let f : Rd → R+ be a k-lipschitz func-tion. Let δ and ε be two positive real numbers.

A point set S is an (ε, δ)-sample of Ω if S ⊂ Ω, andif

Density: ∀x ∈ Ω, ∃y ∈ S, d(x, y) < εf(x),

Sparsity: ∀x, y ∈ S, x 6= y ⇒ d(x, y) ≥ δf(x).

In the two following lemmas, we assume that Ω is(ε, δ)-sampled and that f is k-lipschitz.

7 Two technical lemmas

We want to provide theoretical guarantees that theLDA-α-boundary is a good approximation of theboundary F of Ω.

To do this we have to prove these two results: (1)Any point in the LDA-α-boundary is near F . (2) Anypoint in F is near a point of the LDA-α-boundary.

Next lemma shows that with a right choice of theparameters α, ε, δ, the LDA-α-boundary is close to F .

Lemma 6 (elimination lemma). Suppose ε < 12k

and α > 2εδ(1−2kε) . If p is a vertex of a LDA-α-

eliminated polytope tp, then

d(p, F ) ≤ ε

1− kεf(p) (1)

d(c,Ω) > 0 (2)

where c is the center of any empty ball B such thattp ⊂ conv(S ∩B).

7.1 Statements of main results about (ε, δ)-samples

The first inequality in the elimination lemma showsimmediately that points in the LDA-α-boundary areclose to F :

Corollary 7 (α-boundary). Supposeα > 2ε

δ(1−2kε) and ε < 12k . Then all p ∈ S

that are in the LDA-α-boundary of S, are at adistance from F lower or equal to ε

1−kεf(p).

The second inequality in the elimination lemma im-mediately leads to an information about Del|Ω(S) therestricted Delaunay triangulation of S with respect toΩ:

Corollary 8 Suppose α > 2εδ(1−2kε) and ε < 1

2k .

Then the restricted Delaunay complex Del|Ω(S) is asubcomplex of the LDA-α-complex.

Until now, we only assume the function f to be k-lipschitz. In order to go further, we need to specifythe function f . Let SF be the skeleton of F , that isthe set of centers of maximal open balls that do notmeet F . In the following f : Rd → R is defined byf(x) = d(x, F ) + d(x,SF ). It is a 2-lipschitz function.

With this more precise assumption about the sam-ple set S, we are not able to prove the reverse in-clusion LDA-α-complex⊂ Del|Ω(S). Nevertheless, wecan prove that the LDA-α-complex is included in theDelaunay complex of S restricted to a neighborhoodof Ω.

Proposition 9 (neighborhood). Suppose that ε <120 and 1

4ε > α > 2. Then a maximal ball B(c,R)

such that d(c,Ω) ≥ 2εα1−4εαf(c) is α-eliminator.

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By the first corollary, any point in the LDA-α-boundary is close to F . Conversely any point in Fis close to a point of the LDA-α-boundary of S:

Proposition 10 (α-boundary). Suppose that ε <120 and 1

32ε > α > 2. Then for any p in F thereexists a point p0 in the LDA-α-boundary of S whosedistance to p is ≤ 8αεf(p).

8 A reconstruction result

The last step is to see that with a good choice ofα, ε, and δ, the topological information about F iscontained in the α-boundary. For this, we will use aresult of F. Chazal and A. Lieutier [6]. Their resultallows a reconstruction of a compact manifold Σ ofdimension d − 1 starting with a compact set K closeto Σ. They have proved that if K is close enough toΣ, then the boundary of a union of balls B(p, rp) cen-tered at the points p in K, is the union of two subsetshomeomorphic to Σ. This shows that K carries thetopological informations about Σ.

Notations. Let Σ be a compact set in Rd. Forx in Rd, denote by π(x) a point in Σ nearest to x.This point is unique if x is not in the medial axis ofΣ. Denote by LFS(x) the distance from x to theskeleton of Σ.

Definition 11 Let κ and ρ be positive real numbers,and let Σ be a compact manifold in Rd. A compactset K ⊂ Rd is a (κ, ρ)-approximation of Σ if :i. For all p ∈ K, d(p, π(p)) ≤ κρLFS(p),ii. For all p ∈ Σ, there exists a point q ∈ K such thatd(p, π(q)) < ρLFS(p).

Theorem 12 (Chazal, Lieutier) Let κ, ρ, and 0 <a < b < 1

3 − κρ be such that

(1− a′)2 +

((b′ − a′) +

b(1 + 2b′ − a′)1− b− κρ

)2

<

(1− κρ(1 + 2b′ − a′)

1− b− ρ

)2

with a′ = (a− κρ)(1− ρ)− ρ and b′ = b+κρ1−2(b+κρ) .

Let K be a (κ, ρ)-approximation of Σ. Let (rp)p∈Kbe a family of real numbers such that a ≤ rp

LFS(p) ≤ bfor all p ∈ K and set

K = K((rp)p∈K) =⋃p∈K

B(p, rp).

Then- Σ is a deformation retract of K.- K is homeomorphic to any tubular neighborhoodx ∈ Rd : d(x,Σ) ≤ s where s < reach(Σ) =d(Σ,MΣ).- The boundary ∂K is an isotopic hypersurface toΣs = x ∈ Rd : d(x,Σ) = s.

Remark 1. If Σ is an hypersurface, ∂K is isotopicto 2 copies of Σ (isotopy=⇒ homeorphism).

In order to reconstruct the boundary F = ∂Ω start-ing with the (ε, δ)-sample set S, it is enough to usethis theorem with Σ = F and K the LDA-α-boundaryof S.

Proposition 13 Choose α = 10, ε ≤ 12500 , and δ =

ε4 . Then Chazal and Lieutier’s theorem works withΣ = F and K = the LDA-α-boundary of S

All the proofs are in the full version of this paper.

References

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[4] J.-D. Boissonnat and F. Cazals. Smooth surfacereconstruction via natural neighbour interpolationof distance functions. In Proceedings of the six-teenth annual symposium on Computational ge-ometry, SCG ’00, pages 223–232, New York, NY,USA, 2000. ACM.

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