International audienceThe complexity of the 3D-Delaunay triangulation (tetrahedralization) of n points distributed on a surface ranges from linear to quadratic. When the points are a deterministic good sample of a smooth compact generic surface, the size of the Delaunay triangulation is O(n log n). Using this result, we prove that when points are Poisson distributed on a surface under the same hypothesis, whose expected number of vertices is λ, the expected size is O(λ log^2 λ)

Devillers, Olivier

Duménil, Charles

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HAL Id: hal-02394144https://hal.archives-ouvertes.fr/hal-02394144Submitted on 1 Jul 2020HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.A Poisson sample of a smooth surface is a good sampleOlivier Devillers, Charles DuménilTo cite this version:Olivier Devillers, Charles Duménil. A Poisson sample of a smooth surface is a good sample. EuroCG2019 - 35th European Workshop on Computational Geometry, Mar 2019, Utrecht, Netherlands. ￿hal-02394144￿A Poisson sample of a smooth surface is a goodsampleOlivier Devillers1 and Charles Duménil11 Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, FranceOlivier.Devillers@inria.fr, charles.dumenil@inria.frAbstractThe complexity of the 3D-Delaunay triangulation (tetrahedralization) of n points distributedon a surface ranges from linear to quadratic. When the points are a deterministic good sampleof a smooth compact generic surface, the size of the Delaunay triangulation is O(n logn) [2].Using this result, we prove that when points are Poisson distributed on a surface under the samehypothesis, whose expected number of vertices is λ, the expected size is O(λ log2 λ).1 IntroductionWhile the complexity of the Delaunay triangulation of n points is strictly controlled in twodimensions to be between n and 2n triangles (depending on the size of the convex hull) thegap between the lower and upper bound ranges from linear to quadratic in dimension 3. Theworst case is obtained using points on the moment curve1 and the best case by using thecenter of spheres defining a packing.2To get a more precise result on the size of the 3D Delaunay triangulation, it is possibleto make different kinds of hypotheses on the point set. A first possibility is to assume arandom distribution in 3D and if the points are evenly distributed in a sphere [6], (resp. ina cube [3]), Dwyer (resp. Bienkoswski et al.) proved that the expected size is Θ(n). Butthis hypothesis of random distribution is not relevant for all applications, for example whendealing with 3D reconstruction the Delaunay triangulation is an essential tool and it is muchmore natural to assume that the points are not distributed in space but on a surface [4]. Ifthe points are evenly distributed on the boundary of a polyhedron, the expected size wasproved to be Θ(n) in the convex case [9] and between Ω(n) and Õ(n) in the non convex caseby Golin and Na [8].Instead of using probabilistic hypotheses one can assume that the points are a goodsampling of the surface, namely an (ε, η)-sample where any ball of radius ε centered on thesurface contains at least one and at most η points of the point-set. Under such hypothesisAttali and Boissonnat proved that the complexity of the Delaunay triangulation of a poly-hedron is linear [1]. Attali, Boissonnat, and Lieutier extend this result to smooth surfacesverifying some genericity hypotheses with an upper bound of O(n logn) [2]. The genericityhypothesis is crucial since Erickson proved that there exists good sample of a cylinder with atriangulation of size Ω(n√n) [7]. In the example by Erickson the point set is placed in a veryspecial position on an helix, nevertheless, even with an unstructured point set it is possibleto reach a supra-linear triangulation since Erickson, Devillers, and Goaoc proved that thetriangulation of points evenly distributed on a cylinder has expected size Θ(n logn) [5].1 The moment curve is parameterized by (t, t2, t3). When computing the Delaunay triangulation ofpoints on this curve, any pair of points define a Delaunay edge.2 The kissing number in 3D is 12, thus in such a point set, the number of edges is almost 6n.35th European Workshop on Computational Geometry, Utrecht, The Netherlands, March 19–20, 2019.This is an extended abstract of a presentation given at EuroCG’19. It has been made public for the benefit of thecommunity and should be considered a preprint rather than a formally reviewed paper. Thus, this work is expectedto appear eventually in more final form at a conference with formal proceedings and/or in a journal.40:2 Poisson sample is goodContributionIn this paper we prove that a Poisson sample of parameter λ on a smooth surface of finite areais an (ε, η)-sample for ε = 3»logλλ and η = 1000 log λ with high probability. Using the resultof Attali, Boissonnat, and Lieutier, it yields that the complexity of the Delaunay triangulationof a Poisson sample of a generic surface is O(λ log2 λ) losing an extra logarithmic factor withrespect to the case of good sampling (see Section 3).2 Notation, definitions, previous resultsWe consider a surface Σ embedded in R3, compact, smooth, oriented and without boundary.At a point p ∈ Σ, for a given orientation, we denote by κ1(p) and κ2(p) the principalcurvatures at p with κ1(p) > κ2(p). We assume that the curvature is bounded and defineκsup = supp∈Σ max(|κ1(p)|, |κ2(p)|). We denote by σ(p,R) the sphere of center p and radiusR. We denote by B(σ) the closed ball whose boundary is the sphere σ, by E̊ the interior of aset E and, for p ∈ Σ, by D(p,R) the intersection between Σ and the B̊(σ(p,R)). Abusivelywe call D(p,R) a disk. For a discrete set X, we denote ] (X) the cardinality of X. If X is aset of points, Del (X) denotes the Delaunay triangulation of X. In the 3D case, ] (Del (X)) isthe sum of the number of tetrahedra, triangles, edges and vertices belonging to the Delaunaytriangulation.Without loss of generality, we assume that Area(Σ) = 1 and consider that the set ofpoints X is a Poisson point process with parameter λ > 0 over Σ.We recall classical properties of a Poisson sample:I Observation 2.1. For two regions R and R′ of Σ,P [] (X ∩R) = k] = (λArea(R))kk! e−λArea(R),E [] (X ∩R)] = λArea (R),R ∩R′ = ∅ ⇒ ] (X ∩R) and ] (X ∩R′) are independent random variables.In particular, we have P [] (X ∩R) = 0] = e−λArea(R) and E [] (X)] = λ.We consider the same definition of genericity as Attali, Boissonnat and Lieutier, roughly:the set of points where one of the principal curvatures is locally maximal is a finite set ofcurves whose total length is bounded and, the number of contacts of any medial ball withthe surface is finite.Then we define what is a good-sampling of a surface and precise the result by Attali,Boissonnat and Lieutier.I Definition 2.2 (Good sample). A point-set on a surface is an (ε, η)-sample if any ball ofradius ε centered on the surface contains at least one and at most η points of the sample.I Theorem 2.3 ([2]). The 3D Delaunay triangulation of an (ε, η)-sample of a generic smoothsurface has complexity OÄη2ε2 log1εä.While the result of Attali et al. provides a bound O(N lnN) on complexity of theDelaunay triangulation of an (ε, η)-sample of N points and a constant η, by looking morecarefully at the result [2, Eq.(14)], we notice that the actual complexity can be expressed byC(ηε )2 log(ε−1) for C being a constant of the surface.O. Devillers and C. Duménil 40:3√32κsup1κsupSFigure 1 Illustration of the proof of Lemma 3.1 for the 2D case.3 Is a random sample a good sample?In a Poisson sampling of parameter λ on the surface, a disk of radius ε = 1√λis expected tocontain π points, but with constant probability it can be empty or contains more than η points.Thus with high probability there will be such disks even if their number is limited. Thussuch a sample is likely not to be a good sample with ε2 = 1λ and η constant. Nevertheless, itis possible to not consider η as a constant, namely, we take η = Θ(log(λ)). In a first Lemma,we bound the area of D(p,R), for any p ∈ Σ and R > 0 sufficiently small.I Lemma 3.1. Let Σ be a smooth surface of curvature bounded by κsup, and consider p ∈ Σand R > 0 smaller than 1κsup . The area of D(p,R) is greater than34πR2.Proof. The bound is obtained by considering the fact that the surface must stay in betweenthe two tangent spheres of curvature κsup tangent to the surface at p. The tangent disk at pof radius√32 R >√321κsupis included in the projection of D(p, r) on the tangent plane andthus has a smaller area than D(p,R). JI Lemma 3.2. Let Σ be a C3 surface of curvature bounded by κsup.For R small enough, Area(D(p,R)) < 54πR2.Proof. Let z = f(x, y) := 12κ1x2 + 12κ2y2 +O(x3 + y3) be the Monge of Σ patch [10] at apoint p. We denote by dσ an element of surface and by A(p,R) the projection of D(p,R) onthe xy-plane. Since on D(p,R) the slope of the normal to Σ is bounded, we have:Area (D(p,R)) =∫D(p,R)dσ =∫ ∫A(p,R) 1 + (∂f∂x(x, y))2 + (∂f∂y(x, y))2dxdyThat is smaller than∫ ∫x2+y2≤R2»1 + (∂f∂x (x, y))2 + (∂f∂y (x, y))2dxdy, sinceD(p,R) ⊂ B(p,R).Since f is in C3 and ∂f∂x (x, y) ∼ κ1x, we can say that there exists a neighborhood of p onwhich |∂f∂x | ≤√2κ1|x| ≤√2κsup|x|, i.e.,Ä∂f∂xä2≤ 2(κsupx)2. Applying the same for y, andturning to polar coordinates, we get:Area(D(p,R)) ≤∫ 2πθ=0∫ Rr=0r»1 + 2(rκsup)2drdθ =π3(2(Rκsup)2 + 1)32 − 1κ2supEuroCG’1940:4 Poisson sample is goodppiεε3Figure 2 A disk of radius ε always contains a disk of a maximal set of disks of radius ε3 ,Noticing that (a+ 1) 32 − 1 = aa+√a+1+2√a+1+1 ≤158 a for a < 1, we can conclude that for anyR small enough,Area(D(p,R)) ≤ π3154 (Rκsup)2κ2sup= 54πR2.JI Lemma 3.3. Let Σ be a C3 surface with Area(Σ) = 1. Let MR be a maximal set of kRdisjoint disks D(pi, R) on Σ. If R is small enough then kR ≤ 43πR2 .Proof. By Lemma 3.1, for R small enough, we have D(p,R) ≥ 34πR2. Thus:kR ·34πR2 ≤i=kR∑i=1Area (D(pi, R)) ≤ Area (Σ) = 1,and we can deduce the following bound: kR ≤ 43πR2 . JI Lemma 3.4. Let X be a Poisson sample of parameter λ distributed on a C3 smooth closedsurface Σ of area 1. If λ is large enough, the probability that there exists p ∈ Σ such thatD(p, 3»logλλ)does not contain any point of X is O(λ−1).Proof. We prove that a Poisson sample has no empty disk of radius 3»logλλ with probabilityO(λ−1). In a first part we use a packing argument. On the one hand, for any ε > 0 smallenough and given a maximal set Mε/3 and any point p ∈ Σ, the disk D(p, ε) contains entirelyone of the disksD(pi, ε3 ) belonging toMε/3. Indeed, by maximality ofMε/3, the diskD(p, ε/3)intersects a disk of Mε/3 whose diameter is 2ε/3 so D(p, ε) contains it entirely. On the otherhand, remember from Lemma 3.1 that if ε is small enough then Area(D(p, ε))≥ 34πε2. Thenwe can bound the probability of existence of an empty disk for ε small enough:O. Devillers and C. Duménil 40:5P [∃p ∈ Σ, ] (X ∩D(p, ε)) = 0] ≤ P[∃i < kε/3, ] (X ∩D(pi, ε/3)) = 0]≤ kε/3 P [] (X ∩D(c, ε/3)) = 0] for a point c on Σ≤ 43π(ε/3)2 e−λ 34π(ε3 )2= 12πε2 e−λ 1πε212 .By taking ε = 3»logλλ we get:P[∃p ∈ Σ, ](X ∩D(p, 3»logλλ ))= 0]≤ 4λ3π logλe− 3π logλ4 = O(λ−1).JWe have proved that when a Poisson sample is distributed on a surface, the pointssufficiently cover the surface, i.e., there is no large empty disk on the surface with highprobability. Now we have to verify the other property of a good sample, namely, a Poissonsample does not create large concentration of points in a small area.I Lemma 3.5. Let X be a Poisson sample of parameter λ distributed on a C3 closed surfaceof area 1. If λ is large enough, the probability that there exists p ∈ Σ such that D(p, 3»logλλ )contains more than 1000 log(λ) points of X is O(λ−2).Proof. Consider an Mε maximal set, we can notice that for any p ∈ Σ, the disk D(p, ε) withp ∈ Σ is entirely contained in one disk D(pi, 3ε) that is an augmented disk of Mε. Indeed,by maximality of Mε, the disk D(p, ε) intersects a disk from Mε say D(pj , ε) so D(pj , 3ε)contains entirely D(p, ε).Then we can bound the probability of existence of a disk containing more than η points:P [∃p ∈ Σ, ] (X ∩D(p, ε)) > η] ≤ P [∃i < kε, ] (X ∩D(pi, 3ε)) > η]≤ kε P [] (X ∩D(c, 3ε)) > η] for a point c on Σ≤ 43πε2 P [] (X ∩D(c, 3ε)) > η]We use a Chernoff inequality [11] to bound P [] (X ∩D(c, 3ε)) > η]: If V follows a Poissonlaw of mean v0, then ∀v > v0,P (V > v) ≤ ev−v0(v0v)v.From Lemmas 3.1 and 3.2, we have that: 274 πε2 ≤ Area(D(c, 3ε)) ≤ 454 πε2 for ε smallenough. Consequently we can say that the expected number of points v0 in D(c, 3ε) verifies274 λπε2 ≤ v0 ≤ 454 λπε2.Then we apply the above Chernoff bound with v = 454 eπλε2 (chosen for the convenienceof the calculus)Pï] (X ∩D(c, 3ε)) > 454 eπλε2ò≤ e 454 eπλε2−v0(v0454 eπλε2) 454 eπλε2≤ e 454 eπλε2− 274 πλε2( 454 πλε2454 eπλε2) 454 eπλε2= e− 274 πλε2EuroCG’1940:6 Poisson sample is goodSo for ε = 3»logλλ and η =454 eπλε2 = 4054 eπ log λ, we have:Pï∃p ∈ Σ, ](X ∩D(p, 3»logλλ ))>4054 eπ log λò≤ 4λ27π logλe− 2434 π logλ = O(λ−189)Since 4054 eπ < 1000, it is sufficient for our purpose to say:P[∃p ∈ Σ, ](X ∩D(p, 3»logλλ ))> 1000 log(λ)]= O(λ−2)JI Theorem 3.6. On a C3 closed surface, a Poisson sample of parameter λ large enough is a(3»logλλ , 1000 log λ)-sample with probability 1−O(λ−1).Proof. From Lemmas 3.4 and 3.5, we have that a Poisson sample is not a (3»logλλ , 1000 log λ)-sample with probability O(λ−1). JI Theorem 3.7. For λ large enough, the Delaunay triangulation of a point set Poissondistributed with parameter λ on a closed smooth generic surface of area 1 has O(λ log2 λ)expected size.Proof. Given a Poisson sample X we distinguish two cases:If X is a good sample, i.e., an (ε, η)-sample with ε = 3»logλλ and η = 1000 log λ, weapply the O((ηε )2 log(ε−1)) bound from the paper by Attali et al., that is O(λ log2 λ).If X is not a good sample, which arises with small probability by Lemma 3.6, we boundthe triangulation size by the quadratic bound:∑k∈Nk2 P [] (X) = k] =∑k∈Nk2λkk! e−λ = λ(λ+ 1) = O(λ2)Combining the two results, we getE [] (Del (X))] = E [] (Del (X)) |X good sample]P [X good sample]+ E [] (Del (X)) |X not good sample]P [X not good sample]≤ O(λ log2 λ)× 1 +O(λ2)×O(λ−1)= O(λ log2 λ)JO. Devillers and C. Duménil 40:7References1 Dominique Attali and Jean-Daniel Boissonnat. A linear bound on the complexity of theDelaunay triangulation of points on polyhedral surfaces. 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A Poisson sample of a smooth surface is a good sample

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A Poisson sample of a smooth surface is a good sample

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