Efficient Computation of Clipped Voronoi Diagram for Mesh Generation

Dong-Ming Yana,b,c, Wenping Wanga, Bruno Levyb, Yang Liub,d

aDepartment of Computer Science, The University of Hong Kong, Pokfulam Road, Hong KongbProject ALICE, INRIA/LORIA, Campus scientifique 615, rue du Jardin Botanique, 54600, Villers les Nancy, France

cGeometric Modeling and Scientific Visualization Center, KAUST, Thuwal 23955-6900, Kingdom of Saudi ArabiadMicrosoft Research Asia, Building 2, No. 5 Danling Street, Haidian District, Beijing, 100800, P.R. China

Abstract

The Voronoi diagram is a fundamental geometric structure widely used in various fields, especially incomputer graphics and geometry computing. For a set of points in a compact domain (i.e. a boundedand closed 2D region or a 3D volume), some Voronoi cells of their Voronoi diagram are infinite or partiallyoutside of the domain, but in practice only the parts of the cells inside the domain are needed, as whencomputing the centroidal Voronoi tessellation. Such a Voronoi diagram confined to a compact domain iscalled a clipped Voronoi diagram. We present an efficient algorithm to compute the clipped Voronoi diagramfor a set of sites with respect to a compact 2D region or a 3D volume. We also apply the proposed methodto optimal mesh generation based on the centroidal Voronoi tessellation.

Keywords: clipped Voronoi diagram, Delaunay triangulation, centroidal Voronoi tessellation, meshgeneration.

1. Introduction

The Voronoi diagram is a fundamental geometricstructure which has numerous applications in var-ious fields, such as shape modeling, motion plan-ning, scientific visualization, geography, chemistry,biology and so on.

Suppose that a set of sites in a compact domain inRd is given. Each site is associated with a Voronoicell containing all the points in Rd closer to the sitethan to any other sites; these cells constitute theVoronoi diagram of the set of sites. Voronoi cells ofthose sites on the convex hull are infinite, and someof Voronoi cells may be partially outside of the spec-ified domain. However, in many applications oneusually needs only the parts of Voronoi cells insidethe specific domain. That is, the Voronoi diagramrestricted to the given domain, which is defined asthe intersection of the Voronoi diagram and the do-main, and is therefore called the clipped Voronoidiagram [7]. The corresponding Voronoi cells arecalled the clipped Voronoi cells (see Figure 1).

Computing the clipped Voronoi diagram in a con-vex domain is relatively easy – one just needs tocompute the intersection of each Voronoi cell andthe domain, both being convex. However, directly

computing the clipped Voronoi diagram with re-spect to a complicated input domain is a difficultproblem and there is no efficient solution in the ex-isting literature. There has been no previous workon computing the exact clipped Voronoi diagramfor non-convex domains with arbitrary topology. Abrute-force implementation would be inefficient be-cause of the complexity of the domain.

The motivation of the work is inspired by the re-cent work [23, 38]. They showed in [23] that theCVT energy function is C2-continuous, which canbe minimized by the Newton-like algorithm, suchas the L-BFGS method presented. In [38], an effi-cient CVT-based surface remeshing algorithm waspresented with an exact algorithm for computingthe restricted Voronoi diagram on mesh surfaces.In this paper, we aim at applying the fast CVTremeshing framework to 2D/3D mesh generation.To minimize the CVT energy function, one needs tocompute the clipped Voronoi diagram in the inputdomain for function evaluation and gradient com-putation (see Section 2).

In this paper, we shall present practical algo-rithms for computing clipped Voronoi diagramsbased on several simple operations. The main idea

Preprint submitted to Computer-Aided Design October 20, 2011

(a) (b)

Figure 1: Examples of clipped Voronoi diagram ina circle (a) and a cylinder (b). The clipped Voronoicells on the boundary are shaded.

of our approach is that instead of computing theintersection of Voronoi diagram and the domain di-rectly, we first detect the Voronoi cells that have in-tersections with domain boundary and then applycomputation for those cells only. We use a simpleand efficient algorithm based on connectivity propa-gation for detecting the cells that intersect with thedomain boundary (i.e., polygons in 2D and meshsurfaces in 3D, respectively). We also utilize thepresented techniques for mesh generation as appli-cations. The contributions of this paper include :

• introduce new methods for computing theclipped Voronoi diagram in 2D regions (Sec-tion 3) and 3D volumes (Section 4);

• present practical algorithms for 2D/3D meshgeneration based on the presented clippedVoronoi diagram computation techniques (Sec-tion 5).

1.1. Previous work

The properties of the Voronoi diagram have beenextensively studied in the past decades. Existingtechniques compute the Voronoi diagram for pointsites in 2D and 3D Euclidean spaces efficiently.There are several robust implementations that arepublicly available, such as CGAL [1] and Qhull [6].A thorough survey of the Voronoi diagram is outof the scope of this paper, the reader is referred to[5, 15, 26] for details of theories and applications ofthe Voronoi diagram. We shall restrict our discus-sion to the approaches of computing the Voronoidiagram restricted to a specific 2D/3D domain andtheir applications.

Voronoi diagram of surfaces/volumes. It is natu-ral to use the geodesic metric to define the socalled Geodesic Voronoi Diagram (GVD) on sur-faces. Kunze et al. [20] presented a divide-and-conquer algorithm of computing GVD for paramet-ric surfaces. Peyre and Cohen [28] used the fastmarching algorithm to compute a discrete approx-imated GVD on a mesh surface. However, the costof computing the exact GVD on surfaces is high,for instance, the fast marching method requires tosolve the nonlinear Eikonal equation.

The restricted Voronoi diagram (RVD) [14] is de-fined as the intersection of the 3D Voronoi diagramand the surface, which is applied for computingconstrained/restricted CVT on continuous surfacesby Du et al. [13]. The concept of the constrainedCVT was extended to mesh surfaces in recent work[23, 38] and applied for isotropic surface remesh-ing. Yan et al. [38] proposed an exact algorithm toconstruct the RVD on mesh surfaces which consistof triangle soups. They processed each triangle in-dependently where a kd-tree was used to find thenearest sites of each triangle in order to identify itsincident Voronoi cells and compute the intersection.In this paper, we further improve the efficiency ofthe RVD computation by applying a neighbor prop-agation approach instead of using kd-tree query, as-suming the availability of the mesh connectivity in-formation (Section 4.1).

The clipped Voronoi diagram is defined as theintersection of the 3D (resp. 2D) Voronoi diagramand the given 3D volume (resp. a 2D region). Chanet al. [7] introduced an output-sensitive algorithmfor constructing the 3D clipped Voronoi diagramof a convex polytope. Kyons et al. [24] presentedan O(nlog(n)) algorithm to compute the clippedVoronoi diagram in a 2D square and applied itto network visualization. Yan et al. [39] utilizedthe clipped Voronoi diagram to compute the peri-odic CVT in 2D periodic space. Hudson et al. [18]computed the 3D clipped Voronoi diagram in thebounding box of the sites and used it to improve thetime and space complexities of computing the fullpersistent homological information. However, thehandling of non-convex objects was not addressedin these approaches. Existing algorithms used a dis-crete approximation in specific applications. HoffIII et al. [17] proposed a method for computing thediscrete generalized Voronoi diagram using graph-ics hardware. The Voronoi diagram computationwas formulated as a clustering problem in the dis-crete voxel/pixel space. Sud et al. [33] presented an

2

n-body proximity query algorithm based on com-puting the discrete 2nd order Voronoi diagram onthe GPU. GPU-based algorithms were fast but pro-duced only a discrete approximation of the trueVoronoi diagram. In this paper, we shall presentefficient algorithms to compute the exact clippedVoronoi diagram for both 2D and 3D domains.

Mesh generation. Mesh generation has been ex-tensively studied in meshing community over pastdecades. The detailed reviews of mesh genera-tion techniques are available in [27, 16]. In thefollowing, we will focus on the work based onVoronoi/Delaunay concepts, which are most relatedto ours. We also briefly review the main categoriesof tetrahedral mesh generation techniques.

The concept of Voronoi diagram has been suc-cessfully used for meshing and analyzing point data.Amenta et al. [4] presented a new surface recon-struction algorithm based on Voronoi filtering. Thisalgorithm has provable guarantees when the sam-ple points of a smooth surface satisfy the lfs (localfeature size) property. Alliez et al. [2] proposed asurface reconstruction algorithm from noisy inputdata based on the Voronoi-PCA estimation. Ley-marie and Kimia introduced the medial scaffold ofpoint cloud data [21], which is a hierarchical rep-resentation of the medial axis of 3D objects. Al-though these works deal with point data, they canbe extended further for volumetric meshing.

The medial axis, which is a subset of Voronoidiagram, has been applied in applications such as2D quadrilateral meshing [37] and 3D hexahedralmeshing [29]. Given a closed 2D polygon or 3Dtriangulated surface as the input domain, a set ofdense points is first sampled on the domain bound-ary and the medial axis/surface is computed di-rectly from the Voronoi diagram of samples. Thefinal mesh is generated by first meshing the medialaxis(2D)/surface(3D) and extruding to the domainboundary [30]. The medial axis based method issuitable for models which have well defined medialaxis, such as CAD/CAM models, but the medialaxis computation is sensitive to noise or small fea-tures of the domain boundary.

In this paper, we focus on the tetrahedral mesh-ing as an application of the clipped Voronoi diagramcomputation (see Section 5). The shape quality andboundary preservation are two main issues of tetra-hedral meshing algorithms, since the quality of sim-plices is crucial to finite element applications. Werefer the reader to [31] for the theoretic study of the

relationship between element qualities and interpo-lation error/condition number. In the following, webriefly discuss the main categories of tetrahedralmeshing.

• The octree-based approaches (e.g. [10, 40])subdivide the bounding box of input modelrepeatedly until a pre-specified resolution isreached, then connect those cells to form thetetrahedra. In general, this kind of approachescannot prevent bad elements near the bound-ary.

• Advancing front methods start from the do-main boundary and stuff the interior of the do-main progressively, guided by specified heuris-tic to control the shape/size. Advancing frontmethods are fast but a high-quality triangu-lated boundary is required.

• Delaunay/Voronoi based approaches generatemeshes satisfying Delaunay properties, whichmaximize the minimal angle of shape elements.Given an input domain, Delaunay/Voronoibased methods repeatedly insert Steiner pointsinto the mesh, until all the elements meet theDelaunay property. This approach aims atgenerating meshes which conform to the inputdomain boundary, but often leads to unsatis-fied results if the given domain boundary ispoorly triangulated. An alternative way is toapproximate the boundary instead of conform-ing, which results the better shape/size quality.

• Variational approach is one of the most ef-fective ways of generating isotropic tetrahe-dral meshes. Recent work includes both CVT-based and ODT-based techniques. The CVT-based approach aims at optimizing the dualVoronoi structure of Delaunay triangulation,while ODT tends to optimize the shape of pri-mal elements [8]. The CVT-based mesh gener-ation has been extensively studied in the liter-ature [13], while ODT was recently introducedto graphics community [3, 36]. One of themain difficulties of both CVT and ODT-basedtetrahedral meshing is the boundary conform-ing issue. Alliez et al. [3] used dense quadra-ture samples to approximate restricted Voronoicells on mesh surface. Dardenne et al. [11] useda discrete version of the CVT to generate tetra-hedral meshes from the discrete volume data.The voxels are clustered into n cells via Lloyd

3

iteration, with each cell corresponding to a site.The tetrahedral mesh is obtained from the con-nectivity relations of cells. However, such anapproach is limited to the resolution of voxels.

2. Problem Formulation

We first provide mathematical definitions and no-tations, then introduce the main idea of the clippedVoronoi diagram computation.

2.1. Definitions

Definition 2.1. The Voronoi Diagram of a givenset of distinct sites X = xini=1 in Rd is defined bya collection of Voronoi cells Ωini=1, where

Ωi = x ∈ Rd∣∣ ‖x− xi‖ ≤ ‖x− xj‖,∀j 6= i.

Each Voronoi cell Ωi is the intersection of a set ofhalf-spaces, delimited by the bisectors of the Delau-nay edges incident to the site xi.

Definition 2.2. The Clipped Voronoi Diagram forthe sites X with respect to a connected compact do-main Ω is the intersection of the Voronoi diagramand the domain, denoted as Ωi|Ωni=1, where

Ωi|Ω = x ∈ Ω∣∣ ‖x− xi‖ ≤ ‖x− xj‖,∀j 6= i

Each clipped Voronoi cell is the intersection of theVoronoi cell Ωi and the domain Ω, i.e., Ωi|Ω =Ωi⋂

Ω. We call Ωi|Ω the clipped Voronoi cell withrespect to Ω (see Figure 1 for examples).

Definition 2.3. Centroidal Voronoi Tessellationof a set of distinct sites X with respect to a com-pact domain Ω is the minimizer of the CVT energyfunction [12] :

F (X) =

n∑i=1

∫Ωi|Ω

ρ(x)‖x− xi‖2 dσ. (1)

In the above definition, ρ(x) > 0 is a user-defineddensity function. The partial derivative of the en-ergy function with respect to each site is givenby [19] :

∂F

∂xi= 2mi(xi − x∗i ), (2)

here mi =∫

Ωi|Ω ρ(x) dσ, and x∗i =

∫Ωi|Ω

ρ(x)x dσ∫Ωi|Ω

ρ(x) dσ

is the centroid of the clipped Voronoi cell Ωi|Ω.We use the L-BFGS method [23] for computing theCVT. The clipped Voronoi diagram is used to assistthe function evaluation (Eqn. 1) and the gradientcomputation (Eqn. 2).

2.2. Algorithm overview

There are two types of clipped Voronoi cells of aclipped Voronoi diagram : inner Voronoi cells andboundary Voronoi cells, whose corresponding sitesare called inner sites and boundary sites, respec-tively. The inner Voronoi cells are entirely con-tained in the interior of the domain Ω, which can bededuced from the Delaunay triangulation directly.The boundary Voronoi cells are those cells that in-tersect with the domain boundary ∂Ω, as shown inFigure 1. In the following, we will focus on how tocompute the boundary Voronoi cells.

To compute a clipped Voronoi diagram with re-spect to a given domain, we first need to classify thesites into inner and boundary sites, and then com-pute the clipped Voronoi cells for boundary sites.As discussed above, the boundary cells have inter-sections with the domain boundary ∂Ω (i.e., poly-gons in 2D and mesh surfaces in 3D), which canbe found by intersecting the boundary with theVoronoi diagram. We present efficient algorithmsfor computing the intersection of a Voronoi diagramand 2D polygons or 3D mesh surfaces, respectively.Once the boundary sites are identified, we are ableto compute the clipped Voronoi cells efficiently byclipping the domain Ω against boundary Voronoicells.

In the following sections, we shall present efficientalgorithms for computing clipped Voronoi diagramin 2D (Section 3) and 3D (Section 4) spaces, respec-tively. Furthermore, we show how to utilize the pre-sented clipped Voronoi diagram computation tech-niques for practical mesh generation (Section 5).

3. 2D Clipped Voronoi Diagram Computa-tion

Suppose that the input domain Ω is a compact2D region, whose boundary is represented by a 2Dcounter-clockwise outer polygon, and several clock-wise inner polygons without self-intersections. As-sume that the boundary is represented by a set ofordered edge segments ei. The main steps of ourmethod are illustrated in Figure 2. For a given setof sites inside the given domain, we first computethe Voronoi diagram of the sites. Then we identifythe boundary sites and finally compute the clippedVoronoi cells of boundary sites .

3.1. Voronoi diagram construction

We first construct a Delaunay triangulation frominput sites X = xini=1. The corresponding

4

(a) (b) (c) (d)

Figure 2: Illustration of main steps for computing clipped Voronoi diagram in 2D. (a) Delaunay triangulation,(b) 2D Voronoi diagram, (c) detect boundary sites, (d) compute clipped Voronoi diagram.

Voronoi diagram Ωini=1 is constructed as the dualof the Delaunay triangulation, as defined in Sec-tion 2. Each Voronoi cell is stored as a set of bi-secting planes, which is used for clipping operationsin the following steps.

3.2. Detection of boundary cells

In this step, we shall identify the boundaryVoronoi cells by computing the intersection ofboundary edges and the Voronoi diagram Ωi. Werepeatedly find the incident cell-edge pairs with theassistance of an FIFO queue. An incident Voronoicell of a boundary edge ei is the cell that intersectswith ei, i.e., a boundary Voronoi cell.

We assign a boolean tag to each boundary edgeei which indicates whether ei has been processed ornot. This flag is initialized as false. Once the edgeis visited, the flag is switched to true. Startingfrom an unvisited boundary edge ei, we first findits nearest incident Voronoi cell Ωj , then use thebarycenter (or midpoint) of ei to query the nearestsite xj . Any linear search function can be used herefor the nearest point query.

The FIFO queue is initialized by the initial inci-dent cell-edge pair (Ωj , ei). We repeatedly pop outthe cell-edge pair from the queue and compute theintersection of the current Voronoi cell Ωc and theboundary edge ec. The intersected segment is de-noted as sc. The current boundary edge is markedas visited and the current Voronoi cell is markedas boundarycell. We detect new cell-edge pairsby examining the current intersected segment sc.There are two cases of sc’s endpoints :

(a) if the endpoints of sc contain a boundary ver-tex of the current edge ec (green dots in Fig-ure 2(c)), the adjacent boundary edge who

shares the same vertex with ec is pushed intothe queue together with the current Voronoicell Ωc;

(b) if the endpoints of sc contain an intersectionpoint, i.e., the intersection point between aVoronoi edge of Ωc and ec (yellow dots in Fig-ure 2(c)), the neighboring Voronoi cell whoshares the intersecting Voronoi edge with Ωcis pushed into the queue together with ec.

The boundary detection process terminates whenall the edges have been visited.

3.3. Computation of clipped Voronoi cells

Once the boundary sites are identified, we com-pute the clipped Voronoi cells by clipping the do-main against their corresponding bounding line seg-ments. A straightforward extension of [38] shouldfirst triangulate the boundary polygons and then docomputation on the resulting planar mesh, whichwill be the same as the surface RVD computationdescribed in Section 4.1. Given that the averagenumber of bisectors of 2D Voronoi cells is six [12], itis efficient enough to clip the 2D domain by Voronoicells directly. Here we simply use the Sutherland-Hodgman clipping algorithm [34] to compute theintersection. More examples of 2D clipped Voronoidiagram are given in Section 6.

4. 3D Clipped Voronoi Diagram Computa-tion

In this section we describe an efficient algorithmfor computing the clipped Voronoi diagram of 3Dobjects. Suppose that the input volume Ω is givenby a tetrahedral mesh M = V, T , where V =

5

(a) (b)

Figure 3: Illustration of clipped Voronoi diagramcomputation of 500 sites in a torus. (a) SurfaceRVD of 227 boundary sites, (b) Clipped Voronoidiagram.

vknv

k=1 is the set of mesh vertices and T = timi=1

the set of tetrahedral elements. Each tetrahedron(tet for short in the following) ti stores the informa-tion of its four incident vertices and four adjacenttets. The four vertices are assigned indices 0, 1, 2, 3and so are the four adjacent tets. The index of anadjacent tet is the same as the index of the vertexwhich is opposite to the tet. The boundary ofM isa triangle mesh, denoted as S = fj

nf

j=1, which isassumed to be a 2-manifold. Each boundary trian-gle facet fj stores the indices of three neighboringfacets and the index of its containing tet. Note thatalthough other types of convex primitives can alsobe used for domain decomposition, we use tetrahe-dral mesh here for simplicity.

The 3D clipped Voronoi diagram computation issimilar to the 2D counterpart. After constructingthe 3D Voronoi diagram Ωi of the sites X (seeSection 3.1), there are two main steps, as illustratedin Figure 3 :

1. detect boundary sites by intersecting Voronoidiagram with the boundary surface S, i.e.,compute the surface RVD (Section 4.1);

2. compute the clipped Voronoi cells for all theboundary sites (Section 4.2).

4.1. Detection of boundary sites

For the given set of sites X = xini=1 andthe boundary surface S = fj

nf

j=1, the restrictedVoronoi diagram (RVD) is defined as the intersec-tion of the 3D Voronoi diagram and the surface S,denoted as R = Rini=1, where Ri = Ωi

⋂S [14].

Each Ri is called a restricted Voronoi cell (RVC).The sites corresponding to non-empty RVCs are re-garded as boundary sites.

Ω4

Ω3

Ω1

Ω0

f0

Ω2

Ω5

f3

f2

f1

(a)

Ω4

Ω3

Ω1

Ω0

f0

Ω2

Ω5f3

f2

f1

q0

q4q3q2

q1

q5

(b)

Figure 4: Illustration of the propagation process.The green points are the vertices of input boundarymesh and the white points are the sites. The yellowpoints in (b) are the vertices of RVD.

We use the algorithm presented in [38] for com-puting the surface RVD. The performance of RVDcomputation is improved by using a neighbor prop-agation approach for finding the incident cell-triangles pairs, instead of using a kd-tree structureto query the nearest site for each triangle, as shownby our tests.

Now we are going to explain the propagation step(refer to Figure 4). We assign a boolean flag (initial-ized as false) for each boundary triangle at the ini-tialization step. The flag is used to indicate whethera triangle is processed or not. Starting from anunprocessed triangle and one of its incident cells,which is the cell corresponding to the nearest site ofthe triangle by using the barycenter of the triangleas the query point. Here we assume that a trianglef0 on S is the unprocessed triangle and the Voronoicell Ω0 is the corresponding cell of the nearest siteof f0, as shown in Figure 4(a). We use an FIFOqueue Q to store all the incident cell-triangle pairsto be processed. To start, the initial pair f0,Ω0is pushed into the queue. The algorithm repeatedlypops out the pair in the front of Q and computestheir intersection. During the intersection process,the current triangle is marked as processed, newvalid pairs are identified and pushed back into Q.The process terminates when Q is empty and allthe triangles are processed.

The key issue now is how to identify all the validcell-triangle pairs during the intersection. Assumethat f0,Ω0 is popped out fromQ, as shown in Fig-ure 4. In this case, we clip f0 against the bound-ing planes of Ω0, which has five bisecting planes,i.e.,[x0,x1], [x0,x2]..., [x0,x5]. The resulting poly-gon is represented by q0,q1, ...,q5, as shown in Fig-ure 4(b). Since the line segment q0q1 is the inter-section of f0 and [x0,x1], we know that the oppo-

6

(a) (b) (c) (d)

Figure 5: 2D CVT-based meshing. (a) The clipped Voronoi diagram of initial sites; (b) the result of CVTwith ρ = 1; (c) the result of constrained optimization. Notice that boundary seeds are constrained on theborder; (d) the final uniform 2D meshing.

site cell Ω1 is also an incident cell of f0, thus thepair f0,Ω1 is an incident pair. Since the com-mon edge of [f0, f1] has intersection with Ω0, theadjacent facet f1 also has intersection with cell Ω0,thus the pair f1,Ω0 is also an incident pair. Sois the pair f2,Ω0. The other incident pairs arefound in the same manner. To keep the same pairfrom being processed multiple times, we store theincident facet indices for each cell. Before pushinga new pair into the queue, we add the facet in-dex to the incident facet index set of the cell. Thepair is pushed into the queue only if the facet isnot contained in the incident facet set of the cell;otherwise the pair is discarded. At each time afterintersection computation, the resulting polygon isassociated with the surface RVC of the current site.The surface RVD computation terminates when thequeue is empty. Those sites that have non-emptysurface RVC are marked as the boundary sites, de-noted as Xb = xi|Ri 6= ∅.

4.2. Construction of clipped Voronoi cells

Once the boundary sites Xb are found, we com-pute the clipped Voronoi cells for these sites. Thecomputation of boundary Voronoi cells is similarto the surface RVD computation presented in Sec-tion 4.1, with the difference that we restrict thecomputation on boundary cells only. For eachboundary cell, we have recorded the indices of its in-cident boundary triangles. We know that the neigh-boring tet of each boundary triangle is also incidentto the cell. We also store the indices of the incidenttet for each boundary cell. The incident tet setis initialized as the neighboring tet of the incidentboundary triangle.

We use an FIFO queue to facilitate this process.The queue is initialized by a set of incident cell-tet pairs (Ωi, tj), which can be obtained from theboundary cell and its initial incident tet set.

The pair (Ωi, tj) in front of Q is popped out re-peatedly. We compute the intersection of Ωi andtj again by the Sutherland-Hodgman clipping al-gorithm [34] and identify new incident pairs at thesame time. We clip the tet tj by bounding planesof cell Ωi one by one. If the current boundingplane has intersection with tj , we check the oppo-site Voronoi cell Ωo that shares the current bisect-ing plane with Ωi; if Ωo is a boundary cell and tjis not in the incident set of Ωo, a new pair (Ωo, tj)is found. We also check the neighboring tets whoshare the facets clipped by the current bisectingplane. Those tets that are not in the incident set ofΩi are added to its set, and new pairs are pushedinto the queue. After clipping, the resulting poly-hedron is associated with the clipped Voronoi cellΩi|M of site xi. This process terminates when Q isempty.

5. Applications for mesh generation

We present two applications of the presentedclipped Voronoi diagram computation techniques,including 2D triangular meshing and 3D tetrahe-dral meshing.

5.1. 2D mesh generation

Triangle mesh generation is a well-known appli-cation of CVT optimization. In this section wepresent such an application based on our 2D clippedVoronoi diagram computation. The input domain

7

(a) (b) (c) (d)

Figure 6: Illustration of the CVT-based tetrahedral meshing algorithm. The wireframe is the boundary ofthe input mesh. (a) The clipped Voronoi diagram of the initial sites (the boundary Voronoi cells are shaded);(b) the result of the unconstrained CVT with ρ = 1; (c) the result of the constrained optimization. Noticethat boundary seeds are constrained on the surface S; (d) the final isotropic tetrahedral meshing result.

Ω is a 2D polygon, which can be single connected orwith multiple components. We first sample a set ofinitial points inside the input domain (Figure 5(a))and then compute a CVT (Eqn. 1) from this initialsampling (Figure 5(b)). Once we have a set of welldistributed samples, we snap the seeds correspond-ing to boundary Voronoi cells to the boundary andrun optimization again, with the boundary seedsconstrained on the border (Figure 5(c)). Finally,we keep the primal triangles whose circumscribingcenters are inside the domain as the meshing re-sult (5(d)). Our 2D meshing framework also allowsthe user to insert vertices of input polygon and tagthese vertices as fixed. By doing this, the geomet-ric properties of the input domain can be betterpreserved. More results are given in Section 6.

5.2. Tetrahedral mesh generation

There are three main steps of the CVT-basedmeshing framework: initialization, iterative opti-mization, and mesh extraction, which are illus-trated by the example in Figure 6.

Initialization. In this step, we build a uniformgrid to store the sizing field for adaptive meshing.Following the approach in [3], we first compute thelocal feature size (lfs) for all boundary vertices andthen use a fast matching method to construct asizing field on the grid. This grid is also used forefficient initial sampling (Figure 6(a)). The readeris referred to [3] for details.

Optimization. There are two phases of the globaloptimization: the unconstrained CVT optimizationand the constrained CVT optimization. In the first

phase, we optimize the positions of the sites insidethe input volume without any constraints, whichyields a well-spaced distribution of the sites withinthe domain, with no sites lying on the boundarysurface (Figure 6(b)).

During the second phase of optimization, all theboundary sites will be constrained on the boundary.The partial derivative of the energy function withrespect to each boundary site is computed as:

∂F

∂xi

∣∣∣∣S

=∂F

∂xi−[∂F

∂xi·N(xi)

]N(xi), (3)

where N(xi) is the unit normal vector of the bound-ary surface at the boundary site xi [23]. The partialderivative with respect to an inner site is still com-puted by Eqn. 2. Both boundary and inner siteswill be optimized simultaneously, applying againthe L-BFGS method to minimize the CVT energyfunction (Figure 6(c)).

Sharp features are preserved in a similar way ashow the boundary sites are treated. For example,we project sites on sharp edges on the boundary andallow them to vary only along these edges duringthe second stage of optimization. For details, pleaserefer to [38] where these steps are described in thecontext of surface remeshing.

Final mesh extraction. Once the optimization isfinished, we extract the tetrahedral cells from theprimal Delaunay triangulation (Figure 6(d)). Asdiscussed in [3], the CVT energy cannot eliminatethe slivers from the resulting tetrahedral mesh. Weperform a post-processing to perturb slivers usingthe approach of [35]. The results are given in Sec-tion 6.

8

Figure 7: Results of clipped Voronoi diagram computation.

6. Experimental results

Our algorithm is implemented in C++ on bothWindows and Linux platform. We use the CGAL li-brary [1] for 2D and 3D Delaunay triangulation andTetGen [32] for background mesh generation whenthe input 3D domain is given as a closed trianglemesh. All the experimental results are tested on alaptop with 2.4GHz processor and 2GB memory.

Efficiency. We first demonstrate the performanceof the proposed clipped VD computation algorithm.The 2D version is very efficient. All the exam-ples shown in this paper take only several millisec-onds. To detect the boundary sites, we have im-plemented a propagation based approach for sur-face RVD computation. This new implementationof RVD performs better then the previous kd-treebased approach [38] since there is no kd-tree queryrequired, as shown in Figure 8. The performanceof the 3D clipped Voronoi diagram computation isdemonstrated in Figure 9. We progressively samplethe input domain with number of sites from 10 to6 × 105. Note that the time of surface RVD com-putation is much less than the Delaunay triangula-tion, since only a small portion of all the sites areboundary sites. The time cost of the clipped VDcomputation algorithm is proportional to the to-tal number of incident cell-tet pairs (Section 4.2).Therefore, an input mesh with a small number oftetrahedral elements would help to improve the effi-ciency. In our experiments, all the input tetrahedral

meshes are generated by the robust meshing soft-ware TetGen [32] with the conforming boundary.More results of the clipped Voronoi diagram com-putation of various 3D objects are given in Figure7 and the timing statistics is given in Table 1.

Model |T | |S| |X| |Xb| TimeTwoprism 68 30 1k 572 0.2

Bunny 10k 3k 2k 734 1.8Elk 34.8k 10.4k 2k 1,173 3.1

Block 77.2k 23.4 1k 659 4.7Homer 16.2k 4,594 10k 2,797 6.3

Rockerarm 212k 60.3k 3k 1,722 12.1Bust 68.5k 20k 30k 5k 16.2

Table 1: Statistics of clipped Voronoi diagram com-putation on various models. |T | is the numberof the input tetrahedra. |S| is the number of theboundary triangles. |X| is the number of the sites.|Xb| is the number of the boundary sites. Time (inseconds) is the total time for clipped Voronoi dia-gram computation, including both Delaunay trian-gulation and surface RVD computation.

Robustness. We use exact predicates to predicatethe side of a vertex against a Voronoi plane duringthe clipping process. We use Meyer and Pion’s FGPpredicate generator [25] provided by CGAL in ourimplementation, as also done in [38]. We did notencounter any numerical issue for all the examplesshown in the paper. Our clipped Voronoi diagramis robust even for extreme configurations. We showan example of computing the clipped Voronoi dia-gram on a sphere in Figure 10. The sites are set tothe vertices of the boundary mesh and there is no

9

Figure 8: Comparison of the propagation-basedsurface RVD computation with the kd-tree-basedapproach.

#seed vs time of clipped VD#seed vs time of RVD#seed vs time of DT

#Seed

Tim

e(s)

Figure 9: The timing curve of the clipped Voronoidiagram computation against the number of siteson Bone model.

inner site. Furthermore, we give another exampleof computing clipped Voronoi diagram in a cubicdomain. The boundary mesh of the cube is shownin Figure 11(a). We sample the eight corners ofthe cube as sites, in this case, the bounding planesof Voronoi diagram are passing through the edgesof the boundary mesh. The surface RVD and thevolume clipped Voronoi diagram are shown in Fig-ure 11(b) and (c), respectively.

2D meshing. We show some 2D mesh generationresults based on our fast clipped Voronoi diagramcomputation. Figure 12 demonstrates that our al-gorithm works well for multiple connected domains.Figure 13 shows that we insert original vertices ofinput polygon for the better preservation of the ge-ometric properties.

Figure 10: Clipped Voronoi diagram of a sphere.The sites are the vertices of the sphere. (a) Thesurface RVD, (b) the clipped Voronoi diagram.

(a) (b) (c)

Figure 11: Clipped Voronoi diagram of a cube. Redpoints represent the sites. (a) The input domain,(b) the surface RVD, (c) the clipped Voronoi dia-gram.

Figure 12: CVT-based 2D mesh generation of aring.

Figure 13: CVT-based 2D mesh generation. Theboundary vertices of the input domain are used asconstraints.

10

Tetrahedral meshing. The complete process ofthe proposed tetrahedral meshing framework is il-lustrated in Figure 6. Figure 14 (a)&(b) show twoadaptive tetrahedral meshing examples, using lfsas the density function [3]. Figure 14 (c)&(d) givetwo examples with sharp features preserved. Ourframework can generate high quality meshes effi-ciently and robustly. The running time for obtain-ing final results ranges from seconds to minutes, de-pending on the size of the input tetrahedral meshand the desired number of sites.

Figure 14: Tetrahedral mesh generation results.The histograms show the angle distribution of theresults.

Comparison. We compare our meshing resultswith the Delaunay refinement approach providedby CGAL [1], as well as a recent work that useda discrete version of clipped Voronoi diagram fortetrahedral mesh generation [11]. Four shape qual-ity measurements are used as in [11], i.e.,

• Q1 = θmin, the minimal dihedral angle θmin ofeach tetrahedron;

• Q2 = θmax, the maximal dihedral angle θmaxof each tetrahedron;

• Q3 = 3 rinrcirc

, the radius-ratio of each tetra-hedral, where rin and rcirc are the in-scribed/circumscribed radius, respectively;

• Q4 = 123√

9V 2∑l2i,j

, meshing quality of [22], where

V is the volume of the tetrahedron, and li,jthe length of the edge which connects verticesvi and vj .

Q3 and Q4 are between 0 and 1, where 0 denotes asilver and 1 denotes a regular tetrahedron.

We choose the sphere generated from an iso-surface as input domain. The Hausdorff distance(measured by Metro [9]) between the boundary ofgenerated mesh and the input surface (normalizedby dividing by the diagonal of bounding box) is0.049%, which is 3 times smaller than 0.17% re-ported by [11]. The quality of the tetrahedral meshis shown in Figure.15 and the comparison of eachmeasurement is given in Table 2. Our approachproduces better meshing quality, as well as smallersurface approximation error, attributed to the ex-act clipped Voronoi diagram computation.

method Q1 Q4 min(Q1) min(Q4) HDist

[1] 48.11 0.847 12.05 0.339 0.054%[11] 56.32 0.911 16.31 0.376 0.170%ours 56.37 0.932 24.23 0.560 0.049%

Table 2: Comparison of meshing qualities. HDist

is the Hausdorff distance between the boundary ofgenerated mesh and the input discretized isosur-face.

We also compare our result with an octree-basedapproach [10]. As shown in Figure 16, the CVTbased approach exhibits much better element qual-ity than a standard approach. Our approach out-performs previous work in boundary approximationerror (as shown in Figure 17), attributed to the ex-act clipped Voronoi diagram computation and si-multaneous surface remeshing [38].

11

0 10 20 30 40 50 60 70 800

1000

2000

3000

4000

5000

6000

0 10 20 30 40 50 60 70 800

1000

2000

3000

4000

5000

6000

(a) Q1

60 80 100 120 140 160 1800

1000

2000

3000

4000

(b) Q2

0.0 0.2 0.4 0.6 0.8 1.00

2000

4000

6000

8000

10000

0.0 0.2 0.4 0.6 0.8 1.00

2000

4000

6000

8000

10000

(c) Q3

0.0 0.2 0.4 0.6 0.8 1.00

2000

4000

6000

8000

10000

12000

0.0 0.2 0.4 0.6 0.8 1.00

2000

4000

6000

8000

10000

12000 OursCGAL

(d) Q4

Figure 15: Comparison of the meshing qualities ofthe sphere with the Delaunay refinement approachimplemented in CGAL [1].

7. Conclusion

We have presented efficient algorithms for com-puting the clipped Voronoi diagram for closed 2Dand 3D objects, which has been a difficult problemwithout an efficient solution. As an application, wepresent a new CVT-based mesh generation algo-rithm which combines the clipped VD computationand fast CVT optimization.

In the future, we plan to look for more interdis-ciplinary applications of the clipped Voronoi dia-gram, such as biology and architecture. Applyingour meshing technique to physical simulation appli-cations, and extending the clipped Voronoi diagramto a higher dimension are also interesting directions.

Acknowledgements

We would like to thank anonymous review-ers for their detailed comments and suggestionswhich greatly improve the manuscript. We alsothank one reviewer who pointed out the ref-erence [7]. This work is partially supportedby the Research Grant Council of Hong Kong(project no.: 718209 and 718010), the State KeyProgram of NSFC project (60933008), EuropeanResearch Council (GOODSHAPE FP7-ERC-StG-205693), and ANR/NSFC (60625202, 60911130368)Program (SHAN Project).

0 10 20 30 40 50 60 70 800

2000

4000

6000

8000

10000

12000

(a) Q1

60 80 100 120 140 160 1800

2000

4000

6000

8000

(b) Q2

0.0 0.2 0.4 0.6 0.8 1.00

5000

10000

15000

20000

(c) Q3

0.0 0.2 0.4 0.6 0.8 1.00

2000

4000

6000

8000

10000

12000

14000

16000 OursCulter

(d) Q4

Figure 16: Comparison with the octree based ap-proach [10]. The resulting tetrahedral mesh has200k tetrahedra.

Figure 17: Approximation error of gargoyle model:50K vertices, 256K tetrahedra, mean/max Haus-dorff distance: 0.045%/0.37%. Our approach pro-duces smaller approximation error compared with[3] (mean error: 0.053%) using the same number ofvertices.

References

[1] CGAL, Computational Geometry Algorithms Library.http://www.cgal.org.

[2] Pierre Alliez, David Cohen-Steiner, Yiying Tong, andMathieu Desbrun. Voronoi-based variational recon-struction of unoriented point sets. In Proceedingsof Symposium on Geometry Processing (SGP 2007),pages 39–48, 2007.

[3] Pierre Alliez, David Cohen-Steiner, Mariette Yvinec,and Mathieu Desbrun. Variational tetrahedral meshing.ACM Transactions on Graphics (Proceedings of ACMSIGGRAPH 2005), 24(3):617–625, 2005.

[4] Nina Amenta, Marsahll Bern, and Manolis Kamvys-selis. A new Voronoi-based surface reconstruction al-gorithm. In Proceedings of ACM SIGGRAPH 1998,pages 415–421, 1998.

[5] Franz Aurenhammer. Voronoi diagrams: a survey of afundamental geometric data structure. ACM Comput-ing Surveys, 23(3):345–405, 1991.

[6] C. Bradford Barber, David P. Dobkin, and Hannu Huh-

12

danpaa. The quickhull algorithm for convex hulls. ACMTrans. Math. Software, 22:469–483, 1996.

[7] Timothy M. Y. Chan, Jack Snoeyink, and Chee-KengYap. Output-sensitive construction of polytopes in fourdimensions and clipped Voronoi diagrams in three. InProceedings of the sixth annual ACM-SIAM symposiumon Discrete algorithms (SODA), pages 282–291, 1995.

[8] L. Chen and J. Xu. Optimal Delaunay triangulations.Journal of Computational Mathematics, 22(2):299–308,2004.

[9] P. Cignoni, C. Rocchini, and R. Scopigno. Metro: Mea-suring error on simplified surfaces. Computer GraphicsForum, 17(2):167–174, 1998.

[10] Barbara Cutler, Julie Dorsey, , and Leonard McMillan.Simplification and improvement of tetrahedral modelsfor simulation. In Proceedings of the Eurographics Sym-posium on Geometry Processing, pages 93–102, 2004.

[11] J. Dardenne, S. Valette, N. Siauve, N. Burais, andR. Prost. Variational tetraedral mesh generation fromdiscrete volume data. The Visual Computer (Proceed-ings of CGI 2009), 25(5):401–410, 2009.

[12] Qiang Du, Vance Faber, and Max Gunzburger. Cen-troidal Voronoi tessellations: applications and algo-rithms. SIAM Review, 41(4):637–676, 1999.

[13] Qiang Du, Max. D. Gunzburger, and Lili Ju. Con-strained centroidal Voronoi tesselations for surfaces.SIAM J. Sci. Comput., 24(5):1488–1506, 2003.

[14] Herbert Edelsbrunner and Nimish R. Shah. Triangu-lating topological spaces. Int. J. Comput. GeometryAppl., 7(4):365–378, 1997.

[15] Steven Fortune. Voronoi diagrams and Delaunay trian-gulations. In Computing in Euclidean Geometry, pages193–233, 1992.

[16] Pascal Jean FREY and Paul-Louis GEORGE. MeshGeneration: Application to Finite Elements. HermesScience, 2000.

[17] Kenneth E. Hoff III, John Keyser, Ming C. Lin, andDinesh Manocha. Fast computation of generalizedVoronoi diagrams using graphics hardware. In Proceed-ings of ACM SIGGRAPH 1999, pages 277–286, 1999.

[18] Benoit Hudson, Gary L. Miller, Steve Y. Oudot, andDonald R. Sheehy. Topological inference via meshing.In Proceedings of the 2010 annual symposium on Com-putational geometry (SOCG), pages 277–286, 2010.

[19] Masao Iri, Kazuo Murota, and Takao Ohya. A fastVoronoi diagram algorithm with applications to geo-graphical optimization problems. In Proceedings of the11th IFIP Conference on System Modelling and Opti-mization, pages 273–288, 1984.

[20] R. Kunze, F.-E. Wolter, and T. Rausch. GeodesicVoronoi diagrams on parametric surfaces. In Proceed-ings of Computer Graphics International 1997, pages230–237, 1997.

[21] F. Leymarie and B. Kimia. The medial scaffold of 3Dunorganized point clouds. IEEE Trans. Pattern Anal.Mach. Intell., 29(2):313–330, 2007.

[22] Anwei Liu and Barry Joe. On the shape of tetra-hedra from bisection. mathematics of computation,63(207):141–154, 1994.

[23] Yang Liu, Wenping Wang, Bruno Levy, Feng Sun,Dong-Ming Yan, Lin Lu, and Chenglei Yang. On cen-troidal Voronoi tessellation: Energy smoothness andfast computation. ACM Trans. on Graphics, 28(4):Ar-ticle No. 101, 2009.

[24] Kelly A. Lyons, Henk Meijer, and David Rappaport.

Algorithms for cluster busting in anchored graph draw-ing. J. Graph Algorithms Appl., 2(1):1–24, 1998.

[25] Andreas Meyer and Sylvain Pion. FPG: A code gen-erator for fast and certified geometric predicates. RealNumbers and Computers (RNC), pages 47–60, 2008.

[26] Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, andSung Nok Chiu. Spatial Tessellations: Concepts andApplications of Voronoi Diagrams. Wiley, 2nd edition,2000.

[27] S. Owen. A survey of unstructured mesh generationtechnology. In Proceedings of 7th International MeshingRoundtable, pages 26–28, 1998.

[28] Gabriel Peyre and Laurent D. Cohen. Geodesic remesh-ing using front propagation. Int. J. of Computer Vision,69:145–156, 2006.

[29] W. R. Quadros and K. Shimada. Hex-layer: Layered all-hex mesh generation on thin section solids via chordalsurface transformation. In Proc. of 11th InternationalMeshing Roundtable, pages 169–180, 2002.

[30] Peter Sampl. Semi-structured mesh generation basedon medial axis. In Proceedings of the 9th InternationalMeshing Roundtable, pages 21–32, 2000.

[31] J. R. Shewchuk. What is a good linear element? inter-polation, conditioning, and quality measures. In Pro-ceedings of the 11th International Meshing Roundtable,pages 115–126, 2002.

[32] Hang Si. TetGen: A quality tetrahedral mesh gen-erator and three-dimensional Delaunay triangulator.http://tetgen.berlios.de.

[33] Avneesh Sud, Naga K. Govindaraju, Russell Gayle,Ilknur Kabul, and Dinesh Manocha. Fast proximitycomputation among deformable models using discreteVoronoi diagrams. ACM Trans. on Graphics (Proc.SIGGRAPH), 25(3):1144–1153, 2006.

[34] Ivan E. Sutherland and Gary W. Hodgman. Reen-trant polygon clipping. Communications of the ACM,17(1):32–42, 1974.

[35] Jane Tournois, Rahul Srinivasan, and Pierre Alliez. Per-turbing slivers in 3D Delaunay meshes. In Proceedingsof the 18th International Meshing Roundtable, pages157–173, 2009.

[36] Jane Tournois, Camille Wormser, Pierre Alliez, andMathieu Desbrun. Interleaving Delaunay refinementand optimization for practical isotropic tetrahedronmesh generation. ACM Trans. on Graphics (Proc. SIG-GRAPH), 28(3):Article No. 75, 2009.

[37] S. Yamakawa and K. Shimada. Quad-layer: Layeredquadrilateral meshing of narrow two-dimensional do-main by bubble packing and chordal axis transforma-tion. Journal of Mechanical Design, 124:564–573, 2002.

[38] Dong-Ming Yan, Bruno Levy, Yang Liu, Feng Sun,and Wenping Wang. Isotropic remeshing with fastand exact computation of restricted Voronoi diagram.Computer Graphics Forum (Proceedings of SGP 2009),28(5):1445–1454, 2009.

[39] Dong-Ming Yan, Kai Wang, Bruno Levy, and LaurentAlonso. Computing 2D periodic centroidal Voronoi tes-sellation. In Proc. of 8th International Symposium onVoronoi Diagrams in Science and Engineering (ISVD),pages 177 – 184, 2011.

[40] Yi-Jun Yang, Jun-Hai Yong, and Jia-Guang Sun. Analgorithm for tetrahedral mesh generation based on con-forming constrained Delaunay tetrahedralization. Com-puters & Graphics, 29(4):606–615, 2005.

13