Computers & Graphics - Computer Science Department, Techniongotsman/AmendedPubl/Yin/halftone.pdf · tous in computer graphics [6,3]. The blue noise point distribution generation problem

Computers & Graphics ] (]]]]) ]]]–]]]

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Computers & Graphics

0097-84

doi:10.1

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E-m

gotsma

Pleas(201

journal homepage: www.elsevier.com/locate/cag

SMI 2011: Full Paper

Capacity-Constrained Delaunay Triangulation for point distributions

Yin Xu a, Ligang Liu a,n, Craig Gotsman b, Steven J. Gortler c

a Zhejiang University, Chinab Technion, Israelc Harvard University, USA

a r t i c l e i n f o

Article history:

Received 14 December 2010

Received in revised form

13 March 2011

Accepted 14 March 2011

Keywords:

Blue noise

Poisson disk distribution

Capacity-Constrained Delaunay

Triangulation (CCDT)

Minimal area variance

93/$ - see front matter & 2011 Elsevier Ltd. A

016/j.cag.2011.03.031

esponding author. Tel./fax: þ86 571 8795366

ail addresses: [email protected] (Y. Xu), li

[email protected] (C. Gotsman), [email protected]

e cite this article as: Xu Y, et al. Cap1), doi:10.1016/j.cag.2011.03.031

a b s t r a c t

Sample point distributions possessing blue noise spectral characteristics play a central role in computer

graphics, but are notoriously difficult to generate. We describe an algorithm to very efficiently generate

these distributions. The core idea behind our method is to compute a Capacity-Constrained Delaunay

Triangulation (CCDT), namely, given a simple polygon P in the plane, and the desired number of points

n, compute a Delaunay triangulation of the interior of P with n Steiner points, whose triangles have

areas which are as uniform as possible. This is computed iteratively by alternating update of the point

geometry and triangulation connectivity. The vertex set of the CCDT is shown to have good blue noise

characteristics, comparable in quality to those of state-of-the-art methods, achieved at a fraction of the

runtime. Our CCDT method may be applied also to an arbitrary density function to produce non-

uniform point distributions. These may be used to half-tone grayscale images.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Point distributions with blue noise characteristic are ubiqui-tous in computer graphics [6,3]. The blue noise point distributiongeneration problem can be formulated as follows: given a shapedescribed by its polygonal boundary P and the number of pointsamples n, position n points inside P forming a spatial patternwith a blue noise spectrum. A blue noise spectrum has a signaturewithout low frequency energy and structural bias in the frequencydomain. Actually, a good generator of blue noise sample tends toreplace low frequency aliasing with high frequency noise in orderto avoid be less visually objectionable [12]. Such patterns arewidely used in various graphics applications including importancesampling, rendering, imaging, sensing and geometry processing.The most popular pattern in this family is the so-called Poisson disk

distribution [7], in which all points are separated from each otherby a minimum distance. These distributions have provable bluenoise characteristics, but are notoriously difficult to generate.

Because of its popularity, a large number of alternative methodsto generate point distributions having blue noise spectra have beenproposed over the last two decades, the most recent based on theso-called Capacity-Constrained Voronoi Tessellation (CCVT) [3], whichproduces quality point distributions, but is very slow. In this paper,we present a different approach which can be considered the ‘‘dual’’

ll rights reserved.

8.

[email protected] (L. Liu),

ard.edu (S.J. Gortler).

acity-Constrained Delauna

method to CCVT. It generates point distributions comparable toCCVT, but does it significantly faster. It is based on the Capacity-

Constrained Delaunay Triangulation (CCDT), which is a Delaunaytriangulation of P with n points and has triangle areas which areas uniform as possible. The CCDT is computed by alternatingbetween two complementary phases. The first phase optimizesthe geometry (positions) of the points while keeping its connectivityfixed and the second step optimizes the connectivity (topology) ofthe triangulation, while keeping its geometry fixed. After conver-gence of this procedure, the resulting vertex set is shown to possesssuperior blue noise characteristics.

As an extension of the basic algorithm, we introduce adensity function into the CCDT cost function. This results in pointdistributions having different spatial densities in different regionsof the shape, and can be used in applications requiring non-uniform point distributions, such as image halftoning.

The more difficult component of the CCDT algorithm is thatwhich optimizes the geometry of the points. Intuitively, we wouldlike each point to represent a region of the plane, and that allregions have approximately the same area. The CCVT methodachieves this by generating a generalized Voronoi diagram whosecells all have approximately the same area, and then each cell isrepresented by its site, which is also at its centroid. The rationalebehind using Voronoi diagrams is clear—each cell is convex witha fairly regular shape, thus plays a role similar to the Poisson disk.However, dealing with the complexity of Voronoi cells requiresquite an elaborate algorithm, and is achieved by using extra‘‘samples’’ (apart from the sites) to numerically approximate thecell areas. Since the number of extra samples is typically at least an

y Triangulation for point distributions. Computers and Graphics

Y. Xu et al. / Computers & Graphics ] (]]]]) ]]]–]]]2

order of magnitude larger than the number of sites, this makes for avery slow algorithm. A later implementation of the CCVT algorithm,called Fast CCVT (FCCVT) [11], improved the runtime by an order ofmagnitude, but still uses the same basic technique of discretesamples. We bypass the need to use discrete samples by consideringthe ‘‘dual’’ problem, which is much easier to solve, essentiallyreplacing generalized Voronoi cells with Delaunay triangles on thesame sites. Using triangles allows for a simple scheme to optimizethe geometry of a given triangulation such that a relevant costfunction is optimized. This is done by locally optimizing each vertexin turn, assuming its neighbors’ positions are fixed. In its simplestform, we minimize the variance of the triangle areas, which is aquadratic form, thus boils down to a simple 2�2 linear system inthe (x, y) coordinates of the vertex. When given a density functionon the plane, we may similarly minimize the variance of the‘‘capacities’’—the total density in each triangle. The well-knownregularity of the shapes of the Delaunay triangles prevents artifactsfrom forming in the distribution.

When minimizing the variance of the Delaunay triangle areas,starting from a random distribution of sites, we find that theresulting distributions of sites possess high-quality blue noisecharacteristics with large mutual distances between points andno apparent regularity artifacts, comparable to those generatedby other leading approaches, at a fraction of the cost. Whenminimizing the capacity of a non-uniform density function overthe Delaunay triangulation, the resulting point distributionsclosely mimic the density function with minimal visual artifacts.

2. Related work

Sampling is widely used in various graphics applicationsincluding image rendering and geometry processing. Poisson disksampling was first introduced to solve the aliasing problem in thefield of computer graphics [7]. A Poisson disk distribution con-tains spatially uniform points which are irregular, yet not tooclose to each other. These distributions possess a so-called ‘‘bluenoise’’ power spectrum—one lacking low frequency energy andhaving structural residual peaks.

Many algorithms to generate Poisson disk distributions havebeen proposed in the literature. Cook [6] first proposed an expensive‘‘dart throwing’’ algorithm, but since then, several faster techniqueshave been developed, and an extensive survey of these methodsmay be found in [10]. Yet, none of these methods achieves all goals,namely blue noise characteristics, adaptation to density functionsand efficiency. The most practical methods come from introducingsome discrete ‘‘topology’’ into the point pattern, either a Voronoidiagram or its dual Delaunay triangulation, and optimizing a costfunction based on both geometry and topology.

McCool and Fiume [15] first used Lloyd’s method in an attemptto generate Poisson disk distributions. This method leads directlyto the Centroidal Voronoi Tessellation (CVT), which is a Voronoidiagram such that each site is located at the centroid (centerof mass) of its Voronoi cell. It is a commonly used optimizationmethod for 2D triangulations [8], and has been used for bothisotropic [1] and anisotropic meshing [9]. The Optimal Delaunay

Triangulation (ODT) method is another optimization-based mesh-ing method proposed by Chen and Xu [5] and enhanced by Alliezet al. [2] and Tournois et al. [16]. Being the solution to a non-convex optimization problem, any algorithm to compute a CVTwill converge to some local minimum of the energy. However,these local minima, like the global minimum (the hexagonal grid),are extremely regular. Thus the CVT method fails to satisfy boththe spatial uniformity and low regularity artifact requirements,as does the ODT method, and does not result in distributionssufficiently close to Poisson disk.

Please cite this article as: Xu Y, et al. Capacity-Constrained Delauna(2011), doi:10.1016/j.cag.2011.03.031

Recently, Balzer et al. [3] proposed the Capacity-Constrained

Voronoi Tessellation (CCVT) method for optimizing point distributions.The CCVT is a generalized Voronoi diagram (also called a power

diagram) such that all cells have identical ‘‘capacity’’ and each site is atthe centroid of its cell. In its simplest form, capacity is area. Theresulting distributions of Voronoi sites possess high-quality blue noisecharacteristics. When given a density function, capacity translates tothe total density contained in a cell. However, the CCVT methodapproximates the capacity using a large number of discrete samples,so its efficiency leaves a lot to be desired.

3. The CCDT algorithm

3.1. Uniform CCDT

The CCVT algorithm [3] proposes to generate uniform pointdistributions by partitioning the relevant planar region intogeneralized Voronoi cells having uniform areas, from which thesites are taken as the points. Area uniformity seems to overcomeregularity artifacts, resulting in superior blue noise characteris-tics. Our CCDT algorithm does the same, replacing the complexVoronoi cells with simple Delaunay triangles. So we seek aDelaunay triangulation T of a given planar domain, whosetriangles have areas At, with minimal variance

T ¼ argmin1

m

XtAT

At�1

m

XtAT

At

!2

where m is the number of triangles in T. Our algorithm to generate aCCDT alternates between local area uniformization and Delaunaytriangulation until convergence. The first phase optimizes thegeometry, and the second optimizes the connectivity. Starting froma random initial distribution of n points, the first phase examineseach vertex in turn, and adjusts its position such that the trianglesincident on that vertex have minimal variance in area. Since allneighboring vertices have fixed positions, this is a simple 2�2linear system in the (x, y) coordinates of the vertex, having a closed-form solution (which is a special case of the more general solutionpresented in the next section analyzing the non-uniform case). Thesecond phase runs a standard Delaunay triangulation routine, suchas may be found in CGAL [18]. In practice, a number of successivegeometry optimizations (up to 70 passes over the point set) may berun in the first phase.

Fig. 1 shows the results of running the uniform CCDT algo-rithm on some polygonal boundary shapes, both convex and non-convex, and graphs of their variances of triangle areas over time.Starting from an essentially random distribution, it is quiteevident that the output CCDT’s have much more uniform areasthan those of the input.

The graphs in Fig. 1 show how the triangle area variancedecreases as the algorithm proceeds. The occasional slight increasein energy comes from the Delaunay triangulation phase, due to thefact that some area distribution uniformity is sacrificed to gain ahigher quality connectivity. As can be seen in the graph, typicallyless than 10 iterations are required to reach convergence.

3.2. Non-uniform CCDT

The uniform CCDT algorithm produces point distributions whichhave a uniform distribution, essentially close to a Poisson diskdistribution. However, in many applications, such as the imagehalf-tone problem, it is desirable to generate point distributionswhich conform to a non-uniform density function. Indeed, assumewe are given a density function rðXÞ on the interior of the shape. Wewould like to generate a CCDT such that the total density in each

y Triangulation for point distributions. Computers and Graphics

0 50 100 150 200 250 300 350 4005.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

6

6.1x 10−4

x 10−4

0 20 40 60 80 1004.7

4.8

4.9

5

5.1

5.2

5.3

5.4

5.5initial CCDT area variance

Fig. 1. Generating a uniform CCDT of 1024 sites (top) and 286 sites (bottom). The x-axis of the graph is the number of passes over the image. The red stars indicate a

Delaunay triangulation phase (end of an iteration). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this

article.)

Input Initial CCDT

Fig. 2. Non-uniform CCDT of 4096 sites.

Input 0.83% 9.05% 30.8% 59.3%

CCVT 1.1% 9.1% 30.8% 59.0%

CCDT 0.9% 8.6% 31.4% 59.1%

Fig. 3. Non-uniform CCVT and CCDT of quadratic ramp using 1000 sites.

Y. Xu et al. / Computers & Graphics ] (]]]]) ]]]–]]] 3

triangle – it’s capacity – is more or less the same. This will force theplacement of more points in regions of high density, and less pointsin regions of low density.

The only change to the uniform CCDT algorithm required to dealwith non-uniform distributions is in the geometry optimizationphase. Given a triangulation T, we must first compute the averagepixel density dt within each triangle t, and then locally optimize theposition of each vertex v such that the total triangle capacityDt¼Atdt has minimal variance. To do this we assume that, oncecomputed, the per-pixel density dt is fixed and we must optimizethe coordinates of v such that the capacity variance is minimized

1

m

XtAT

Atdt�1

m

XtAT

Atdt

!2

This is a simple 2�2 linear system for the coordinates (x,y) of thevertex v as a function of the coordinates of its k one-ring neighborsvi�(xi,yi) and densities di of its one-ring triangles. For simplicity, wewill denote by Ai and di the area and density of the triangle whose

Please cite this article as: Xu Y, et al. Capacity-Constrained Delauna(2011), doi:10.1016/j.cag.2011.03.031

vertices are v, vi and viþ1 (see embedded figure).

y Triangulation for point distributions. Computers and Graphics

Y. Xu et al. / Computers & Graphics ] (]]]]) ]]]–]]]4

Denoting ai ¼ ðyi�yiþ1Þ, bi ¼ ðxiþ1�xiÞ and ci ¼ ðxiyiþ1�xiþ1yiÞ,we have Ai ¼ aixþbiyþci. Minimizing the variance

Xk

i ¼ 1

diAi�1

k

Xk

j ¼ 1

ðdjAjÞ

0@

1A

2

which is a now quadratic form in x and y, leads to a 2�2 linearsystem for X¼(x y)t: RtR

� �X ¼�RtS, where R is the k�2 matrix:

Ri,1 ¼ diai�1

k

Xk

j ¼ 1

ðdjajÞ

Ri,2 ¼ dibi�1

k

Xk

j ¼ 1

ðdjbjÞ

and S is the k�1 column vector:

Si ¼ dici�1

k

Xk

j ¼ 1

ðdjcjÞ

Fig. 4. Spectral analysis of different methods generating uniform distributions of 1024 s

bottom: CVT, ODT, CCVT and CCDT.

Please cite this article as: Xu Y, et al. Capacity-Constrained Delauna(2011), doi:10.1016/j.cag.2011.03.031

Computation of the average pixel density per triangle di isdone in one pass over the image, immediately after the Delaunaytriangulation. Each pixel is classified as belonging to a triangle(using a hierarchical point location routine), and the intensityvalues are summed per triangle. The average density per pixel isthis value normalized by the triangle area (in pixels).

The CCDT algorithm starts from an initial random distribu-tion of n sites conforming to r, generated in a manner similarto how CCVT generates its sample points. In a nutshell, a point ispositioned randomly at (x, y) in the (continuous) 2D region ofinterest. A uniformly distributed random number rA[0,1] isthen generated. If ror(x,y), the point is kept, otherwise dis-carded. This process is repeated until n sites are kept. This initialdistribution is a very rough approximation of r. If r is given as adiscrete grayscale image, this just means that r(x,y) is piecewiseconstant.

Fig. 2 shows an example of a binary half-tone generated by thenon-uniform CCDT algorithm for a simple grayscale image, andFig. 3 compares the non-uniform CCVT and CCDT on the quadratic

ites. The graphs and timings were averaged over 10 point distributions. From top to

y Triangulation for point distributions. Computers and Graphics

Y. Xu et al. / Computers & Graphics ] (]]]]) ]]]–]]] 5

density ramp used in [3]. As opposed to the uniform version, thenon-uniform version of CCDT performs only 5 passes over theimage in the geometry optimization phase (after that the di areoutdated).

4. Experimental results

In this section we demonstrate the usefulness of the CCDTapproach, both uniform and non-uniform, to quickly generatesuperior point distributions. We compare the CCDT, implementedin Cþþ, to those of the main competing algorithms—CVT, ODT andCCVT, whose Cþþ implementations were provided by the authors.Our experiments were run on a PC with a Intel Core 2 Duo T7200@ 2.0 GHz processor and 2 GB RAM under Windows 7. The CCVTalgorithm was run with 256 samples per site.

4.1. Blue noise spectra

The point samplings generated by Uniform CCDT may be used asPoisson disk distributions. We measure the quality of these dis-tributions, and those generated by the competition, by examiningtheir blue noise characteristics in the frequency domain, as advo-cated by Ulichney [17]. We use Barlett’s method [4] to estimate thepower spectrum by averaging periodograms of distributions, deter-mined by their Fourier transforms. Due to its radial symmetry, thepower spectrum may be characterized by two one-dimensionalmeasures: radially averaged power spectrum and anisotropy. In agood distribution, the radially averaged power spectrum shouldexhibit typical blue noise characteristics and the anisotropy curveshould be low and stable, implying high symmetry.

To compare the blue noise characteristics, we apply all methodsto generate point sets in the unit square. For each method, itsperiodogram is averaged using the results of 10 different outputs of1024 sites generated using different initial (random) inputs. Fig. 4summarizes the results. The point distribution of CVT in the top row

3

0

Fig. 5. Evolution of the CCDT process. Result after 0–

Please cite this article as: Xu Y, et al. Capacity-Constrained Delauna(2011), doi:10.1016/j.cag.2011.03.031

has quite visible regularity artifacts. Thus the radially averagedpower spectrum and anisotropy are turbulent. The outputs of theother three methods contain significantly less regularity artifacts,but the power spectrum and radially averaged power spectrum ofODT are still less smooth and more turbulent than those of CCVTand CCDT. The spectra of CCVT and CCDT are quite similar except fora little less symmetry shown in anisotropies. Both of their powerspectra and radially averaged power spectra exhibit the high-qualityblue noise characteristic, thus both are good candidates to bePoisson disk distributions. However, CCVT required 2.8 s to computevs. 0.3 s for CCDT. Generating a uniform CCVT and uniform CCDT for4096 sites required 24.8 and 1.4 s, respectively, and for 16,384 sitesrequired 497 vs. 4.3 s respectively.

4.2. Binary halftoning

Fig. 5 shows the evolution of our halftoning algorithm basedon the non-uniform CCDT, displaying the result after each of thefive iterations required for convergence. The entire process took3.4 s. The input image is the first of those used in Fig. 6, where wecompare the results of binary halftoning of a number of grayscaleimages using non-uniform CCVT and CCDT. Each image is accom-panied by the final halftones generated by CCVT and CCDT for4096 sites, and the runtime statistics.

4.3. Algorithm complexity

All algorithms studied in this paper are iterative, thus not veryfast. CVT uses the classical Lloyd algorithm [14], for which effectiveaccelerated versions exist [13]. Unfortunately, CVT has been shownto generate inferior point distributions. The original implementationof CCVT [3] was very slow, and an improved implementation [11]reports acceleration by up to an order of magnitude, using variousparallelization techniques. CCVT seems to be an effective, albeitrather complex, algorithm, requiring approximately 100 iterations

2 1

4 5

5 iterations. The entire process consumed 3.4 s.

y Triangulation for point distributions. Computers and Graphics

pixels CCVT (74 iters, 39.4 secs) CCDT (5 iters, 3.4 secs)

468 pixels CCVT (81 iters, 48 .3 secs) CCDT (5 iters, 5.7 secs)

pixels CCVT (93 iters, 986 secs) CCDT (5 iters, 9 secs)

Fig. 6. Binary halftoning of a number of grayscale images using CCVT and CCDT. The top two images used 4096 sites, and the bottom one 16,384 sites. These images are

best viewed on a high-resolution display device (or printer).

Y. Xu et al. / Computers & Graphics ] (]]]]) ]]]–]]]6

until convergence, versus just 5 iterations for CCDT, where eachiteration consists of 5 geometry optimization passes and oneDelaunay triangulation (and subsequent triangle capacity computa-tion). In the uniform case, the timings quoted in Section 4.1 wouldindicate a runtime which is linear in the number of sites for CCDT,while CCVT seems to be super-linear. In the non-uniform case, thesituation is similar—the speedup achieved by CCDT relative to theoriginal CCVT algorithm can reach two orders of magnitude fortypical number of sites (whereas Fast CCVT provides a speedup ofjust one order of magnitude). Thus CCDT strikes an excellencebalance between complexity and quality, while being extremelysimple to describe and implement.

5. Conclusion

In this paper, we propose a new approach for efficientlygenerating point distributions with superior blue-noise character-istics using an optimization-based meshing approach. The method is

Please cite this article as: Xu Y, et al. Capacity-Constrained Delauna(2011), doi:10.1016/j.cag.2011.03.031

based on the generation of a Capacity-Constrained Delaunay Trian-gulation (CCDT), in which the triangle areas are as uniform aspossible. We formulate the problem as the minimum of an energyfunction and describe a simple and efficient iterative algorithm tooptimize it. Spectral analysis shows that CCDT can produce pointdistributions with superior blue noise characteristics. We also showhow to introduce a non-uniform density function into the CCDTenergy function and optimize it using a similar algorithm. Thisallows rapid generation of point distributions for the image half-tone problem at a speed which is an order of magnitude faster thanthe best competitor (Fast CCVT). Further optimizations to theCCDT algorithm are possible by using OpenGL to perform thetriangle capacity computation, which is currently implementedusing hierarchical point location in a triangulation per pixel, andby straightforward parallelization of the geometry optimizationroutine, which is run locally per vertex.

The CCDT method may be generalized in a straightforwardmanner to three dimensions to produce tetrahedralizations of 3Ddomains having uniform and non-uniform volume distributions.

y Triangulation for point distributions. Computers and Graphics

Y. Xu et al. / Computers & Graphics ] (]]]]) ]]]–]]] 7

Acknowledgments

This work is partly supported by the National Natural ScienceFoundation of China (61070071) and the Fundamental ResearchFunds for the Central Universities (2010QNA3039).

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y Triangulation for point distributions. Computers and Graphics