Computers & Graphics

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1 Computers & Graphics ] (]]]]) ]]] ]]] Contents lists available at ScienceDirect Computers & Graphics journal homepage: SMI 211: 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, China b Technion, Israel c Harvard University, USA article info Article history: Received 14 December 21 Received in revised form 13 March 211 Accepted 14 March 211 Keywords: Blue noise Poisson disk distribution Capacity-Constrained Delaunay Triangulation (CCDT) Minimal area variance abstract 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 nonuniform point distributions. These may be used to half-tone grayscale images. & 211 Elsevier Ltd. All rights reserved. 1. Introduction Point distributions with blue noise characteristic are ubiquitous in computer graphics [6,3]. The blue noise point distribution generation problem can be formulated as follows: given a shape described by its polygonal boundary P and the number of point samples n, position n points inside P forming a spatial pattern with a blue noise spectrum. A blue noise spectrum has a signature without low frequency energy and structural bias in the frequency domain. Actually, a good generator of blue noise sample tends to replace low frequency aliasing with high frequency noise in order to avoid be less visually objectionable [12]. Such patterns are widely used in various graphics applications including importance sampling, 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 other by a minimum distance. These distributions have provable blue noise characteristics, but are notoriously difficult to generate. Because of its popularity, a large number of alternative methods to generate point distributions having blue noise spectra have been proposed over the last two decades, the most recent based on the so-called Capacity-Constrained Voronoi Tessellation (CCVT) [3], which produces quality point distributions, but is very slow. In this paper, we present a different approach which can be considered the dual n Corresponding author. Tel./fax: þ addresses: yinxu.zju@gmail.com (Y. Xu), ligangliu@zju.edu.cn (L. Liu), gotsman@cs.technion.ac.il (C. Gotsman), sjg@cs.harvard.edu (S.J. Gortler). method to CCVT. It generates point distributions comparable to CCVT, but does it significantly faster. It is based on the Capacity- Constrained Delaunay Triangulation (CCDT), which is a Delaunay triangulation of P with n points and has triangle areas which are as uniform as possible. The CCDT is computed by alternating between two complementary phases. The first phase optimizes the geometry (positions) of the points while keeping its connectivity fixed and the second step optimizes the connectivity (topology) of the triangulation, while keeping its geometry fixed. After convergence of this procedure, the resulting vertex set is shown to possess superior blue noise characteristics. As an extension of the basic algorithm, we introduce a density function into the CCDT cost function. This results in point distributions having different spatial densities in different regions of the shape, and can be used in applications requiring nonuniform point distributions, such as image halftoning. The more difficult component of the CCDT algorithm is that which optimizes the geometry of the points. Intuitively, we would like each point to represent a region of the plane, and that all regions have approximately the same area. The CCVT method achieves this by generating a generalized Voronoi diagram whose cells all have approximately the same area, and then each cell is represented by its site, which is also at its centroid. The rationale behind using Voronoi diagrams is clear each cell is convex with a fairly regular shape, thus plays a role similar to the Poisson disk. However, dealing with the complexity of Voronoi cells requires quite an elaborate algorithm, and is achieved by using extra samples (apart from the sites) to numerically approximate the cell areas. Since the number of extra samples is typically at least an /$ - see front matter & 211 Elsevier Ltd. All rights reserved. doi:1.116/j.cag

2 2 order of magnitude larger than the number of sites, this makes for a very slow algorithm. A later implementation of the CCVT algorithm, called Fast CCVT (FCCVT) [11], improved the runtime by an order of magnitude, but still uses the same basic technique of discrete samples. We bypass the need to use discrete samples by considering the dual problem, which is much easier to solve, essentially replacing generalized Voronoi cells with Delaunay triangles on the same sites. Using triangles allows for a simple scheme to optimize the geometry of a given triangulation such that a relevant cost function is optimized. This is done by locally optimizing each vertex in turn, assuming its neighbors positions are fixed. In its simplest form, we minimize the variance of the triangle areas, which is a quadratic form, thus boils down to a simple 2 2 linear system in the (x, y) coordinates of the vertex. When given a density function on the plane, we may similarly minimize the variance of the capacities the total density in each triangle. The well-known regularity of the shapes of the Delaunay triangles prevents artifacts from forming in the distribution. When minimizing the variance of the Delaunay triangle areas, starting from a random distribution of sites, we find that the resulting distributions of sites possess high-quality blue noise characteristics with large mutual distances between points and no apparent regularity artifacts, comparable to those generated by other leading approaches, at a fraction of the cost. When minimizing the capacity of a non-uniform density function over the Delaunay triangulation, the resulting point distributions closely mimic the density function with minimal visual artifacts. 2. Related work Sampling is widely used in various graphics applications including image rendering and geometry processing. Poisson disk sampling was first introduced to solve the aliasing problem in the field of computer graphics [7]. A Poisson disk distribution contains spatially uniform points which are irregular, yet not too close to each other. These distributions possess a so-called blue noise power spectrum one lacking low frequency energy and having structural residual peaks. Many algorithms to generate Poisson disk distributions have been proposed in the literature. Cook [6] first proposed an expensive dart throwing algorithm, but since then, several faster techniques have been developed, and an extensive survey of these methods may be found in [1]. Yet, none of these methods achieves all goals, namely blue noise characteristics, adaptation to density functions and efficiency. The most practical methods come from introducing some discrete topology into the point pattern, either a Voronoi diagram or its dual Delaunay triangulation, and optimizing a cost function based on both geometry and topology. McCool and Fiume [15] first used Lloyd s method in an attempt to generate Poisson disk distributions. This method leads directly to the Centroidal Voronoi Tessellation (CVT), which is a Voronoi diagram such that each site is located at the centroid (center of mass) of its Voronoi cell. It is a commonly used optimization method for 2D triangulations [8], and has been used for both isotropic [1] and anisotropic meshing [9]. The Optimal Delaunay Triangulation (ODT) method is another optimization-based meshing method proposed by Chen and Xu [5] and enhanced by Alliez et al. [2] and Tournois et al. [16]. Being the solution to a nonconvex optimization problem, any algorithm to compute a CVT will 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 both the spatial uniformity and low regularity artifact requirements, as does the ODT method, and does not result in distributions sufficiently close to Poisson disk. 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 at the centroid of its cell. In its simplest form, capacity is area. The resulting distributions of Voronoi sites possess high-quality blue noise characteristics. When given a density function, capacity translates to the total density contained in a cell. However, the CCVT method approximates 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 point distributions by partitioning the relevant planar region into generalized Voronoi cells having uniform areas, from which the sites are taken as the points. Area uniformity seems to overcome regularity artifacts, resulting in superior blue noise characteristics. Our CCDT algorithm does the same, replacing the complex Voronoi cells with simple Delaunay triangles. So we seek a Delaunay triangulation T of a given planar domain, whose triangles have areas A t, with minimal variance T ¼ argmin 1 m X t A T! A t 1 2 X A t m t A T where m is the number of triangles in T. Our algorithm to generate a CCDT alternates between local area uniformization and Delaunay triangulation until convergence. The first phase optimizes the geometry, and the second optimizes the connectivity. Starting from a random initial distribution of n points, the first phase examines each vertex in turn, and adjusts its position such that the triangles incident on that vertex have minimal variance in area. Since all neighboring vertices have fixed positions, this is a simple 2 2 linear system in the (x, y) coordinates of the vertex, having a closedform solution (which is a special case of the more general solution presented in the next section analyzing the non-uniform case). The second phase runs a standard Delaunay triangulation routine, such as may be found in CGAL [18]. In practice, a number of successive geometry optimizations (up to 7 passes over the point set) may be runinthefirstphase. Fig. 1 shows the results of running the uniform CCDT algorithm on some polygonal boundary shapes, both convex and nonconvex, and graphs of their variances of triangle areas over time. Starting from an essentially random distribution, it is quite evident that the output CCDT s have much more uniform areas than those of the input. The graphs in Fig. 1 show how the triangle area variance decreases as the algorithm proceeds. The occasional slight increase in energy comes from the Delaunay triangulation phase, due to the fact that some area distribution uniformity is sacrificed to gain a higher quality connectivity. As can be seen in the graph, typically less than 1 iterations are required to reach convergence Non-uniform CCDT The uniform CCDT algorithm produces point distributions which have a uniform distribution, essentially close to a Poisson disk distribution. However, in many applications, such as the image half-tone problem, it is desirable to generate point distributions which conform to a non-uniform density function. Indeed, assume we are given a density function rðxþ on the interior of the shape. We would like to generate a CCDT such that the total density in each

3 initial 3 CCDT 5.5 x 1 4 area variance x Fig. 1. Generating a uniform CCDT of 124 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 Input Initial.83% 9.5% 3.8% 59.3% CCVT 1.1% 9.1% 3.8% 59.% CCDT.9% 8.6% 31.4% 59.1% CCDT Fig. 2. Non-uniform CCDT of 496 sites. triangle it s capacity is more or less the same. This will force the placement of more points in regions of high density, and less points in regions of low density. The only change to the uniform CCDT algorithm required to deal with non-uniform distributions is in the geometry optimization phase. Given a triangulation T, we must first compute the average pixel density dt within each triangle t, and then locally optimize the position of each vertex v such that the total triangle capacity Dt ¼Atdt has minimal variance. To do this we assume that, once computed, the per-pixel density dt is fixed and we must optimize the coordinates of v such that the capacity variance is minimized 1X 1X At dt At dt m tat m tat Fig. 3. Non-uniform CCVT and CCDT of quadratic ramp using 1 sites. vertices are v, vi and vi þ 1 (see embedded figure).!2 This is a simple 2 2 linear system for the coordinates (x,y) of the vertex v as a function of the coordinates of its k one-ring neighbors vi (xi,yi) and densities di of its one-ring triangles. For simplicity, we will denote by Ai and di the area and density of the triangle whose

4 4 Denoting a i ¼ðy i y i þ 1 Þ, b i ¼ðx i þ 1 x i Þ and c i ¼ðx i y i þ 1 x i þ 1 y i Þ, we have A i ¼ a i xþb i yþc i. Minimizing the variance 1 d i A i 1 ðd k j A j ÞA i ¼ 1 which is a now quadratic form in x and y, leads to a 2 2 linear system for X¼(x y) t : R t R X ¼ R t S, where R is the k 2 matrix: R i,1 ¼ d i a i 1 k R i,2 ¼ d i b i 1 k ðd j a j Þ ðd j b j Þ and S is the k 1 column vector: S i ¼ d i c i 1 k ðd j c j Þ Computation of the average pixel density per triangle d i is done in one pass over the image, immediately after the Delaunay triangulation. Each pixel is classified as belonging to a triangle (using a hierarchical point location routine), and the intensity values are summed per triangle. The average density per pixel is this value normalized by the triangle area (in pixels). The CCDT algorithm starts from an initial random distribution of n sites conforming to r, generated in a manner similar to how CCVT generates its sample points. In a nutshell, a point is positioned randomly at (x, y) in the (continuous) 2D region of interest. A uniformly distributed random number ra[,1] is then generated. If ror(x,y), the point is kept, otherwise discarded. This process is repeated until n sites are kept. This initial distribution is a very rough approximation of r. Ifr is given as a discrete grayscale image, this just means that r(x,y) is piecewise constant. Fig. 2 shows an example of a binary half-tone generated by the non-uniform CCDT algorithm for a simple grayscale image, and Fig. 3 compares the non-uniform CCVT and CCDT on the quadratic Fig. 4. Spectral analysis of different methods generating uniform distributions of 124 sites. The graphs and timings were averaged over 1 point distributions. From top to bottom: CVT, ODT, CCVT and CCDT.

5 density ramp used in [3]. As opposed to the uniform version, the non-uniform version of CCDT performs only 5 passes over the image in the geometry optimization phase (after that the di are outdated). 4. Experimental results In this section we demonstrate the usefulness of the CCDT approach, both uniform and non-uniform, to quickly generate superior point distributions. We compare the CCDT, implemented in Cþþ, to those of the main competing algorithms CVT, ODT and CCVT, whose Cþþ implementations were provided by the authors. Our experiments were run on a PC with a Intel Core 2 Duo 2. GHz processor and 2 GB RAM under Windows 7. The CCVT algorithm was run with 256 samples per site Blue noise spectra The point samplings generated by Uniform CCDT may be used as Poisson disk distributions. We measure the quality of these distributions, and those generated by the competition, by examining their blue noise characteristics in the frequency domain, as advocated by Ulichney [17]. We use Barlett s method [4] to estimate the power spectrum by averaging periodograms of distributions, determined by their Fourier transforms. Due to its radial symmetry, the power spectrum may be characterized by two one-dimensional measures: radially averaged power spectrum and anisotropy. In a good distribution, the radially averaged power spectrum should exhibit typical blue noise characteristics and the anisotropy curve should be low and stable, implying high symmetry. To compare the blue noise characteristics, we apply all methods to generate point sets in the unit square. For each method, its periodogram is averaged using the results of 1 different outputs of 124 sites generated using different initial (random) inputs. Fig. 4 summarizes the results. The point distribution of CVT in the top row 5 has quite visible regularity artifacts. Thus the radially averaged power spectrum and anisotropy are turbulent. The outputs of the other three methods contain significantly less regularity artifacts, but the power spectrum and radially averaged power spectrum of ODT are still less smooth and more turbulent than those of CCVT and CCDT. The spectra of CCVT and CCDT are quite similar except for a little less symmetry shown in anisotropies. Both of their power spectra and radially averaged power spectra exhibit the high-quality blue noise characteristic, thus both are good candidates to be Poisson disk distributions. However, CCVT required 2.8 s to compute vs..3 s for CCDT. Generating a uniform CCVT and uniform CCDT for 496 sites required 24.8 and 1.4 s, respectively, and for 16,384 sites required 497 vs. 4.3 s respectively Binary halftoning Fig. 5 shows the evolution of our halftoning algorithm based on the non-uniform CCDT, displaying the result after each of the five iterations required for convergence. The entire process took 3.4 s. The input image is the first of those used in Fig. 6, where we compare the results of binary halftoning of a number of grayscale images using non-uniform CCVT and CCDT. Each image is accompanied by the final halftones generated by CCVT and CCDT for 496 sites, and the runtime statistics Algorithm complexity All algorithms studied in this paper are iterative, thus not very fast. CVT uses the classical Lloyd algorithm [14], for which effective accelerated versions exist [13]. Unfortunately, CVT has been shown to generate inferior point distributions. The original implementation of CCVT [3] was very slow, and an improved implementation [11] reports acceleration by up to an order of magnitude, using various parallelization techniques. CCVT seems to be an effective, albeit rather complex, algorithm, requiring approximately 1 iterations Fig. 5. Evolution of the CCDT process. Result after 5 iterations. The entire process consumed 3.4 s.

6 6 pixels 468 CCVT (74 iters, 39.4 secs) pixels CCVT (81 iters, 48.3 secs) pixels CCVT (93 iters, 986 secs) CCDT (5 iters, 3.4 secs) CCDT (5 iters, 5.7 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 496 sites, and the bottom one 16,384 sites. These images are best viewed on a high-resolution display device (or printer). until convergence, versus just 5 iterations for CCDT, where each iteration consists of 5 geometry optimization passes and one Delaunay triangulation (and subsequent triangle capacity computation). In the uniform case, the timings quoted in Section 4.1 would indicate a runtime which is linear in the number of sites for CCDT, while CCVT seems to be super-linear. In the non-uniform case, the situation is similar the speedup achieved by CCDT relative to the original CCVT algorithm can reach two orders of magnitude for typical number of sites (whereas Fast CCVT provides a speedup of just one order of magnitude). Thus CCDT strikes an excellence balance between complexity and quality, while being extremely simple to describe and implement. 5. Conclusion In this paper, we propose a new approach for efficiently generating point distributions with superior blue-noise characteristics using an optimization-based meshing approach. The method is based on the generation of a Capacity-Constrained Delaunay Triangulation (CCDT), in which the triangle areas are as uniform as possible. We formulate the problem as the minimum of an energy function and describe a simple and efficient iterative algorithm to optimize it. Spectral analysis shows that CCDT can produce point distributions with superior blue noise characteristics. We also show how to introduce a non-uniform density function into the CCDT energy function and optimize it using a similar algorithm. This allows rapid generation of point distributions for the image halftone problem at a speed which is an order of magnitude faster than the best competitor (Fast CCVT). Further optimizations to the CCDT algorithm are possible by using OpenGL to perform the triangle capacity computation, which is currently implemented using hierarchical point location in a triangulation per pixel, and by straightforward parallelization of the geometry optimization routine, which is run locally per vertex. The CCDT method may be generalized in a straightforward manner to three dimensions to produce tetrahedralizations of 3D domains having uniform and non-uniform volume distributions.

7 7 Acknowledgments This work is partly supported by the National Natural Science Foundation of China (61771) and the Fundamental Research Funds for the Central Universities (21QNA339). References [1] Alliez P, de Verdiere EC, Devillers O, Isenburg M. Centroidal Voronoi diagrams for isotropic surface remeshing. Graphical Models 25;67(3): [2] Alliez P, Steiner DC, Yvinec M, Desbrun M. Variational tetrahedral meshing. ACM Transactions on Graphics 25;24(3): [3] Balzer M, Schlomer T, Deussen O. Capacity-constrained point distributions: a variant of Lloyd s method. ACM Transactions on Graphics 29;28(3): [4] Bartlett M. An introduction to stochastic processes with special reference to methods and applications. Cambridge University Press; [5] Chen L, Xu J. Optimal Delaunay triangulations. Journal of Computational Mathematics 24;22(2): [6] Cook RL. Stochastic sampling in computer graphics. ACM Transactions on Graphics 1986;5(1): [7] Crow FC. The aliasing problem in computer-generated shaded images. Communications of the ACM 1977;2(11): [8] Du Q, Faber V, Gunzburger M. Centroidal Voronoi tessellations: applications and algorithms. SIAM Review 1999;41(4): [9] Du Q, Wang D. Anisotropic centroidal Voronoi tessellations and their applications. SIAM Journal on Scientific Computing 25;26(3): [1] Lagae A, Dutre P. A comparison of methods for generating Poisson disk distributions. Computer Graphics Forum 28;27(1): [11] Li H, Nehab D, Wei LY, Sander PV, Fu CW. Fast capacity-constrained Voronoi tessellation. In: Proceedings of the ACM SIGGRAPH symposium interactive 3D graphics and games (I3D); 21. p [12] Li H, Wei LY, Sander PV, Fu CW. Anisotropic blue noise sampling. ACM Transaction on Graphics 21;29:6. [13] Liu Y, Wang W, Lévy B, Sun F, Yan DM, Lu L, et al. On centroidal Voronoi tessellation energy smoothness and fast computation. ACM Transaction on Graphics 29;28(4):1 17. [14] Lloyd SP. Least square quantization in PCM. IEEE Transactions on Information Theory 1982;28(2): [15] McCool M, Fiume E. Hierarchical Poisson disk sampling distributions. In: Proceedings of the graphics interface; p [16] Tournois J, Wormser C, Alliez P, Desbrun M. Interleaving Delaunay refinement and optimization for practical isotropic tetrahedron mesh generation. ACM Transactions on Graphics 29;28(3):75:1 9. [17] Ulichney R. Digital halftoning. MIT Press; [18] /