SCIENTIA SINICA Informationis, Volume 47 , Issue 4 : 442-454(2017) https://doi.org/10.1360/N112016-00167

Maximal Poisson-disk sampling by sampling radius optimization

• AcceptedOct 23, 2016
• PublishedMar 14, 2017
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Abstract

In the field of computer graphics, maximal Poisson-disk sampling (MPS) is a fundamental research topic. An ideal sampling set should satisfy unbiased sampling property, minimal distance property, and maximal sampling property. In general, MPS is obtained by Dart Throwing, as we all know, the drawback of this method is unable to precisely control the number of samples. In view of the above problem, this work proposes a novelty algorithm that can precisely control the number of samples of two-dimensional radius-equal MPS, and satisfy other properties simultaneously. Unlike existing methods, the proposed method controls the number of samples by adjusting sampling radius. Firstly, according to user-specified the number of samples and sampling domain (closed polygon), initial samples are randomly obtained, then Delaunay triangulation is conducted, and taking as current sampling radius the shortest edge length of the triangulation. Secondly, iteratively removing the endpoint of global shortest edge with larger neighborhood-averaged edge length, and then using Dart Throwing to randomly insert it into gap region that is calculated at current sampling radius. By iteratively adjusting the position of points, the sampling radius increases gradually, finally, MPS with fixed number of sampling point can be achieved. Experimental results show that this method generates point sets with high quality, at the same time, ensures the excellent blue-noise property for MPS.

Funded by

• Figure 1

(Color online) The workflow of our method. (a) Initial sampling; (b) and (c) show the sampling after 100 and 300 iterations, respectively; (d) final result

• Figure 2

(Color online) Illustration of gap primitives. (a) Triangle; (b) quadrilateral; (c) pentagon; (d) hexagon

• Figure 3

(Color online) Illustration of one iteration

• Figure 4

Comparison of the sampling quality

• Figure 5

Sampling and triangulation in polygonal domains

• Figure 6

Histograms of the relative radius of FPO and our method

• Figure 7

PCF analysis of points with different numbers. (a) 0.5k; (b) 1k; (c) 2k; (d) 20k

• Figure 10

(Color online) Timing comparison

• Table 1   Statistics of the sampling and the triangulation quality in periodic domain
 Method $\delta_X$ $Q_{\rm min}$ $Q_{\rm avg}$ $\theta_{\rm min}$ $\overline{\theta}_{\rm min}$ $\theta_{\rm max}$ $\theta_{<30^\circ}$ $\theta_{>90^\circ}$ $V_{567}%$ FPO 0.925 0.567 0.856 35.12 50.90 107.51 0.00 6.50 99.61 SER 0.781 0.509 0.813 30.54 46.11 114.41 0.00 14.50 96.48 MPS 0.780 0.487 0.806 30.19 45.30 117.11 0.00 15.07 96.53 OURS 0.855 0.514 0.833 30.21 47.97 113.85 0.00 12.60 98.30 OURS CV($%$) 0.19 3.44 0.38 0.22 0.47 1.76 0.00 2.46 0.31
•

Algorithm 1 基于采样半径优化的最大化Poisson圆盘采样算法

输入: 采样点数$N$和采样区域$\Omega$

输出: 采样点集$X$

初始化: 在$\Omega$中随机产生$N$个采样点, 并进行Delaunay三角化(DT);

while TRUE do

寻找DT的全局最短边, 删除其中邻域平均边长较大的端点;

局部更新DT, 更新采样半径为当前DT的最短边长;

执行gap检测操作;

if 不存在gap三角形 then

将删除的顶点插入到DT的最大外接圆的圆心;

BREAK; //算法结束;

end if

执行gap基元提取操作;

在空隙区域中随机产生一个采样点, 将其插入到$X$中;

动态更新DT;

end while

• Table 2   Statistics of the sampling and the triangulation quality in arbitrary boundary conditions
 Boundary name $\delta_X$ $Q_{\rm min}$ $Q_{\rm avg}$ $\theta_{\rm min}$ $\overline{\theta}_{\rm min}$ $\theta_{\rm max}$ $\theta_{<30^\circ}$ $\theta_{>90^\circ}$ $V_{567}%$ Wavy 0.816 0.513 0.813 30.62 45.82 114.01 0.00 14.25 94.85 Face 0.822 0.502 0.818 31.12 45.30 115.33 0.00 13.96 94.12 Dolphin 0.810 0.490 0.813 30.70 46.03 116.84 0.00 14.23 93.95 S 0.830 0.512 0.813 31.02 46.12 113.08 0.00 13.32 94.33

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