Parallel Bluenoise Sampling by Constrained Farthest Point Optimization

Parallel Blue-noise Sampling by Constrained Farthest Point Optimization Renjie Chen Craig Gotsman Technion – Israel Institute of Technology

Blue-noise distribution • AKA Poisson disk distribution • Uniform Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO • Uniform point density • Large minimal mutual distance • Irregularity • No correlations between points 2 Random Blue-noise Hexagonal grid

• Introduction & related work • FPD & FPO • Local characterization of FPD • Constrained FPO • Experimental results & conclusion Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Outline 3

Blue-noise distribution • Power spectrum analysis • Periodograms of Fourier transform on point distributions Radially averaged power spectrum Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO • Ulichney, 1987: study of the frequency domain characteristics Anisotropy 4 Power spectrum Blue-noise spectra

structural residual peaks stably flat Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Blue-noise Spectra 5 lacking lowfrequency

Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Applications 6

Related Work Relaxation based Precomputed tiles Poisson disk distribution Lloyd’s relaxation Poisson disk tiles (Crow, 1977) (Mc. Cool et al, 1992) (Shade et al, 2000) Dart-throwing ODT Edge-based tiles (Cook, 1986) (Chen et al, 2004) (Lagae and Dutre, 2005 a) Boundary Sampling CCVT Template tiles (Dunbar and Humphreys 2006) (Balzer et al, 2009) (Lagae and Dutre, 2005 b) Parallel Poisson disk sampling FPO Corner-based tiles (Schlömer et al, 2011) (Lagae and Dutre, 2006) CCDT Recursive Wang Tiles (Xu et al, 2011) (Kopf et al, 2006) …… …… Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Dart-throwing based (Wei et al, 2008) Maximal Poisson-disk sampling (Ebeida et al, 2011, 2012) …… 7

Dart-throwing • Hard to control sampling size • Computationally expensive All points are separated from each other by a minimum distance Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO • Incrementally generate samples randomly 8

• Starting from an initial distribution • Move points following some criteria until converge • Hard to possess blue noise characteristic Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Relaxation Approach 9

Precomputed tiles • Seamlessly tile the tiles • Inferior Blue-noise quality • Lacking variety Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO • Construct a few tiles of Blue-noise pattern 10

Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO FPD - Farthest Point Distribution • 11

Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO FPD - Farthest Point Distribution l Blue-noise Spectra 12

Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO - Farthest Point Optimization • 13

FPO - Farthest Point Optimization Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO • Farthest Point X – the point set Largest empty circle DT(X) Priority queue 14

Local? FPO – Linear complexity? • Con – Requires maintaining a global DT – Slow convergence – Difficult to parallelize Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO • Pro 15

Constrained Farthest Point Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO • Local FP and global FP can be different 16

• Local farthest point property • Local covering property Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Local characterization of FPD 17

• Local farthest point property • Local covering property Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Local characterization of FPD 18

• Local farthest point property • Local covering property Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Local characterization of FPD 19

• Local farthest point x' Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO CFPO – constrained farthest point optimization 20

Uniform random initialization 1 iteration Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO CFPO – constrained farthest point optimization • Local covering 2 iterations 21

1 iteration 20 iterations Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Parallel CFPO 22

LARGEST EMPTY CIRCLE Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Local CFPO 23

Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Convergence 24

FPO CFPO simplified CFPO Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Results – Blue-noise spectra 25

8 x 80 x Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Results - Performance 26

• CFPO • Local optimization • Global property • Equivalency with FPO • Easy parallelization • Future Work • Non-uniform/anisotropic distribution • Higher dimension • Non-Euclidean manifold Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO Conclusion 27

28 Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO

CCDT CCVT DT BS CFPO Parallel Blue-noise Sampling by 2021/2/28 Constrained FPO 29