Blue Noise Sampling via Delaunay Triangulation

1. ZoltanSzego†*, Yoshihiro Kanamori‡, TomoyukiNishita† †The University of Tokyo, *Google Japan Inc., ‡University of Tsukuba Blue Noise Sampling via Delaunay Triangulation

2. Contents • Background • Related Work • Our Method • Results • Conclusions and Future Work

3. Contents • Background • Related Work • Our Method • Results • Conclusions and Future Work

4. Background • Sampling is essential in CG • rendering, image processing, object placement etc. Light sampling on HDR environment maps Halftoning

5. Background • Desired sampling patterns • Equally distant samples … e.g. Poisson disk • Low energy in low frequency of the Fourier spectrum … Blue noise Equally distant → Blue noise cf. Totally random → White noise

6. Background • Blue noise property • Observed in natural objects • Considered optimal for human eyes Layout of human eye photoreceptors [Yellott, 1983]

7. Background • Quality measures for blue noise spectra • Radial average power spectrum • The larger the central ring, the better • Anisotropy • The lower and flatter, the better ring Radial averagepower spectrum ring Spectrum Anisotropy

8. Our Goal • Efficient, high-quality blue noise sampling • Adaptive sampling should be supported Uniform Adaptive

9. Our Goal • Support for sampling in various domains • 2D • 3D (volumetric sampling) • On curved surfaces (spheres, polygonal meshes) 2D 3D On curved surfaces

10. Contents • Background • Related Work • Our Method • Results • Conclusions and Future Work

11. Related Work • Two major approaches • Dart throwing • Random sampling of equidistant samples • Tiling • Tiling of precomputed samples

12. Related Work • Dart throwing[Cook, 1986] • Used for distributed ray tracing • High computational cost • Quality improvement: Lloyd’s relaxation … more costly • Parallel Poisson disk [Wei, 2008] • GPU-based acceleration • # of samples cannot be determined • Only supports 2D and 3D • Our method • # of samples can be specified • Supports 2D, 3D, and curved surfaces

13. Related Work • Wang tiles [Kopf et al., 2006] • Requires precomputation • Low quality • Polyominoes [Ostromoukhov, 2007] • Requires complicated precomputation • Our method • High quality • No precomputation

14. Contents • Background • Related Work • Our Method • Results • Conclusions and Future Work

15. Overview • Input: seed points • Given by the user • Output: blue noise samples • Features: • Deterministic (reproducible with the same seeds) • No precomputation • Supports various sampling domains

16. Overview • Sequentially sample atthe most sparse region • The largest emptycircle problem[Okabe et al., 2000] • Can be solved using Delaunay triangulation • Correspond to finding the largest circumcircle in Delaunay triangles 2D example

17. Basic Algorithm • Loop: • Find the largest empty circle • Add a sample at the center 2D example

18. Basic Algorithm • Loop: • Find the largest empty circle • Add a sample at the center • UpdateDelaunay triangles 2D example

19. Basic Algorithm • Acceleration for search: Use of heap • To find the largest circumcirclein O(1) • Costs for insert / delete: O(log N) • Support for adaptive sampling • Scale the radii stored in the heapusing density functions • The greater the density, the higher the priority Heap of circumcircles’ radii Density function

20. Artifact #1 • Regular patterns peaks in the spectrum

21. Modification #1 • Reason of the artifacts • Iterative subdivisions of equilateral triangles • Our solution: • Detect an equilateral triangle • Displace the new samplefrom the center of its circumcircle(see our paper for details)

22. Artifact #2 • Sparse samplesat boundaries • Reason • Very thin trianglesaround boundaries • Our solution: • Use of periodic boundaries Tiled samples(tiled just for illustration)

23. Modification #2 • Periodic boundaries • Toroidal (torus-like) domain

24. Modification #2 • Pros: • Sparse regions disappear • Edge lengths of triangles become balanced • Overall centers of circumcircles lie within their triangles • Allows us to specify the position of the new sample in O(1) • Cons: • A little additional cost for modifying coordinates

25. Parallelization • Exploit multi-core CPUs • Uniform subdivision of 2D domain • Further subdivision • Costs: O(N log N)4 M log M < N log N (if M = N/4) • 4x4 subdivision is the fastest for a 4-core CPU • 1.69 times faster for 100K samples 1 2 3 4 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4

26. Sampling in 3D • 3D domain: [0, 1)3 • 2D → 3D • Triangles → Tetrahedra(Delaunay Tetrahedralization) • Circumcircles→ Circumspheres • Similar to 2D algorithm Delaunay tetrahedralization

27. Sampling on Curved Surfaces • Sampling domain: • Spherical surfaces • Polygonal mesh surfaces • Initial seeds: • Vertices of simplified mesh • Similar to 2D • New samples are projectedonto the surface Simplified Given mesh Initial seeds Samples on a sphere

28. Contents • Background • Related Work • Our Method • Results • Conclusions and Future Work

29. Results • Uniform sampling # of samples： 20KTime： 92 ms Experimental environment:Intel Core 2 Quad Q6700 2.66GHz,2GB RAM

30. Comparison – 50,000 samples – Radial average Anisotropy Radial average Anisotropy Our method: 378 msec Wang tiles [2006]: 1.35 msec

31. Comparison – 50,000 samples – Radial average Anisotropy Radial average Anisotropy ours Our method: 378 msec Dart throwing [2007]: 420 msec

32. Results • 20K samples in 3D

33. Results • Spectra for 10K samples in 3D Low energy spheres in the center → blue noise property

34. Results • Sampling on a sphere • Initial mesh: an equilateral octahedron Dense Sparse Density function

35. Results • Sampling on HDR environment maps • Blighter region → denser samples

36. Contents • Background • Related Work • Our Method • Results • Conclusions and Future Work

37. Conclusions • High-quality blue noise samplingusing Delaunay triangulation • Find centers of largest circumcirclesof Delaunay triangles • Adaptive sampling by scaling circumcircles’ radii • Support for sampling on various domains:2D, 3D, and curved surfaces

38. Future Work • GPU acceleration using CUDA • Fast Lloyd’s relaxation using the connectivity of Delaunay triangles

39. Thank you