Poisson Sphere Distributions

1. Department of Computer Science Katholieke Universiteit Leuven 11th International Fall Workshop VISION, MODELING, AND VISUALIZATION 2006 Poisson Sphere Distributions Ares Lagae Philip Dutré Friday 24 November 2006

2. Poisson Sphere Distributions • Definition • a 3D Poisson distribution in which all points are separated by a minimum distance 2r • if a sphere of radius r is centered at each point, no two spheres will overlap • Goal • efficiently generating Poisson sphere distributions • Motivation • existing applications of Poisson disk distributions • sampling, procedural modeling, procedural texturing

3. Poisson Disk Distributions • Definition • a 2D Poisson distribution in which all points are separated by a minimum distance 2r • if a disk of radius r is centered at each point, no two disks will overlap Poisson disk distribution minimum distance criterion

4. Poisson Disk Distributions • Applications • Sampling (Yellot 1982, Dippé 1985, Cook 1986, Mitchell 1987) • Procedural modeling (Deussen 1998) • Procedural texturing (Worley 1996, Lagae 2005) • … procedural modeling procedural texturing sampling

5. Poisson Disk Distributions • Generation • Dart throwing (Cook 1986, McCool 1992, Dunbar 2006) • Lloyd’s relaxation scheme (Lloyd 1982, McCool 1992) initial point set relaxation final point set

6. Poisson Disk Distributions • Generation • Tile-based methods (Shade 2000, Hiller 2001, Cohen 2003 Ostromoukhov 2004, Lagae 2005, Lagae 2006, Kopf 2006) tiling Poisson disk distribution

7. Corner Tiles • Tile Set • unit cube tiles, fixed orientation, colored corners • similar to Wang tiles and corner tiles (Cohen 2003, Lagae 2006) • 2 colors, 256 tiles

8. Corner Tiles • Tiling • efficient direct stochastic tiling algorithm • using hash function defined over the integer lattice (see poster) Problem: generating a Poisson sphere distribution over a set of corner tiles such that every possible tiling results in a valid Poisson sphere distribution

9. Poisson Sphere Tiles • Poisson sphere tile regions • determined by the Poisson sphere radius r corner regions edge regions face regions interior region

10. Poisson Sphere Tiles • Modified Poisson sphere tile regions • enlarge regions to make distance between regions of the same kind at least 2r modified modified modified modified corner regions edge regions face regions interior region

11. Poisson Sphere Tiles • Dual tiling • combine corner tiles with modified Poisson disk regions

12. Poisson Sphere Tiles • Dual tiling • combine corner tiles with modified Poisson disk regions

13. Poisson Sphere Tiles • Dual tile set • 4 kinds of tiles, fixed orientation • 2 corner tiles, 3x4 edge tiles, 3x16 face tiles, 256 interior tiles corner tile edge tile face tile interior tile (8 mod. corner regions) (4 mod. edge regions) (2 mod. face regions) (1 mod. interior region) Problem: generating a Poisson sphere distribution over a dual tile set

14. Poisson Sphere Tiles • Construct Poisson sphere distribution over corner tile • for each of the 2 corner tiles constraints dart throwing relaxation clip

15. Poisson Sphere Tiles • Construct Poisson sphere distribution over edge tile • for each of the 3x4 edge tiles constraints dart throwing relaxation clip

16. Poisson Sphere Tiles • Construct Poisson sphere distribution over face tile • for each of the 3x16 face tiles constraints dart throwing relaxation clip

17. Poisson Sphere Tiles • Construct Poisson sphere distribution over interior tile • for each of the 256 tiles constraints dart throwing relaxation clip

18. Poisson Sphere Tiles • Efficiently generating Poisson sphere distributions • construct Poisson sphere tiles (off-line) • generate stochastic tiling (on-line) • fast • local evaluation

19. Applications • Procedural modeling, procedural object distribution, geometry instancing

20. Applications • A 3D procedural object distribution function • outputs of the texture basis function boolean distance unique ID

21. Applications • A 3D procedural object distribution function • solid textures modeled using the texture basis function Polka dots Granite Mondriaan

22. Thanks! • Acknowledgements • Fonds Wetenschappelijk Onderzoek - Vlaanderen • Björn Jónsson • Scott Hudson grid boolean distance unique ID texture

23. Video • A 3D procedural object distribution function • integration into a commercial rendering system

24. Relative Radius Specification • Absolute radius • difficult to work with • Relative radius • intuitive • quality measure • Maximum radius

25. Spectral Analysis • Poisson sphere distribution, dart throwing power spectrum coordinate plane slices slice slice slice yz plane slice zx plane slice xy plane slice radially averaged power spectrum anisotropy

26. Spectral Analysis • Tiled Poisson sphere distribution power spectrum coordinate plane slices slice slice slice yz plane slice zx plane slice xy plane slice radially averaged power spectrum anisotropy