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Poisson Sphere Distributions
Ares Lagae Philip Dutré
Department of Computer ScienceKatholieke Universiteit Leuven
11th International Fall Workshop
VISION, MODELING, AND VISUALIZATION 2006
Friday 24 November 2006
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
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
r
2r r
Poisson disk distribution minimum distance criterion
Poisson Disk Distributions
• Applications– Sampling (Yellot 1982, Dippé 1985, Cook 1986, Mitchell 1987)
– Procedural modeling (Deussen 1998)
– Procedural texturing (Worley 1996, Lagae 2005)
– …
sampling procedural modeling procedural texturing
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
Poisson Disk Distributions
• Generation– Tile-based methods (Shade 2000, Hiller 2001, Cohen 2003
Ostromoukhov 2004, Lagae 2005, Lagae 2006, Kopf 2006)
Poisson disk distributiontiling
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
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
Poisson Sphere Tiles
• Poisson sphere tile regions– determined by the Poisson sphere radius r
corner regions edge regions face regions interior region
Poisson Sphere Tiles
• Modified Poisson sphere tile regions– enlarge regions to make distance between regions of the same
kind at least 2r
corner regions edge regions face regions interior regionmodified modified modified modified
Poisson Sphere Tiles
• Dual tile set– 4 kinds of tiles, fixed orientation– 2 corner tiles, 3x4 edge tiles, 3x16 face tiles, 256 interior tiles
(8 mod. corner regions) (4 mod. edge regions) (2 mod. face regions) (1 mod. interior region)
corner tile edge tile face tile interior tile
Problem: generating a Poisson sphere distribution over a dual tile set
Poisson Sphere Tiles
• Construct Poisson sphere distribution over corner tile– for each of the 2 corner tiles
constraints dart throwing relaxation clip
Poisson Sphere Tiles
• Construct Poisson sphere distribution over edge tile– for each of the 3x4 edge tiles
constraints dart throwing relaxation clip
Poisson Sphere Tiles
• Construct Poisson sphere distribution over face tile– for each of the 3x16 face tiles
constraints dart throwing relaxation clip
Poisson Sphere Tiles
• Construct Poisson sphere distribution over interior tile– for each of the 256 tiles
constraints dart throwing relaxation clip
Poisson Sphere Tiles
• Efficiently generating Poisson sphere distributions•
– construct Poisson sphere tiles (off-line)
– generate stochastic tiling (on-line)
– fast– local evaluation
Applications
• A 3D procedural object distribution function– outputs of the texture basis function
boolean distance unique ID
Applications
• A 3D procedural object distribution function– solid textures modeled using the texture basis function
Polka dots Granite Mondriaan
Thanks!
• Acknowledgements– Fonds Wetenschappelijk Onderzoek - Vlaanderen– Björn Jónsson– Scott Hudson
grid boolean distance unique ID texture
Video
• A 3D procedural object distribution function– integration into a commercial rendering system
Relative Radius Specification
• Absolute radius– difficult to work with
• Relative radius– intuitive– quality measure
• Maximum radius
r
maxr
maxr r
0.740518
3max
1
4 2r
N
0.65 0.85
Spectral Analysis
• Poisson sphere distribution, dart throwing
power spectrum coordinate plane slices
slice slice slice
yz plane slice zx plane slice xy plane slice anisotropyradially averaged power spectrum
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