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 tiling– combine corner tiles

with modified Poisson disk regions

Poisson Sphere Tiles

• Dual tiling– combine corner tiles

with modified Poisson disk regions

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

• Procedural modeling, procedural object distribution, geometry instancing

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

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