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Wang Tiles for Image and Texture Generation
Michael Cohen Microsoft Research
Jonathan Shade University of Washington and WildTangent
Stefan Hiller University of Constance
Oliver Deussen University of Constance
Wang Tiles
• Rules – Tilings are assembled by matching like edges– Tiles cannot be rotated
• In 1961 Wang asserted that no aperiodic set of tiles exists, i.e., a set that– COULD tile the plane– could NOT tile it periodically
• He was wrong, but still got to keep his name on them
• Aperiodic sets can be used to – Simulate Turing machines– Act as life forms in science fiction (“Wang’s
Carpets” by Greg Egan)
Wang Tiles – not new to CG
• Jos Stam– Showed the use of aperiodic tile sets for
texture assembly
Wang Tiles – not new to CG
• Jos Stam– Showed the use of aperiodic tile sets for
texture assembly
• Neyret and Cani– Used triangular tiles with coded edges to
assure continuous texture
Wang Tiles
Mirrored Wang Tiles
Wang Tiles
Wang Tiles
• We introduce non-periodic tilings– set can trivially tile the plane periodically– simple stochastic algorithm → non-periodic
aperiodic non-periodic
• 2 colors = 8 tiles
• 3 colors = 18 tiles• C colors ?
Wang Tiles
2 x C2
Why Wang Tiles?
• Once you fill a set of Wang Tiles with stuff
• You can create as much of that stuff as you want for almost free.
Free stuff!Free stuff!Demo
Wang Tiles for Texture Synthesis
Wang Tiles for Texture Synthesis
Wang Tiles for Texture Synthesis
Wang Tiles for Texture Synthesis
Wang Tiles for Texture Synthesis
Wang Tiles for Texture Synthesis
Demo
Wang Tiles for Texture Synthesis
Wang Tiles: the corner problem
?
Wang Tiles: the corner problem
Wang Tiles: the corner problem
A better solution? Lefebvre and Neyret, I3D 2003
Poisson Disc Distributions
Wang Tiles for Distributions
Wang Tiles for Distributions
Wang Tiles for Distributions
Wang Tiles for Distributions
Wang Tiles for Distributions
Wang Tiles for Distributions
Wang Tiles for Distributions
Wang Tiles for Distributions
Wang Tiles for Distributions
Wang Tiles for Distributions
3D Wang Tiles
• Volumetric tiles filled with 3D “texture”
• In this example, 2.5D
• Layered Depth Image samples
3D Wang Tiles
• Hierarchical
• View Dependent
3D Wang Tiles
Wang Tiles
• Simple stochastic assembly: non-periodic
• Texture synthesis
• Distribution synthesis
• The corner problem
• Some 3D applications
Free stuff!Free stuff!
Wang Tiles
T H A N KY O U
Don Mitchell
Alex Colburn
Reviewers
