Wang Tiles for Image and Texture Generation

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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

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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 (Wangs
  Carpets by Greg Egan)

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Wang Tiles not new to CG

• Jos Stam
• Showed the use of aperiodic tile sets for texture
  assembly

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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

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Wang Tiles
Wang Tiles
Mirrored
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Wang Tiles
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Wang Tiles

• We introduce non-periodic tilings
• set can trivially tile the plane periodically
• simple stochastic algorithm ? non-periodic

aperiodic
non-periodic
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Wang Tiles

• 2 colors 8 tiles
• 3 colors 18 tiles
• C colors ?

2 x C2
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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
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Wang Tiles for Texture Synthesis
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Wang Tiles for Texture Synthesis
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Wang Tiles for Texture Synthesis
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Wang Tiles for Texture Synthesis
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Wang Tiles for Texture Synthesis
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Wang Tiles for Texture Synthesis
Demo
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Wang Tiles for Texture Synthesis
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Wang Tiles the corner problem
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Wang Tiles the corner problem
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Wang Tiles the corner problem
A better solution? Lefebvre and Neyret, I3D 2003
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Wang Tiles for Distributions
Poisson Disc Distributions
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Wang Tiles for Distributions
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Wang Tiles for Distributions
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Wang Tiles for Distributions
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Wang Tiles for Distributions
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Wang Tiles for Distributions
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Wang Tiles for Distributions
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Wang Tiles for Distributions
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Wang Tiles for Distributions
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Wang Tiles for Distributions
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3D Wang Tiles

• Volumetric tiles filled with 3D texture
• In this example, 2.5D
• Layered Depth Image samples

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3D Wang Tiles

• Hierarchical
• View Dependent

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3D Wang Tiles
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Wang Tiles

• Simple stochastic assembly non-periodic
• Texture synthesis
• Distribution synthesis
• The corner problem
• Some 3D applications

Free stuff!
Free stuff!
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Wang Tiles
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Don Mitchell Alex Colburn Reviewers