wang tiles for image and texture generation michael cohen microsoft research jonathan shade...

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