1. 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

2. 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)

3. Wang Tiles – not new to CG
Jos Stam
Showed the use of aperiodic tile sets for texture assembly

4. 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

5. Wang Tiles
Wang Tiles
Mirrored

6. Wang Tiles

7. Wang Tiles aperiodic non-periodic We introduce non-periodic tilings
set can trivially tile the plane periodically
simple stochastic algorithm → non-periodic
aperiodic
non-periodic

8. Wang Tiles
2 colors = 8 tiles
3 colors = 18 tiles
C colors ?
2 x C2

9. Free stuff! Free stuff! 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

10. Wang Tiles for Texture Synthesis

11. Wang Tiles for Texture Synthesis

12. Wang Tiles for Texture Synthesis

13. Wang Tiles for Texture Synthesis

14. Wang Tiles for Texture Synthesis

15. Wang Tiles for Texture Synthesis
Demo

16. Wang Tiles for Texture Synthesis

17. Wang Tiles: the corner problem?

18. Wang Tiles: the corner problem

19. Wang Tiles: the corner problem
A better solution? Lefebvre and Neyret, I3D 2003

20. Wang Tiles for Distributions
Poisson Disc Distributions

21. Wang Tiles for Distributions

22. Wang Tiles for Distributions

23. Wang Tiles for Distributions

24. Wang Tiles for Distributions

25. Wang Tiles for Distributions

26. Wang Tiles for Distributions

27. Wang Tiles for Distributions

28. Wang Tiles for Distributions

29. Wang Tiles for Distributions

30. 3D Wang Tiles Volumetric tiles filled with 3D “texture”
In this example, 2.5D
Layered Depth Image samples

31. 3D Wang Tiles
Hierarchical
View Dependent

32. 3D Wang Tiles

33. Free stuff! Free stuff! Wang Tiles
Simple stochastic assembly: non-periodic
Texture synthesis
Distribution synthesis
The corner problem
Some 3D applications
Free stuff!
Free stuff!

34. Wang Tiles T H A N K Y O U
Don Mitchell
Alex Colburn
Reviewers