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Wang Tiles for Image and Texture Generation

Wang Tiles for Image and Texture Generation

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Wang Tiles for Image and Texture Generation

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  1. Wang Tiles for Image and Texture Generation Michael CohenMicrosoft Research Jonathan Shade University of Washington and WildTangent Stefan Hiller University of Constance Oliver DeussenUniversity 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 • We introduce non-periodic tilings • set can trivially tile the plane periodically • simple stochastic algorithm → non-periodic non-periodic aperiodic

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

  9. 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. 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