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 Wang Tiles Mirrored

Wang Tiles

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

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

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

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

Wang Tiles for Distributions 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

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

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!

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