Wang Tiles for image and texture generation

We present a simple stochastic system for non-periodically tiling the plane with a small set of Wang Tiles. The tiles may be filled with. patterns, or geometry that when assembled create a continuous representation. The primary advantage of using Wang Tiles is that once the tiles are filled, large expanses of non-periodic texture (or patterns or geometry) can be created as needed very efficiently at runtime. Wang Tiles are squares in which each edge is assigned a color. A valid tiling requires all shared edges between tiles to have matching colors. We present a new stochastic algorithm to nonperiodically tile the plane with a small set of Wang Tiles at runtime. Furthermore, we present new methods to fill the tiles with 2D texture, 2D Poisson distributions, or 3D geometry to efficiently create at runtime as much non-periodic texture (or distributions, or geometry) as needed. We leverage previous texture synthesis work and adapt it to fill Wang Tiles. We demonstrate how to fill individual tiles with Poisson distributions that maintain their statistical properties when combined. These are used to generate a large arrangement of plants or other objects on a terrain. We show how such environments can be rendered efficiently by pre-lighting the individual Wang Tiles containing the geometry. We also extend the definition of Wang Tiles to include a coding of the tile corners to allow discrete objects to overlap more than one edge. The larger set of tiles provides increased degrees of freedom.