Symposium MEIS2015 : Mathematical Progress in Expressive Image Synthesis

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two images of daisies, one with a high daisy density and the otherone with low density. The texture quilting optimization process inthis case lasts longer (about 1 minute on a 2 GHz PC). But, once thisis run, generating large amounts of textures at runtime is relativelyfree both in texture memory and time.

4 Three Dimensional Applications

Figure 12: A Wang tiled field filled with sunflowers

The manual effort required to model natural environments and thecomputational cost required to render such scenes can both be veryhigh. Systems that attempt to render realistic looking natural envi-ronments have solved either one problem, or the other, but not both.The systems presented in Weber and Penn [Weber and Penn 1995],Mech and Prusinkiewicz [Mech and Prusinkiewicz 1996], Deussenet al. [Deussen et al. 1998], and Prusinkiewicz et al. [Prusinkiewiczet al. 2001] are capable of creating realistic plants and environments.However, the models created by these systems are of such high detailthat they can not be rendered in real-time. Deussen et al. [Deussenet al. 2002] achieve frame rates from 4-8 fps for complex vegeta-tion, but they need lots of memory to store their plant models. Thismakes their approach not so well suited for larger areas. At the otherend of the spectrum, geometric level-of-detail algorithms [Hoppe1996; Perbert and Cani 2001] have increased the effective geometriccomplexity that can be rendered in real-time, but provide no directsupport for realistic shading.We address both problems with the use of Wang Tiles as con-

tainers for 3D geometry. In our example application of a sunflowerfield, we use tiles to group a set of flowers into a single lightingenvironment. We use eleven variations of a model of a sunflowerand position them stochastically in each of 8 Wang Tiles based onthe Poisson disc distribution discussed earlier. Sunflowers that inter-sect a tile boundary are repeated in all tiles with opposite matchingcolors. A tiling draped over a height field can be rendered with astandard ray tracer as seen in Figure 12.

4.1 Layered Depth Image TilesA lower quality, but real-time rendering can be achieved by WangTiles in which the geometry within each tile is pre-shaded. This ispossible since Wang Tiles are not allowed to be reoriented whenconstructing the tilings. We discuss one possible represention inwhich the geometry is sampled into a hierarchy of 3D textures. The

(a) (b)

(c) (d)

(e)

Figure 13: LOD selection. (a) A typical portion of the sunflowerfield. (b) This view straddles two of the 8 wedges that divide thespace of directions around a tile. (c) Levels of detail false-colored.Red (not seen below us) is highest detail, orange is lowest detail.(d) Layout of directions and resolution levels from viewpoint; e) AFrame from interactive LDI renderer with 20 flowers per tile. Sky isenvironment mapped, mountains are geometry. Frames rendered atbetween 2.7 and 4.0 fps on a P4 1.7 GHz PC.

3D textures are each represented as Layered Depth Images (LDIs)that convert complex geometry into a sparse set of pre-shaded sam-ples [Shade et al. 1998].Real time rendering is enhanced by extending LDIs with a novel

view-dependent multi-resolution formulation of the representation.Since tiles further from the viewpoint require fewer samples, we cre-ate 7 levels of detail of each tile, each with successively fewer sam-ple points. Tiles viewed from a particular direction need includeonly those points of geometry seen from that direction. To addressthis we create 8 variations of each tile specialized for viewing fromeach of 8 compass directions.The complete system consists of an offline and an online compo-

nent. Offline, we create and pre-shade the hierarchical set of tilescomposed of instances of the eleven sunflower models. At runtime,we compute a tiling of the plane that is draped over a terrain and

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タイルと辺のデータ構造

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方に着目してもいいのでは？+

•  イメージとしてはこんな感じ．+

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