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W = g Heightmap . 0 − 1. 1. Tree-density map pixel shader. generated forest will be too sparse. To solve this issue, we have to increase the trees-per-texel ratio; therefore, we need one more placement technique. We assign each tree type a different radius that determines how much space this type of tree owns (in world space). It can be compared to the situation in a real environment when bigger trees take more resources and prevent smaller trees from growing in their neighborhood. Also, we want our trees to be evenly but randomly distributed across a patch corresponding to one density map texel.
If the surface is highly slanted, then the true penumbra size will be much larger than the one we used during pre-integration. For this reason we add a second dimension to our shadow falloff texture, which represents the size of the penumbra in world space. This is similar to the way we pre-integrate lighting against all sphere sizes. In the end, our two-dimensional shadow-lookup integration is a simple convolution: PS (s, w) = ∞ -∞ P (P −1 (s) + x)R(x/w)dx , ∞ R(x/w)dx -∞ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 1. Pre-Integrated Skin Shading 51 where P −1 () is the inverse of our representative falloff, P () is the new, smaller penumbra, R() is the diffusion profile, and s and w are the shadow value and penumbra width in world space, respectively.
If we don’t, we might use all “available” trees for the farthest patches, and there won’t be any close to the camera. In this situation, we may use billboards for the farthest trees, or use a smooth transition into the fog color. The last part of our procedural generation is the Rayleigh-Mie atmospheric scattering simulation [West 08, O’Neil 05]. Our implementation follows the technique described in GPU Gems 2. We first calculate optical depth to use it as a lookup table for further generating Mie and Rayleigh textures.