``Bump'' mapping is a variant of texture mapping where the texture information is used to alter the surface normal. Current techniques to pre-filter textures are all relying on the fact that the texture information can be linearly "factored out" of the shading equation, and therefore can be pre-averaged in some way. This is not the case with bump maps, and those techniques fail to filter them correctly. We propose here a technique to pre-filter bump maps by building a pyramid where each level stores distributions of normal vectors reconstructed from the distribution given by the underlying bump map. The distributions are represented as sums of a small number of Phong-like spreads of normal vectors. The technique, besides allowing an effective and smooth transition between a bump map and a single surface description, gives rise to the concept of a multiple surface, where each point of the single surface is characterized by more than one normal vector. This allows the description of visually complex surfaces by a trivial modification of current local illumination models. When a surface has an underlying microstructure, masking and self-shadowing are important factors in its appearance. Along with the filtering of normals we include the filtering of the masking and self-shadowing information. This is accomplished by computing the limiting angles of visibility and their variance along the two texture axes for the reconstructed distribution of normals. These techniques allow the modeling of any surface whose microstructure we can model geometrically. This includes complex but familiar surfaces such as anisotropic surfaces, many woven cloth, and stochastic surfaces.
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