A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Reflectance and Visibility <s> ``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. <s> BIB001 </s> A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Reflectance and Visibility <s> A method is described for switching smoothly between rendering algorithms as required by the amount of visible surface detail. The result will be more realism with less computation for displaying objects whose surface detail can be described by one or more bump maps. The three rendering algorithms considered are a BRDF, bump mapping, and displacement mapping. The bump mapping has been modified to make it consistent with the other two. For a given viewpoint, one of these algorithms will show a better trade-off between quality, computation time, and aliasing than the other two. The decision as to which algorithm is appropriate is a function of distance, viewing angle, and the frequency of bumps in the bump map. <s> BIB002 </s> A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Reflectance and Visibility <s> Extremely dense spatial sampling is often needed to prevent aliasing when rendering objects with high frequency variations in geometry and reflectance. To accelerate the rendering process, we introduce characteristic point maps (CPMs), a hierarchy of view-independent points, which are chosen to preserve the appearance of the original model across different scales. In preprocessing, randomized matrix column sampling is used to reduce an initial dense sampling to a minimum number of characteristic points with associated weights. In rendering, the reflected radiance is computed using a weighted average of reflectances from characteristic points. Unlike existing techniques, our approach requires no restrictions on the original geometry or reflectance functions. <s> BIB003
Correlations between reflectance and visibility frequently occur, for instance, with rough surfaces whose hills and valleys have different reflectances (e.g., due to weathering). As with colors, the main effect of these correlations is that the average surface reflectance, i.e., the NDF, becomes view and light dependent (many diffuse materials become specular at grazing angles). For instance, if the bumps of the complex surface in Fig. 2 have a diffuse reflectance, while the flat areas are purely specular, then the specular component disappears for grazing view angles (because then the flat areas are masked). Again, these correlation effects only happen when self-masking or self-shadowing occur, i.e., for "grazing" view or light directions, but these "grazing" angles can be quite large, e.g., with^-shaped bumps. The prefiltering method proposed by Fournier BIB001 ,  can handle these correlation effects with a low precision, at the price of huge memory requirements (see Section 4). Becker and Max BIB002 get view-dependent normal distributions by using a precomputed redistribution function, transforming the normal distribution for a vertical view into the distribution of normals in some other view direction (taking self-masking into account). Wu et al. BIB003 can also handle these correlations in offline renderings, at the price of huge memory requirements.