A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Multiple Lobes Algorithms <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> Multiple Lobes Algorithms <s> Physically-based reflectance models typically represent light scattering as a function of surface geometry at the pixel level. With changes in viewing resolution, the geometry imaged within a pixel can undergo significant variations that result in changing reflectance characteristics. To address these transformations, we present a multiresolution reflectance framework based on microfacet normal distributions within a pixel over different scales. Since these distributions must be efficiently determined with respect to resolution, they are recorded at multiple resolution levels in mipmaps. The main contribution of this work is a real-time mipmap filtering technique for these distribution-based parameters that not only provides smooth reflectance transitions in scale, but also minimizes aliasing. With this multiresolution reflectance technique, our system can rapidly and accurately incorporate fine reflectance detail that is customarily disregarded in multiresolution rendering methods. <s> BIB002 </s> A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Multiple Lobes Algorithms <s> Filtering is critical for representing detail, such as color textures or normal maps, across a variety of scales. While MIP-mapping texture maps is commonplace, accurate normal map filtering remains a challenging problem because of nonlinearities in shading---we cannot simply average nearby surface normals. In this paper, we show analytically that normal map filtering can be formalized as a spherical convolution of the normal distribution function (NDF) and the BRDF, for a large class of common BRDFs such as Lambertian, microfacet and factored measurements. This theoretical result explains many previous filtering techniques as special cases, and leads to a generalization to a broader class of measured and analytic BRDFs. Our practical algorithms leverage a significant body of work that has studied lighting-BRDF convolution. We show how spherical harmonics can be used to filter the NDF for Lambertian and low-frequency specular BRDFs, while spherical von Mises-Fisher distributions can be used for high-frequency materials. <s> BIB003 </s> A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Multiple Lobes Algorithms <s> The reflectance of a surface depends upon the resolution at which it is imaged. In this work, we propose to represent this resolution-dependent reflectance as a mixture of multiple conventional reflectance models and present a framework for efficiently rendering the reflectance effects of such mixture models over different resolutions. To rapidly determine reflectance at runtime with respect to resolution, we record the mixture model parameters at multiple resolution levels in mipmaps and propose a technique to minimize aliasing in the filtering of these mipmaps. This framework can be applied to several widely used parametric reflectance models and can be implemented in graphics hardware for real-time processing, using a presented hardware-accelerated technique for nonlinear filtering of mixture model parameters. In addition, shadowing and masking effects can be included into this framework to increase the realism of rendering. With this mixture model filtering and rendering framework, our system can efficiently render the fine reflectance detail that is customarily disregarded in conventional rendering methods. <s> BIB004
Fournier BIB001 ,  represents the NDFf A with a sum of Blinn-Phong  lobes of the form: He uses up to seven lobes per texel, which gives 28 parameters per texel. Each level of the MIP-map pyramid is not computed from the previous level, but directly from the base level, using a nonlinear least-squares method to find the lobes that best fit the base level NDFs. The a k , k , and k parameters cannot be linearly interpolated. Thus, for rendering, Fournier interpolates the lighting due to the neighboring texels. This means evaluating (20) with 28 terms, or 56 terms with trilinear filtering. This representation is quite accurate but not very adapted for real-time rendering. Tan et al. BIB002 , BIB004 extend this approach. Instead of Blinn-Phong lobes, they use a mixture of isotropic Gaussian lobes to represent the NDF: They also improve the nonlinear optimization phase to ensure that the lobe directions k and 0 k in any two neighboring texels are as close as possible, for each index k. Thanks to this "alignment," they can use hardware bilinear or trilinear filtering to interpolate the means k and second moments 2 of corresponding lobes, before using  to compute the lighting. Tan et al. use four lobes per texel, which give 12 parameters per texel. A drawback of this method is that it is not always possible to align the lobes in neighboring texels, for instance, at sharp edges. Han et al. BIB003 use a convolution method (see Section 3.1.2). They model the macroscopic NDF p A with its decomposition into spherical harmonics. This gives p lm coefficients that can be linearly MIP-mapped and interpolated. Then, the spherical harmonics coefficients of the NDFf A are simply the spherical harmonics coefficients of p A , times the zonal harmonics coefficients f l of the NDFf x . This gives Han et al. use up to 64p lm coefficients per texel. For a BlinnPhong exponent , about 4ð þ ffiffiffi ffi p Þ coefficients are needed. This limits this method to cases where the NDFs and the NDFs are of low frequency (i.e., no specular reflections, and NDF without narrow lobes). In the other cases, Han et al. BIB003 model the macroscopic NDF p A with a mixture of von Mises-Fisher (vMF) lobes: Like Tan et al. BIB002 , they propose a nonlinear optimization method to find the lobes corresponding to an NDF, and to align corresponding lobes in neighboring texels (which is not always possible, as said above). They can then interpolate their coefficients ( k and k do not interpolate linearly but a k and a k ðcoth k À À1 k Þ k do). The result is