A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> TABLE 1 Important Symbols Used in This Paper <s> Different methods for prefiltered environment maps have been proposed, each of which has different advantages and disadvantages. We present a general notation for prefiltered environment maps, which will be used to classify and compare the existing methods. Based on that knowledge we develop three new algorithms: 1. A fast hierarchical prefiltering method that can be utilized for all previously proposed prefiltered environment maps. 2. A technique for hardware-accelerated prefiltering of environment maps that achieves interactive rates even on low-end workstations. 3. Anisotropic environment maps using the Banks model. <s> BIB001 </s> A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> TABLE 1 Important Symbols Used in This Paper <s> A framework for discussing the motion blur image generation process is formulated. Previous work is studied in the context of this framework. Due to the implicit assumptions on low temporal frequencies in most motion blur algorithms, issues involved in large screen space movements and fast illumination changes in time have not been adequately addressed so far. A new approach that does not make these assumptions is introduced to solve the spatial-temporal geometric and shading aliasing problems separately. Based on newly developed adaptive algorithms in the spatial-temporal domain, an implementation of the new approach is developed to efficiently deliver high-quality motion blurred images in general computer graphics production environments. <s> BIB002
For a pinhole camera with an instantaneous exposure (see Fig. 1 ), the intensity I of a given screen pixel is a convolution of the incident radiance Lðo; ! !Þ with a low-pass filter w over its angular support Á! ! [1], BIB002 . This support corresponds to a single pixel for a box filter, or to several pixels for more efficient antialiasing filters: In order to get an equation in terms of surface maps, the first step is to rewrite I as an integral over the footprint S, that we define as the intersection of the surface with the Á! ! cone (we assume here, without loss of generality, that S fully covers this cone-if not, split it in two parts). In the case of self-masking, by definition of S, a view ray intersects S several times, but only the nearest intersection point contributes to I. We exclude the others with the function V p ðvÞ equal to 1 if p is visible from direction v, and 0 otherwise. The change of variables from ! ! to p yields the Jacobian d! ! ¼ nv r 2 dp where r ¼ kp À ok is the distance to the viewer. Thus, (1) becomes: wðpÞLðp; ! !ÞV p ðvÞnv dp r 2 with Z S wðpÞV p ðvÞnv dp Since the solid angle of a pixel is small, r and v can generally be considered as constants in these integrals. Then, r can be eliminated by moving it outside the integrals. We can now use the local illumination equation Lðp; ! !Þ ¼ R Lðp; lÞRðl; vÞ nl dl to express the outgoing radiance Lðp; ! !Þ as a function of the incident radiance Lðp; lÞ and of the surface reflectance R: . We write the incident radiance as the product EðlÞV p ðlÞU p ðlÞ. EðlÞ is the distant environment radiance coming from direction l, supposed quasiconstant over S, or uncorrelated with local features. V p ðlÞ is a self-shadowing term equal to 1 where p is visible from direction l, and 0 otherwise. Finally, U p ðlÞ is a shadowing term equal to 1 where p is unoccluded by other objects in direction l, and 0 otherwise (soft shadows result from an integral over l). Note that V p ðlÞ excludes self-interreflections. The local illumination equation also excludes subsurface scattering. These two effects are discussed in Section 8. . We write the surface reflectance R as the product k p p ðl; vÞ, with k p a color and p a normalized BRDF. In general, a sum of such terms is used, for instance to decompose the reflectance into diffuse and specular components, but this does not impact our discussion here. Keep also in mind that both terms implicitly depend on the wavelength . Putting all this together in (2), we get The last step to get an equation in terms of surface maps is to rewrite I as an integral over the coarse mesh, on which the maps are defined. This change of variables yields the Jacobian dx ¼ ng dp, which gives This is the "prefiltering equation" that we will consider in the rest of this paper. It gives I as a function of the texel values of the color, normal, BRDF, visibility, shadow, and environment maps k x , n x , x , V x , U x , and EðlÞ-with V x often represented with a horizon map (see Section 4) . Note that the parallax offset is implicit in this equation. Indeed, A is not the footprint of the pixel on the coarse mesh, but is related to it via a view-dependent offset (see Section 8.3 and Fig. 7) . The goal of the prefiltering methods is to avoid a numerical evaluation of (4) during rendering, over all the texels x. Instead, the goal is to estimate I directly from prefiltered versions of the surface maps. The problem is that (4) gives I as a nonlinear and nonseparable function of the map texels. Thus, a simple separate linear prefiltering of each map is not sufficient. Note that I includes a spatial integral over A, and a directional integral over l. Thus, both the spatial maps and the environment map EðlÞ should be prefiltered for efficient rendering. In this survey, we restrict ourselves to the prefiltering of the spatial maps, i.e., to the computation of A . A survey of the environment map prefiltering methods can be found in BIB001 .