A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Convolution Methods <s> Logic embedded memory is an emerging technology that combines high transfer rates and computing power. Texram implements this technology and a new filtering algorithm to achieve high speed, high quality texture mapping. Integrating arithmetic units and large memory arrays on the same chip and thus exploiting the enormous internal transfer rates provides an elegant solution to the memory access bottleneck of high quality texture mapping. Using this technology, we can not only achieve higher texturing speed at lower system costs, we can also incorporate new functionalities such as detail mapping and footprint assembly to produce higher quality images at real time rendering speeds. Environment and video mapping are also integrated on the Texram, which therefore represents an autonomous and versatile texturing coprocessor. Logic enhanced memories might become the computing paradigm of the future, not just in graphics applications. Technological advances will foster this trend by providing an ever increasing amount of memory capacity and chip space for arithmetic units. As the ultimate solution, we can expect a complete 3D graphics pipeline including all memory systems integrated on a single chip. <s> BIB001 </s> A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Convolution Methods <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> BIB002
The convolution methods  , BIB002 use the additional hypothesis that the NDFf x only depends on n, i.e., that f x ð Þ ¼ fðn; Þ. This allows them to replace the integral over A in  with an integral using the distribution function of the normals n in A. This NDF, noted p A ð Þ, is defined by the ergodicity equation: Reporting (15) below in BIB001 shows that the NDF in a footprint A is a weighted average of the NDFs at each point x, which are Dirac distributions: Then, by using BIB001 in (13), the NDF in a footprint A becomes the convolution of the punctual NDFf with the NDF p A in A: The prefiltering problem is thus decomposed in two subproblems: prefiltering the NDFs p A , and computing the above convolution. In practice, the convolution methods  , BIB002 assume that f x ð Þ ¼ fðn Þ-we have f L ¼ n and f BP ¼ ðn Þ for the Lambert and Blinn-Phong models, respectively. This restricts them to uniform, isotropic BRDFs (while the direct methods support anisotropic BRDFs varying with x). A generic method to reformulate f as a linear function is to introduce the probability distribution of its m parameters, that we call here a parameter distribution function, or PDF, noted p x ðmÞ ¼ ðm À M x Þ: where M is the range of m ( for normals, ½0; 2 for horizon angles, ½0; 1½ for depths, etc.). The reformulated function g, which can be seen as a convolution, is now a linear function of the PDF. These distributions can thus be linearly prefiltered, with p A ðmÞ ¼ def R Aŵ ðxÞp x ðmÞdx, to compute the average of f in a footprint: The only problem is that the PDF is a function: it cannot be stored directly in a map like a scalar or a vector. The solution is to store linear parameters characterizing this distribution, using the same tools as for the direct methods above, which are presented in the next sections.