A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Frequency Clamping <s> An object space method is given for interpolating between sampled and locally averaged signals, resulting in an antialiasing filter which provides a continuous transition from a sampled signal to its selectively dampened local averages. This method is applied to the three standard Euclidean dimensions and time, resulting in spatial and frame to frame coherence. The theory allows filtering of a variety of functions, including continuous and discrete representations of planar texture. <s> BIB001 </s> A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Frequency Clamping <s> This chapter describes various fractal procedural textures serving as models of natural phenomena. They are divided into four elements of the ancients: air, fire, water, and earth. The chapter describes the most common, quick, and easy cloud texture and discusses a few two-dimensional models that are significant aesthetically. One of the simplest and most often used fractal textures is a simple representation of thin, wispy clouds in a blue sky. The fractals can be constructed from any basis function; the basis function that is most often chosen is the Perlin noise function. The composition of noise functions to provide distortion is useful in another aspect of modeling clouds: emulating the streaming of clouds that are stretched by winds. Modeling of the clouds is presented in the chapter on different scales. One of these scales is a distorted large-scale distribution comprising a weighting function, which is applied to smaller-scale cloud features. The undistorted small features correspond to the phenomenon of viscosity, which damps turbulence at small scales. This may serve as the first step in the direction of multifractal models, as the fractal behavior is different at different scales, and therefore may require more than one value or measure to characterize the fractal behavior. <s> BIB002 </s> A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Frequency Clamping <s> From the Publisher: ::: This the classic text/tutorial/reference introducing and defining the procedural approach to texturing and modeling and computer graphics in general. This Second Edition is completely revised and updated to the current computer graphics marketplace. It contains a toolbox of procedures upon which programmers can build a library of textures and objects, and includes extensive explanations of how these functions work and how to design new functions. New chapters include: Cellular Texture Generation, Volumetric Cloud Modeling with Implicit Functions, Interacting with Virtual Actors, and Rendering with Adaptive Level of Detail. <s> BIB003 </s> A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Frequency Clamping <s> Procedural solid texturing was introduced fourteen years ago, but has yet to find its way into consumer level graphics hardware for real-time operation. To this end, a new model is introduced that yields a parameterized function capable of synthesizing the most common procedural solid textures, specifically wood, marble, clouds and fire. This model is simple enough to be implemented in hardware, and can be realized in VLSI with as little as 100,000 gates. The new model also yields a new method for antialiasing synthesized textures. An expression for the necessary box filter width is derived as a function of the texturing parameters, the texture coordinates and the rasterization variables. Given this filter width, a technique for efficiently box filtering the synthesized texture by either mip mapping the color table or using a summed area color table are presented. Examples of the antialiased results are shown. <s> BIB004 </s> A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Frequency Clamping <s> In this paper we investigate the effects of function composition in the form g(f(x)) = h(x) by means of a spectral analysis of h. We decompose the spectral description of h(x) into a scalar product of the spectral description of g(x) and a term that solely depends on f(x) and that is independent of g(x). We then use the method of stationary phase to derive the essential maximum frequency of g(f(x)) bounding the main portion of the energy of its spectrum. This limit is the product of the maximum frequency of g(x) and the maximum derivative of f(x). This leads to a proper sampling of the composition h of the two functions g and f. We apply our theoretical results to a fundamental open problem in volume rendering - the proper sampling of the rendering integral after the application of a transfer function. In particular, we demonstrate how the sampling criterion can be incorporated in adaptive ray integration, visualization with multi-dimensional transfer functions, and pre-integrated volume rendering <s> BIB005 </s> A Survey of Nonlinear Prefiltering Methods for Efficient and Accurate Surface Shading <s> Frequency Clamping <s> Procedural noise functions are widely used in Computer Graphics, from off-line rendering in movie production to interactive video games. The ability to add complex and intricate details at low memory and authoring cost is one of its main attractions. This state-of-the-art report is motivated by the inherent importance of noise in graphics, the widespread use of noise in industry, and the fact that many recent research developments justify the need for an up-to date survey. Our goal is to provide both a valuable entry point into the field of procedural noise functions, as well as a comprehensive view of the field to the informed reader. In this report, we cover procedural noise functions in all their aspects. We outline recent advances in research on this topic, discussing and comparing recent and well established methods. We first formally define procedural noise functions based on stochastic processes and then classify and review existing procedural noise functions. We discuss how procedural noise functions are used for modeling and how they are applied on surfaces. We then introduce analysis tools and apply them to evaluate and compare the major approaches to noise generation. We finally identify several directions for future work. <s> BIB006
The third method is specific to the case where the color function f is an explicit sum of terms with a band limited spectrum. This is the case, for instance, when summing Fig. 6 . Procedural map filtering. The color of the complex surface S (bottom) can be described procedurally with the analytical function fðxÞ ¼ b2xc Ä 2 (middle), using f brown þ ð1 À fÞ green. The average color in the footprint A can be computed efficiently by using the indefinite integral F ðxÞ ¼ sinusoids, or with Perlin noise  (which is not strictly band limited, but other noise functions are BIB006 ). In this case, an approximate filtered function can be obtained by clamping the sum to the terms whose frequency is less than the low-pass filter cutoff frequency, and by progressively attenuating the others to avoid popping  , BIB003 , BIB001 . This method works well with a basic sum of terms fðxÞ ¼ P i b i ðxÞ. In many cases however, f is used in a color lookup table c to get the final pixel color. When c is not linear, the average final color R Aŵ ðxÞcðfðxÞÞdx is not equal to cð R Aŵ ðxÞfðxÞdxÞ. To solve this problem, Hart et al. BIB004 use the following approximation: where f 0 and f 1 are two samples of f on the boundary of A (this corresponds to the hypothesis that f varies linearly between f 0 and f 1 in A). The right-hand side is then evaluated with a prefiltered lookup table, either using MIPmaps or SATs. Worley BIB003 proposes an extension to this method using more samples of f to get more accurate results. Bergner et al. BIB005 use a spectral analysis to tackle this problem. More generally, it is frequent to compose f, or each of its term b i , with arbitrary functions g or g i . For instance, Musgrave BIB002 uses fðgðxÞÞ to distort a noise pattern f. Perlin and Hoffert  use gðf ðxÞÞ to distort a density field g, and fðxÞ ¼ P i jb i ðxÞj to get a turbulence pattern. These extensions often add high frequencies to f, and the frequency clamping method must be adapted to account for this. However, we are not aware of any generic algorithm for doing that.