## Further Reading

The cone-tracing method of Amanatides (1984) was one of the first techniques for automatically estimating filter footprints for ray tracing. The beam-tracing algorithm of Heckbert and Hanrahan (1984) was another early extension of ray tracing to incorporate an area associated with each image sample rather than just an infinitesimal ray. The pencil-tracing method of Shinya, Takahashi, and Naito (1987) is another approach to this problem. Other related work on the topic of associating areas or footprints with rays includes Mitchell and Hanrahan’s paper (1992) on rendering caustics and Turkowski’s technical report (1993).

Collins (1994) estimated the ray footprint by keeping a tree of all rays
traced from a given camera ray, examining corresponding rays at the same
level and position. The ray differentials used in
`pbrt` are based on Igehy’s (1999) formulation, which was extended by
Suykens and Willems (2001) to handle glossy reflection in addition to
perfect specular reflection.
Worley’s chapter in *Texturing and Modeling* (Ebert et al. 2003) on
computing differentials for filter
regions presents an approach similar to ours.
See Elek et al. (2014) for an extension of ray differentials to
include wavelength, which can improve results with full-spectral rendering.

Two-dimensional texture mapping with images was first introduced to graphics by Blinn and Newell (1976). Ever since Crow (1977) identified aliasing as the source of many errors in images in graphics, quite a bit of work has been done to find efficient and effective ways of antialiasing image maps. Dungan, Stenger, and Sutty (1978) were the first to suggest creating a pyramid of prefiltered texture images; they used the nearest texture sample at the appropriate level when looking up texture values, using supersampling in screen space to antialias the result. Feibush, Levoy, and Cook (1980) investigated a spatially varying filter function, rather than a simple box filter. (Blinn and Newell were aware of Crow’s results and used a box filter for their textures.)

Williams (1983) used a MIP map image pyramid for texture filtering with trilinear interpolation. Shortly thereafter, Crow (1984) introduced summed area tables, which make it possible to efficiently filter over axis-aligned rectangular regions of texture space. Summed area tables handle anisotropy better than Williams’s method, although only for primarily axis-aligned filter regions. Heckbert (1986) wrote a good general survey of texture mapping algorithms through the mid-1980s.

Greene and Heckbert (1986) originally developed the elliptically weighted average technique, and Heckbert’s master’s thesis (1989) put the method on a solid theoretical footing. Fournier and Fiume (1988) developed an even higher quality texture filtering method that focuses on using a bounded amount of computation per lookup. Nonetheless, their method appears to be less efficient than EWA overall. Lansdale’s master’s thesis (1991) has an extensive description of EWA and Fournier and Fiume’s method, including implementation details.

More recently, a number of researchers have investigated generalizing Williams’s original method using a series of trilinear MIP map samples in an effort to increase quality without having to pay the price for the general EWA algorithm. By taking multiple samples from the MIP map, anisotropy is handled well while preserving the computational efficiency. Examples include Barkans’s (1997) description of texture filtering in the Talisman architecture, McCormack et al.’s (1999) Feline method, and Cant and Shrubsole’s (2000) technique. Manson and Schaefer (2013, 2014) have recently shown how to accurately approximate a variety of filter functions with a fixed small number of bilinearly interpolated sample values. This approach is particularly useful on GPUs, where hardware-accelerated bilinear interpolation is available.

Gamma correction has a long history in computer graphics. Poynton (2002a, 2002b) has written comprehensive FAQs on issues related to color representation and gamma correction. Most modern displays are based on the sRGB color space, which has a gamma of roughly 2.2 (International Electrotechnical Commission (IEC) 1999). See Gritz and d’Eon (2008) for a detailed discussion of the implications of gamma correction for rendering and how to correctly account for it in rendering systems.

Smith’s (2002) Web site and document
on audio resampling gives a good overview of
resampling signals in one dimension. Heckbert’s (1989)
`zoom` source code is the canonical reference for image
resampling. His implementation carefully avoids
feedback without using auxiliary storage, unlike ours in this chapter,
which allocates additional temporary buffer space to do so.

Three-dimensional solid texturing was originally developed by Gardner
(1984, 1985), Perlin
(1985a), and Peachey (1985). Norton, Rockwood, and Skolmoski (1982) developed the *clamping* method that is
widely used for antialiasing textures based on solid texturing. The
general idea of procedural texturing was introduced by Cook
(1984), Perlin (1985a), and Peachey
(1985).

Peachey’s chapter in *Texturing and Modeling* (Ebert et al. 2003) has
a thorough summary of approaches to noise functions. After Perlin’s
original noise function, both Lewis (1989) and van Wijk (1991) developed
alternatives that made different time/quality trade-offs. Worley (1996)
has developed a quite different noise function for procedural texturing
that is well suited for cellular and organic patterns. Perlin (2002)
revised his noise function to correct a number of subtle shortcomings.

Noise functions have received additional attention from the research community in recent years. (Lagae et al. (2010) have a good survey of work up to that year.) Building on Lewis’s observation that individual bands of Perlin’s noise function actually have frequency content over a fairly wide range (Lewis 1989), Cook and DeRose (2005) also identified the problem that 2D slices through 3D noise functions aren’t in general band limited, even if the original 3D noise function is. They proposed a new noise function that addresses both of these issues. Goldberg et al. (2008) developed a noise function that makes efficient anisotropic filtering possible, leading to higher quality results than just applying the clamping approach for antialiasing. Their method is also well suited to programmable graphics hardware. Kensler et al. (2008) suggested a number of improvements to Perlin’s revised noise function.

Lagae et al. (2009) have developed a noise function that has good frequency control and can be mapped well to surfaces even without a surface parameterization. Lagae and Drettakis (2011) showed how to compute high quality anisotropically filtered values of this noise function. More recently, Galerne et al. (2012) showed how to automatically determine parameters to this noise function so that the result matches example images. Further work on this topic was done by Du et al. (2013) and Gilet et al. (2014).

The first languages and systems that supported the idea of user-supplied
procedural shaders were developed by Cook (1984) and Perlin
(1985a). (The texture composition model in this chapter is
similar to Cook’s shade trees.) The RenderMan shading language, described
in a paper by Hanrahan and Lawson (1990), remains the
classic shading language in graphics, though a more modern shading language
is available in Open Shading Language (OSL) (Gritz et al. 2010), which is open source and
increasingly used for production rendering. It follows the model of the
shader returning a representation of the material rather than a final color
value, like the approach introduced in Chapter 9. See also Karrenberg et al. (2010), who introduced the *AnySL* shading language, which was
designed for both high performance as well as portability across multiple
rendering systems (including `pbrt`).

See Ebert et al. (2003) and Apodaca and Gritz (2000) for techniques for writing procedural shaders; both of those have excellent discussions of issues related to antialiasing in procedural shaders.

The “Further Reading” section in Chapter 9 described approaches for anti-aliasing bump maps; a number of researchers have looked at the closely related issue of antialiasing surface reflection functions. Van Horn and Turk (2008) developed an approach to automatically generate MIP maps of reflection functions that represent the characteristics of shaders over finite areas in order to antialias them. Bruneton and Neyret (2012) surveyed the state of the art in this area, and Jarabo et al. (2014b) also considered perceptual issues related to filtering inputs to these functions. See also Heitz et al. (2014) for recent work on this topic.

Many creative methods for computing texture on surfaces have been
developed. A sampling of our favorites includes reaction diffusion, which
simulates growth processes based on a model of chemical interactions over
surfaces and was simultaneously introduced by Turk (1991) and Witkin and
Kass (1991); Sims’s (1991)
genetic algorithm-based approach, which
finds programs that generate interesting textures through random
mutations from which users select their favorites;
Fleischer et al.’s (1995) cellular texturing algorithms that generate
geometrically accurate scales and spike features on
surfaces; and Dorsey et al.’s (1996) flow simulations that
model the effect of weathering on buildings and encode the results in
image maps that stored the relative wetness, dirtiness, and so on, at points
on the surfaces of structures. Porumbescu et al. (2005)
developed *shell maps*, which make it possible to map geometric
objects onto a surface in the manner of texture mapping.

A variety of *texture synthesis* algorithms have been developed in the
past decade; these approaches take an example texture image and then
synthesize larger texture maps that appear similar to the original texture
while not being exactly the same. The survey article by Wei et al. (2009)
describes work in this area through 2009 as well as the main approaches
that have been developed so far. For more recent work in this area, see Kim et al. (2012), who developed an effective approach based on finding
symmetries in textures, and Lefebvre et al. (2010), who attacked the
specialized (but useful) problem of synthesizing textures for building
facades.

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