#### Ray Footprints

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 et al. (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. Belcour
et al. (2017) applied Fourier analysis to the light
transport equation in order to accurately and efficiently track ray footprints
after scattering.

Twelve floating-point values are required to store ray differentials, and
Belcour et al.’s approach has similar storage requirements. This poses no
challenge in a CPU ray tracer that only operates on one or a few rays at a
time, but can add up to a considerable amount of storage (and consequently,
bandwidth consumption) on the GPU. To address this issue,
Akenine-Möller et al. (2019) developed a number
of more space-efficient alternatives and showed their effectiveness for
antialiasing that was further improved in subsequent work
(Akenine-Möller et al. 2021; Boksansky et al. 2021). The approach we have
implemented in `CameraBase::Approximate_dp_dxy()` was described by Li
(2018).

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 spectral rendering.

#### Image Texture Maps

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, much 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 survey of early texture mapping algorithms.

Greene and Heckbert (1986) originally developed the elliptically weighted average technique, and Heckbert’s master’s thesis (1989b) 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.

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 shown how to accurately approximate a variety of filter functions with a fixed small number of bilinearly interpolated sample values. An algorithm to convert an arbitrary filter into a set of bilinear lookups over multiple passes subject to a specified performance target was given by Schuster et al. (2020). These sorts of approaches are particularly useful on GPUs, where hardware-accelerated bilinear interpolation is available.

For scenes with many image textures where reading them all into memory
simultaneously has a prohibitive memory cost, an effective approach can be
to allocate a fixed amount of memory for image maps (a *texture
cache*), load textures into that memory on demand, and discard the image
maps that have not been accessed recently when the memory fills up
(Peachey 1990). To enable good performance with small texture caches,
image maps should be stored in a *tiled* format that makes it possible
to load in small square regions of the texture independently of each other.
Tiling techniques like these are used in graphics hardware to improve the
performance of their texture memory caches (Hakura and Gupta 1997; Igehy
et al. 1998, 1999). High-performance
texture caching with parallel execution can be challenging because the cache
contents may be frequently updated; it is desirable to minimize mutual
exclusion in the cache implementation so that threads do not stall while
others are updating the cache. For an effective approach to this problem,
see Pharr (2017), who applied the *read-copy update*
technique (McKenney and Slingwine 1998) to accomplish this.

Smith’s (2002) website and document on audio
resampling gives a good overview of resampling signals in one
dimension. Heckbert’s (1989a) `zoom` source code is
the canonical reference for image resampling. His implementation carefully
avoids feedback without using auxiliary storage.

A variety of *texture synthesis* algorithms have been developed that
take an example texture image and then synthesize larger texture images that
appear similar to the original texture while not being exactly the same.
Survey articles by Wei et al. (2009) and Barnes and Zhang
(2017) summarize work in this area. Convolutional neural
networks have been applied to this task (Gatys et al. 2015;
Sendik and Cohen-Or 2017), giving impressive results, and Frühstück
et al. (2019) have showed the effectiveness of
generative adversarial networks for this problem.

#### Solid Texturing and Noise Functions

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, where texture is generated via
computation rather than via looking up values from images, was introduced by Cook
(1984), Perlin (1985a), and Peachey
(1985).

*Noise functions*, which randomly vary while still having limited
frequency content, have been a key ingredient for many procedural texturing
techniques. Perlin (1985a) introduced the first such noise
function, and later revised it to correct a number of subtle shortcomings
(Perlin 2002). (See also Kensler et al. (2008) for
further improvements.) Many more noise functions have been developed; see
Lagae et al. (2010) for a survey of work up to that year.
Tricard et al. (2019) recently introduced a noise
function (“phasor noise”) that can be filtered anisotropically and allows
control of the orientation, frequency, and contrast of the noise function.
Their paper also includes citations to other recent work on this topic.

In recent years, the *Shadertoy* website,
shadertoy.com, has become a hub of creative application of
procedural modeling and texturing, all of it running interactively in web
browsers. *Shadertoy* was developed by Quilez and Jeremias
(2021).

#### Shading Languages

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 `pbrt`’s model of the shader returning a
representation of the material rather than a final color value. See also Karrenberg
et al. (2010), who introduced the *AnySL* shading
language, which was designed for 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.

#### Normal Mapping, Bump Mapping, and Shading Normals

Blinn (1978) invented the bump-mapping technique. Kajiya
(1985) generalized the idea of bump mapping the normal to
*frame mapping*, which also perturbs the surface’s primary tangent
vector and is useful for controlling the appearance of anisotropic
reflection models. Normal mapping was introduced by Cohen et
al. (1998).

Mikkelsen’s thesis (2008) carefully investigates a number of the assumptions underlying bump mapping and normal mapping, proposes generalizations, and addresses a number of subtleties with respect to its application to real-time rendering.

One visual shortcoming of normal and bump mapping is that those techniques do not naturally account
for self-shadowing, where bumps cast shadows on the surface and prevent
light from reaching nearby points. These shadows can have a significant
impact on the appearance of rough surfaces. Max (1988)
developed the *horizon mapping* technique, which efficiently accounts
for this effect through precomputed information about each bump map.
More recently, Conty Estevez et al. and Chiang et al. have
introduced techniques based on microfacet shadowing functions to improve
the visual fidelity of bump-mapped surfaces at shadow terminators
(Conty Estevez et al. 2019, Chiang et al. 2019).

Another challenging issue is that antialiasing bump and normal maps that have higher-frequency detail than can be represented in the image is quite difficult. In particular, it is not enough to remove high-frequency detail from the underlying function, but in general the BSDF needs to be modified to account for this detail. Fournier (1992) applied normal distribution functions to this problem, where the surface normal was generalized to represent a distribution of normal directions. Becker and Max (1993) developed algorithms for blending between bump maps and BRDFs that represented higher-frequency details. Schilling (1997, 2001) investigated this issue particularly for application to graphics hardware.

Effective approaches to filtering bump maps were developed by Han et al. (2007) and Olano and Baker (2010). Both Dupuy et al. (2013) and Hery et al. (2014) developed techniques that convert displacements into anisotropic distributions of Beckmann microfacets. Further improvements to these approaches were introduced by Kaplanyan et al. (2016), Tokuyoshi and Kaplanyan (2019), and Wu et al. (2019).

A number of researchers have looked at the issue of antialiasing surface reflection functions. Early work in this area was done by Amanatides, who developed an algorithm to detect specular aliasing for a specific BRDF model (Amanatides 1992). 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 further work on this topic.

#### Displacement Mapping

An alternative to bump mapping is displacement mapping, where the bump function is used to actually modify the surface geometry, rather than just perturbing the normal (Cook 1984; Cook et al. 1987). Advantages of displacement mapping include geometric detail on object silhouettes and the possibility of accounting for self-shadowing. Patterson and collaborators described an innovative algorithm for displacement mapping with ray tracing where the geometry is unperturbed, but the ray’s direction is modified such that the intersections that are found are the same as would be found with the displaced geometry (Patterson et al. 1991; Logie and Patterson 1994). Heidrich and Seidel (1998) developed a technique for computing direct intersections with procedurally defined displacement functions.

One approach for displacement mapping has been to use an implicit function
to define the displaced surface and to then take steps along rays until
a zero crossing with the implicit function is found—this point is an
intersection. This approach was first introduced by Hart
(1996); see Donnelly (2005) for information
about using this approach for displacement mapping on the GPU. (This
approach was more recently popularized by Quilez (2015) on the
*Shadertoy* website.)

Another option is to finely tessellate the scene geometry and displace its vertices to define high-resolution meshes. Pharr and Hanrahan (1996) described an approach to this problem based on geometry caching, and Wang et al. (2000) described an adaptive tessellation algorithm that reduces memory requirements. Smits, Shirley, and Stark (2000) lazily tessellate individual triangles, saving a substantial amount of memory.

Measuring fine-scale surface geometry of real surfaces to acquire bump or displacement maps can be challenging. Johnson et al. (2011) developed a novel handheld system that can measure detail down to a few microns, which more than suffices for these uses.

#### Material Models

Burley’s (2012) course notes describe a material model developed at Disney for feature films. This write-up includes extensive discussion of features of real-world reflection functions that can be observed in Matusik et al.’s (2003b) measurements of one hundred BRDFs and analyzes the ways that existing BRDF models do and do not fit these features well. These insights are then used to develop an “artist-friendly” material model that can express a wide range of surface appearances. The model describes reflection with a single color and ten scalar parameters, all of which are in the range and have fairly predictable effects on the appearance of the resulting material. An earlier material model designed to have intuitive parameters for artistic control was developed by Strauss (1990).

The *bidirectional texture function* (BTF) is a generalization of the
BRDF that was introduced by Dana et al. (1999). (BTFs are
also referred to as spatially varying BRDFs (SVBRDFs).) It is a
six-dimensional reflectance function that adds two dimensions to account for
spatial variation to the BSDF. `pbrt`’s material model can thus be seen as
imposing a particular factorization of the BTF where variation due to the
spatial dimension is incorporated into textures that in turn provide values
for a parametric BSDF that defines the directional distribution. The BTF
representation is especially useful for material acquisition, as it does
not impose a particular representation or specific factorization of the six
dimensions. The survey articles on BTF acquisition and representation by
Müller et al. (2005) and Filip and Haindl
(2009) have good coverage of earlier work in this area.

Rainer et al. (2019) recently trained a neural network to represent a given BTF; network evaluation took the position and lighting directions as parameters and returned the corresponding BTF value. This work was subsequently generalized with a technique based on training a single network that provides a parameterization to which given BTFs can easily be mapped (Rainer et al. 2020). Kuznetsov et al. (2021) also used a neural approach, developing a compact representation that allowed 7D queries of position, two directions, and a filter size.

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