## Further Reading

Phong (1975) developed an early empirical reflection model for glossy surfaces in computer graphics. Although neither reciprocal nor energy conserving, it was a cornerstone of the first synthetic images of non-Lambertian objects. The Torrance–Sparrow microfacet model is described in Torrance and Sparrow (1967); it was first introduced to graphics by Blinn (1977), and a variant of it was used by Cook and Torrance (1981, 1982). The Oren–Nayar Lambertian model is described in their 1994 paper (Oren and Nayar 1994).

Hall’s (1989) book collected and described the state of the art in physically based surface reflection models for graphics; it remains a seminal reference. It discusses the physics of surface reflection in detail, with many pointers to the original literature and with many tables of useful measured data about reflection from real surfaces. Burley’s (2012) more recent paper includes a thorough annotated bibliography of more recent work on reflection models for computer graphics.

Heitz’s paper on microfacet shadowing-masking functions (2014a) provides a very well-written introduction to microfacet BSDF models in general, with many useful figures and explanations about details of the topic. See the papers by Beckmann and Spizzichino (1963) and Trowbridge and Reitz (1975) for the introduction of their respective microfacet distribution functions. Kurt et al. (2010) developed an anisotropic Beckmann–Spizzichino distribution function; see Heitz (2014a) for anisotropic variants of many other microfacet distribution functions. Early anisotropic BRDF models for computer graphics were developed by Kajiya (1985) and Poulin and Fournier (1990).

The microfacet masking-shadowing function in Equation (8.14) was introduced by Smith (1967), who used the assumption of no correlation between the height of the microsurface at nearby points to derive this result. Smith also first derived the normalization constraint in Equation (8.12). (This result was derived independently by Ashikhmin, Premoze, and Shirley (2000).) See Heitz (2014a) for further discussion of derivations of these functions. A more accurate function for Gaussian microfacet surfaces that better accounts for the effects of correlation between the two directions was developed by Heitz et al. (2013), and the rational approximation to the Beckmann–Spizzichino function used in this chapter is due to Heitz (2014a), which is derived from an approximation developed by Walter et al. (2007). Our derivation of the function, Equation (8.13), is also due to Heitz (2015).

Stam (2001) developed a generalization of the Cook–Torrance model for transmission, and more recently Walter et al. (2007) revisited this problem. Weyrich et al. (2009) have developed methods to infer a microfacet distribution that matches a measured or desired reflection distribution. Remarkably, they show that it’s possible to manufacture actual physical surfaces that match a desired reflection distribution fairly accurately. Simonot (2009) has developed a model that spans Oren–Nayar and Torrance–Sparrow: microfacets are modeled as Lambertian reflectors with a layer above them that ranges from perfectly transmissive to a perfect specular reflector. However, this model doesn’t account for masking-shadowing effects and can’t be evaluated in closed form.

The microfacet reflection models in this chapter are all based on the assumption that so many microfacets are visible in a pixel that they can be accurately described by their aggregate statistical behavior. This assumption isn’t true for many real-world surfaces, where a relatively small number of microfacets may be visible in each pixel; examples of such surfaces include car paint and glittery plastics. Both Yan et al. (2014) and Jakob et al. (2014b) have developed techniques that model these cases well.

It can be useful to be able to find BSDFs for layered materials, such as a metal base surface tarnished with patina, or wood with a varnish coating. Hanrahan and Krueger (1993) modeled the layers of skin accounting for just a single scattering event in each layer, and Dorsey and Hanrahan (1996) rendered layered materials using the Kubelka–Munk theory, which accounts for multiple scattering within layers but assumes that radiance distribution doesn’t vary as a function of direction.

Pharr and Hanrahan (2000) showed that Monte Carlo integration could be used
to solve the *adding equations* to efficiently compute BSDFs for
layered materials without needing either of these simplifications. The
adding equations are integral equations that accurately describe the effect
of multiple scattering in layered media that were derived by van De Hulst
(1980) and Twomey et al. (1966). Weidlich and Wilkie (2007) rendered
layered materials more efficiently by making a number of simplifying
assumptions, and Jakob et al. (2014a) efficiently computed
scattering in layered materials using the Fourier basis representation
implemented here as the `FourierBSDF`.

A number of researchers have investigated BRDFs based on modeling the small-scale geometric features of a reflective surface. This work includes the computation of BRDFs from bump maps by Cabral, Max, and Springmeyer (1987), Fournier’s normal distribution functions (Fournier 1992), and Westin, Arvo, and Torrance (1992), who applied Monte Carlo ray tracing to statistically model reflection from microgeometry and represented the resulting BRDFs with spherical harmonics. More recently, Wu et al. (2011) developed a system that made it possible to model microgeometry and specify its underlying BRDF while interactively previewing the resulting macro-scale BRDF.

Improvements in data-acquisition technology have led to increasing amounts of detailed real-world BRDF data, even including BRDFs that are spatially varying (sometimes called “bidirectional texture functions,” BTFs) (Dana et al. 1999). Matusik et al. (2003a, 2003b) assembled an early database of measured isotropic BRDF data. See Müller et al. (2005) for a survey of work in BRDF measurement until the year 2005. Sun et al. (2007) measured BRDFs as they change over time—for example, due to paint drying, a wet surface becoming dry, or dust accumulating. While most BRDF measurement has been based on measuring reflected radiance due to a given amount of incident irradiance, Zhao et al. (2011) showed that CT imaging of the structure of fabrics led to very accurate reflection models.

Fitting measured BRDF data to parametric reflection models is a difficult problem. Rusinkiewicz (1998) made the influential observation that reparameterizing the measured data can make it substantially easier to compress or fit to models; this topic has been further investigated by Stark et al. (2005) and in Marscher’s Ph.D. dissertation (1998). Ngan et al. (2005) analyzed the effectiveness of a variety of BRDF models for fitting measured data and showed that models based on the half-angle vector, rather than a reflection vector, tended to be more effective. See also the paper on this topic by Edwards et al. (2005).

Zickler et al. (2005) developed a method for representing BRDFs based on radial basis functions (RBFs)—they use them to interpolate irregularly sampled 5D spatially varying BRDFs. Weistroffer et al. (2007) have shown how to efficiently represent scattered reflectance data with RBFs without needing to resample them to have regular spacing. Wang et al. (2008a) demonstrated a successful approach for acquiring spatially varying anisotropic BRDFs. Pacanowski et al. (2012) developed a representation that could guarantee a given error bound between measured and fit data, and Bagher et al. (2012) introduced a parametric BRDF that accurately fit a wide variety of reflection function distributions using just six coefficients per color channel. More recently, Brady et al. (2014) found new analytic BRDF models that fit measured BRDFs well using genetic programming. Dupuy et al. (2015) developed an efficient and easily implemented approach for fitting measured BRDFs to microfacet distributions based on using power iterations to compute eigenvectors.

Kajiya and Kay (1989) developed an early reflection model for hair based on a model of individual hairs as cylinders with diffuse and glossy reflection properties. Their model determines the overall reflection from these cylinders, accounting for the effect of variation in surface normal over the hemisphere along the cylinder. For related work, see also the paper by Banks (1994), which discusses shading models for 1D primitives like hair. Goldman (1997) developed a probabilistic shading model that models reflection from collections of short hairs. Marschner et al. (2003) developed an accurate model of light scattering from human hair fibers that decomposes the reflected light into three components with distinct directional profiles based on the number of internal refraction events. Sadeghi et al. (2010) developed intuitive controls for physically based hair reflection models that made it easier to achieve a desired visual appearence. Further improvements to hair scattering models were introduced by d’Eon et al. (2011). Finally, see Ward et al.’s survey (2007) for extensive coverage of research in modeling, animating, and rendering hair.

Modeling reflection from a variety of specific types of surfaces has received attention from researchers, leading to specialized reflection models. Examples include Marschner et al.’s (2005) work on rendering wood, Günther et al.’s (2005) investigation of car paint, and Papas et al.’s (2014) model for paper.

Cloth remains particularly challenging material to render. Work in this area includes papers by Sattler et al. (2003), Irawan (2008), Schröder et al. (2011), Irawan and Marschner (2012), Zhao et al. (2012), and Sadeghi et al. (2013).

Nayar, Ikeuchi, and Kanade (1991) have shown that some reflection models based on physical (wave) optics have substantially similar characteristics to those based on geometric optics. The geometric optics approximations don’t seem to cause too much error in practice, except on very smooth surfaces. This is a helpful result, giving experimental basis to the general belief that wave optics models aren’t usually worth their computational expense for computer graphics applications.

The effect of the polarization of light is not modeled in `pbrt`, although for
some scenes it can be an important effect; see, for example, the paper by
Tannenbaum, Tannenbaum, and Wozny (1994) for information about how to
extend a renderer to account for this effect. Similarly, the fact that
indices of refraction of real-world objects usually vary as a function of
wavelength is also not modeled here; see both Section 11.8 of Glassner’s
book (1995) and Devlin et al.’s survey article for
information about these issues and references to previous work (Devlin et al. 2002). Fluorescence, where light is reflected at different wavelengths
than the incident illumination, is also not modeled by `pbrt`; see Glassner (1994) and Wilkie et al. (2006) for more information on this topic.

Moravec (1981) was the first to apply a wave optics model to graphics. This area has also been investigated by Bahar and Chakrabarti (1987) and Stam (1999), who applied wave optics to model diffraction effects. For more recent work in this area, see the papers by Cuypers et al. (2012) and Musbach et al. (2013), who also provide extensive references to previous work on this topic.

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