Chapter 24

Reflection Models

As we discussed in Chapter 18, the reflective properties of a surface can be summarized using the BRDF (Nicodemus, Richmond, Hsia, Ginsberg, & Limperis, 1977; Cook & Torrance, 1982). In this chapter, we discuss some of the most visually important aspects of material properties and a few fairly simple models that are useful in capturing these properties. There are many BRDF models in use in graphics, and the models presented here are meant to give just an idea of nondiffuse BRDFs.

24.1 Real-World Materials

Many real materials have a visible structure at normal viewing distances. For example, most carpets have easily visible pile that contributes to appearance. For our purposes, such structure is not part of the material property but is, instead, part of the geometric model. Structure whose details are invisible at normal viewing distances, but which do determine macroscopic material appearance, are part of the material property. For example, the fibers in paper have a complex appearance under magnification, but they are blurred together into an homogeneous appearance when viewed at arm’s length. This distinction between microstructure that is folded into BRDF is somewhat arbitrary and depends on what one defines as “normal” viewing distance and visual acuity, but the distinction has proven quite useful in practice.

In this section, we define some categories of materials. Later in the chapter, we present reflection models that target each type of material. In the notes at the end of the chapter, some models that account for more exotic materials are also discussed.

24.1.1 smooth Dielectrics and Metals

Dielectrics are clear materials that refract light; their basic properties were summarized in Chapter 4. Metals reflect and refract light much like dielectrics, but they absorb light very, very quickly. Thus, only very thin metal sheets are transparent at all, e.g., the thin gold plating on some glass objects. For a smooth material, there are only two important properties:

1. How much light is reflected at each incident angle and wavelength.
2. What fraction of light is absorbed as it travels through the material for a given distance and wavelength.

The amount of light transmitted is whatever is not reflected (a result of energy conservation). For a metal, in practice, we can assume all the light is immediately absorbed. For a dielectric, the fraction is determined by the constant used in Beer’s Law as discussed in Chapter 13.

The amount of light reflected is determined by the Fresnel equations as discussed in Chapter 4. These equations are straightforward, but cumbersome. The main effect of the Fresnel equations is to increase the reflectance as the incident angle increases, particularly near grazing angles. This effect works for transmitted light as well. These ideas are shown diagrammatically in Figure 24.1. Note that the light is repeatedly reflected and refracted as shown in Figure 24.2. Usually only one or two of the reflected images is easily visible.

24.1.2 Rough Surfaces

If a metal or dielectric is roughened to a small degree, but not so small that diffraction occurs, then we can think of it as a surface with microfacets (Cook & Torrance, 1982). Such surfaces behave specularly at a closer distance, but viewed at a further distance seem to spread the light out in a distribution. For a metal, an example of this rough surface might be brushed steel, or the “cloudy” side of most aluminum foil.

For dielectrics, such as a sheet of glass, scratches or other irregular surface features make the glass blur the reflected and transmitted images that we can normally see clearly. If the surface is heavily scratched, we call it translucent rather than transparent. This is a somewhat arbitrary distinction, but it is usually clear whether we would consider a glass translucent or transparent.

24.1.3 Diffuse Materials

A material is diffuse if it is matte, i.e., not shiny. Many surfaces we see are diffuse, such as most stones, paper, and unfinished wood. To a first approximation, diffuse surfaces can be approximated with a Lambertian (constant) BRDF. Real diffuse materials usually become somewhat specular for grazing angles. This is a subtle effect, but can be important for realism.

24.1.4 Translucent Materials

Many thin objects, such as leaves and paper, both transmit and reflect light diffusely. For all practical purposes no clear image is transmitted by these objects. These surfaces can add a hue shift to the transmitted light. For example, red paper is red because it filters out non-red light for light that penetrates a short distance into the paper, and then scatters back out. The paper also transmits light with a red hue because the same mechanisms apply, but the transmitted light makes it all the way through the paper. One implication of this property is that the transmitted coefficient should be the same in both directions.

24.1.5 Layered Materials

Many surfaces are composed of “layers” or are dielectrics with embedded particles that give the surface a diffuse property (Phong, 1975). The surface of such materials reflects specularly as shown in Figure 24.3, and thus obeys the Fresnel equations. The light that is transmitted is either absorbed or scattered back up to the dielectric surface where it may or may not be transmitted. That light that is transmitted, scattered, and then retransmitted in the opposite direction forms a diffuse “reflection” component.

Note that the diffuse component also is attenuated with the degree of the angle, because the Fresnel equations cause reflection back into the surface as the angle increases as shown in Figure 24.4. Thus, instead of a constant diffuse BRDF, one that vanishes near the grazing angle is more appropriate.

24.2 Implementing Reflection Models

A BRDF model, as described in Section 18.1.6, will produce a rendering which is more physically based than the rendering we get from point light sources and Phong-like models. Unfortunately, real BRDFs are typically quite complicated and cannot be deduced from first principles. Instead, they must either be measured and directly approximated from raw data, or they must be crudely approximated in an empirical fashion. The latter empirical strategy is what is usually done, and the development of such approximate models is still an area of research. This section discusses several desirable properties of such empirical models.

First, physical constraints imply two properties of a BRDF model. The first constraint is energy conservation:

$\text{for}\text{all}{k}_i,R\left(k_i\right)={\displaystyle {\int }_{\text{all}{k}_o}\rho \left(k_i,k_o\right)}\cos {\theta }_{o}d{\sigma }_o\le 1.$

If you send a beam of light at a surface from any direction ki, then the total amount of light reflected over all directions will be at most the incident amount. The second physical property we expect all BRDFs to have is reciprocity:

$\text{for}\text{all}{k}_i,k_o,\rho \left(k_i,k_o\right)=\rho \left(k_o,k_i\right).$

Second, we want a clear separation between diffuse and specular components. The reason for this is that, although there is a mathematically clean delta function formulation for ideal specular components, delta functions must be implemented as special cases in practice. Such special cases are only practical if the BRDF model clearly indicates what is specular and what is diffuse.

Third, we would like intuitive parameters. For example, one reason the Phong model has enjoyed such longevity is that its diffuse constant and exponent are both clearly related to the intuitive properties of the surface, namely surface color and highlight size.

Finally, we would like the BRDF function to be amenable to Monte Carlo sampling. Recall from Chapter 14 that an integral can be sampled by N random points xi ~ p where p is defined with the same measure as the integral:

${\displaystyle \int f\left(x\right)d\mu \approx \frac{1}{N}{\displaystyle \sum _{j=1}^N\frac{f\left(x_j\right)}{p\left(x_j\right)}.}}$

Recall from Section 18.2 that the surface radiance in direction ko is given by a transport equation:

$L_s\left(k_o\right)={\displaystyle {\int }_{\text{all}{k}_{\text{i}}}\rho \left(k_i,k_o\right)}{L}_f\left(k_i\right)\cos {\theta }_{i}d{\sigma }_i.$

If we sample directions with pdf p(ki) as discussed in Chapter 23, then we can approximate the surface radiance with samples:

$L_s\left(k_o\right)\approx \frac{1}{N}{\displaystyle \sum _{j=1}^N\frac{\rho \left(k_j,k_o\right)L_f\left(k_j\right)\cos {\theta }_j}{p\left(k_j\right)}.}$

This approximation will converge for any p that is nonzero where the integrand is nonzero. However, it will only converge well if the integrand is not very large relative to p. Ideally, p(k) should be approximately shaped like the integrand ρ(kj, ko)Lf (kj) cosθj. In practice, Lf is complicated, and the best we can accomplish is to have p(k) shaped somewhat like ρ(k, ko)Lf(k) cos θ.

For example, if the BRDF is Lambertian, then it is constant and the “ideal” p(k) is proportional to cos θ. Because the integral of p must be one, we can deduce the leading constant:

${\displaystyle {\int }_{\text{all}k\text{with}\theta <\pi /2}C\cos \theta d\sigma =1.}$

This implies that C = 1/π, so we have

$p\left(k\right)=\frac{1}{\pi }\cos \theta .$

An acceptably efficient implementation will result as long as p doesn’t get too small when the integrand is nonzero. Thus, the constant pdf will also suffice:

$p\left(k\right)=\frac{1}{2\pi }.$

This emphasizes that many pdfs may be acceptable for a given BRDF model.

24.3 Specular Reflection Models

For a metal, we typically specify the reflectance at normal incidence R0(λ). The reflectance should vary according to the Fresnel equations, and a good approximation is given by (Schlick, 1994a)

$R\left(\theta ,\lambda \right)=R_0\left(\lambda \right)+\left(1-R_0\left(\lambda \right)\right){\left(1-\cos \theta \right)}^5.$

This approximation allows us to just set the normal reflectance of the metal either from data or by eye.

For a dielectric, the same formula works for reflectance. However, we can set R0(λ) in terms of the refractive index n(λ):

$R_0\left(\lambda \right)={\left(\frac{n\left(\lambda \right)-1}{n\left(\lambda \right)+1}\right)}^2.$

Typically, n does not vary with wavelength, but for applications where dispersion is important, n can vary. The refractive indices that are often useful include water (n = 1.33), glass (n = 1.4 to n = 1.7), and diamond (n = 2.4).

24.4 smooth-Layered Model

Reflection in matte/specular materials, such as plastics or polished woods, is governed by Fresnel equations at the surface and by scattering within the subsurface. An example of this reflection can be seen in the tiles in the renderings in Figure 24.5. Note that the blurring in the specular reflection is mostly vertical due to the compression of apparent bump spacing in the view direction. This effect causes the vertically streaked reflections seen on lakes on windy days; it can either be modeled using explicit microgeometry and a simple smooth-surface reflection model or by a more general model that accounts for this asymmetry.

We could use the traditional Lambertian-specular model for the tiles, which linearly mixes specular and Lambertian terms. In standard radiometric terms, this can be expressed as

$\rho \left(\theta ,\varphi ,{\theta }^{\prime },{\varphi }^{\prime }\lambda \right)=\frac{Rd\left(\lambda \right)}{\pi }+R_s{\rho }_s\left(\theta ,\varphi ,{\theta }^{\prime },{\varphi }^{\prime }\right),$

where Rd(λ) is the hemispherical reflectance of the matte term, Rs is the specular reflectance, and ρs is the normalized specular BRDF (a weighted Dirac delta function on the sphere). This equation is a simplified version of the BRDF where Rs is independent of wavelength. The independence of wavelength causes a highlight that is the color of the luminaire, so a polished rather than a metal appearance will be achieved. Ward (G. J. Ward, 1992) suggests to set Rd(λ) + Rs ≤ 1 in order to conserve energy. However, such models with constant Rs fail to show the increase in specularity for steep viewing angles. This is the key point: in the real world the relative proportions of matte and specular appearance change with the viewing angle.

One way to simulate the change in the matte appearance is to explicitly dampen Rd(λ) as Rs increases (Shirley, 1991):

$\rho \left(\theta ,\varphi ,{\theta }^{\prime },{\varphi }^{\prime }\lambda \right)=R_f\left(\theta \right){\rho }_s\left(\theta ,\varphi ,{\theta }^{\prime },{\varphi }^{\prime }\right)+\frac{Rd\left(\lambda \right)\left(1-R_f\left(\theta \right)\right)}{\pi },$

where Rf (θ) is the Fresnel reflectance for a polish-air interface. The problem with this equation is that it is not reciprocal, as can been seen by exchanging θ and θ'; this changes the value of the matte damping factor because of the multiplication by (1 — Rf(θ)). The specular term, a scaled Dirac delta function, is reciprocal, but this does not make up for the non-reciprocity of the matte term. Although this BRDF works well, its lack of reciprocity can cause some rendering methods to have ill-defined solutions.

We now present a model that produces the matte/specular tradeoff while remaining reciprocal and energy conserving. Because the key feature of the new model is that it couples the matte and specular scaling coefficients, it is called a coupled model (Shirley, Smits, Hu, & Lafortune, 1997).

Surfaces which have a glossy appearance are often a clear dielectric, such as polyurethane or oil, with some subsurface structure. The specular (mirror-like) component of the reflection is caused by the smooth dielectric surface and is independent of the structure below this surface. The magnitude of this specular term is governed by the Fresnel equations.

The light that is not reflected specularly at the surface is transmitted through the surface. There, either it is absorbed by the subsurface, or it is reflected from a pigment or a subsurface and transmitted back through the surface of the polish. This transmitted light forms the matte component of reflection. Since the matte component can only consist of the light that is transmitted, it will naturally decrease in total magnitude for increasing angle.

To avoid choosing between physically plausible models and models with good qualitative behavior over a range of incident angles, note that the Fresnel equations that account for the specular term, Rf(θ), are derived directly from the physics of the dielectric-air interface. Therefore, the problem must lie in the matte term. We could use a full-blown simulation of subsurface scattering as implemented, but this technique is both costly and requires detailed knowledge of subsurface structure, which is usually neither known nor easily measurable. Instead, we can modify the matte term to be a simple approximation that captures the important qualitative angular behavior shown in Figure 24.4.

Let us assume that the matte term is not Lambertian, but instead is some other function that depends only on θ, θ' and λ: ρm(θ, θ', λ). We discard behavior that depends on ϕ or ϕ' in the interest of simplicity. We try to keep the formulas reasonably simple because the physics of the matte term is complicated and sometimes requires unknown parameters. We expect the matte term to be close to constant, and roughly rotationally symmetric (He et al., 1992).

An obvious candidate for the matte component ρm(θ, θ', λ) that will be reciprocal is the separable form kRm(λ)f(θ)f(θ') for some constant k and matte reflectance parameter Rm(λ). We could merge k and Rm(λ) into a single term, but we choose to keep them separated because this makes it more intuitive to set Rm(λ) —which must be between 0 and 1 for all wavelengths. Separable BRDFs have been shown to have several computational advantages, thus we use the separable model:

$\rho \left(\theta ,\varphi ,{\theta }^{\prime },{\varphi }^{\prime }\lambda \right)=R_f\left(\theta \right){\rho }_s\left(\theta ,\varphi ,{\theta }^{\prime },{\varphi }^{\prime }\right)+k{R}_m\left(\lambda \right)f\left(\theta \right)f\left({\theta }^{\prime }\right).$

We know that the matte component can only contain energy not reflected in the surface (specular) component. This means that for Rm(λ) = 1, the incident and reflected energy are the same, which suggests the following constraint on the BRDF for each incident θ and λ:

$R_f\left(\theta \right)+2\pi kf\left(\theta \right){\displaystyle {\int }_0^{\frac{\pi }{2}}f\left({\theta }^{\prime }\right)\cos {\theta }^{\prime }\sin {\theta }^{\prime }d{\theta }^{\prime }=1.}\text{ (24.1)}$

We can see that f(θ) must be proportional to (1 - Rf(θ)). If we assume that matte components that absorb some energy have the same directional pattern as this ideal, we get a BRDF of the form

$\rho \left(\theta ,\varphi ,{\theta }^{\prime },{\varphi }^{\prime }\lambda \right)=R_f\left(\theta \right){\rho }_s\left(\theta ,\varphi ,{\theta }^{\prime },{\varphi }^{\prime }\right)+k{R}_m\left(\lambda \right)\left[1-R_f\left(\theta \right)\right]\left[1-R_f\left({\theta }^{\prime }\right)\right].$

We could now insert the full form of the Fresnel equations to get Rf(θ), and then use energy conservation to solve for constraints on k. Instead, we will use the approximation discussed in Section 24.1.1 We find that

$f\left(\theta \right)\propto \left(1-{\left(1-\cos \theta \right)}^5\right).$

Applying Equation (24.1) yields

$k=\frac{21}{20\pi \left(1-R_0\right)}.\text{ (24.2)}$

The full coupled BRDF is then

$\rho \left(\theta ,\varphi ,{\theta }^{\prime },{\varphi }^{\prime }\lambda \right)=\left[R_0+{\left(1-\cos \theta \right)}^5\left(1-R_0\right)\right]{\rho }_s\left(\theta ,\varphi ,{\theta }^{\prime },{\varphi }^{\prime }\right)+k{R}_m\left(\lambda \right)\left[1-{\left(1-\cos \theta \right)}^5\right]\left[1-{\left(1-\cos {\theta }^{\prime }\right)}^5\right].\text{ (24.3)}$

The results of running the coupled model is shown in Figure 24.5. Note that for the high viewpoint, the specular reflection is almost invisible, but it is clearly visible in the low-angle photograph image, where the matte behavior is less obvious.

For reasonable values of refractive indices, R0 is limited to approximately the range 0.03 to 0.06 (the value R0 = 0.05 was used for Figure 24.5). The value of Rs in a traditional Phong model is harder to choose, because it typically must be tuned for viewpoint in static images and tuned for a particular camera sequence for animations. Thus, the coupled model is easier to use in a “hands-off” mode.

24.5 Rough-Layered Model

The previous model is fine if the surface is smooth. However, if the surface is not ideal, some spread is needed in the specular component. An extension of the coupled model to this case is presented here (Ashikhmin & Shirley, 2000). At a given point on a surface, the BRDF is a function of two directions, one in the direction toward the light and one in the direction toward the viewer. We would like to have a BRDF model that works for “common” surfaces, such as metal and plastic, and has the following characteristics:

1. Plausible. As defined by Lewis (R. R. Lewis, 1994), this refers to the BRDF obeying energy conservation and reciprocity.
2. Anisotropy. The material should model simple anisotropy, such as seen on brushed metals.
3. Intuitive parameters. For material, such as plastics, there should be parameters Rd for the substrate and Rs for the normal specular reflectance as well as two roughness parameters nu and nv.
4. Fresnel behavior. Specularity should increase as the incident angle decreases.
5. Non-Lambertian diffuse term. The material should allow for a diffuse term, but the component should be non-Lambertian to assure energy conservation in the presence of Fresnel behavior.
6. Monte Carlo friendliness. There should be some reasonable probability density function that allows straightforward Monte Carlo sample generation for the BRDF.

A BRDF with these properties is a Fresnel-weighted, Phong-style cosine lobe model that is anisotropic.

We again decompose the BRDF into a specular component and a diffuse component (Figure 24.6). Accordingly, we write our BRDF as the classical sum of two parts:

$\rho \left(k_1,k_2\right)={\rho }_s\left(k_1,k_2\right)+\rho d\left(k_1,k_2\right),\text{ (24.4)}$

where the first term accounts for the specular reflection (this will be presented in the next section). While it is possible to use the Lambertian BRDF for the diffuse term ρd(k1, k2) in our model, we will discuss a better solution in Section 24.5.2 and how to implement the model in Section 24.5.3. Readers who just want to implement the model should skip to that section.

24.5.1 Anisotropic Specular BRDF

To model the specular behavior, we use a Phong-style specular lobe but make this lobe anisotropic and incorporate Fresnel behavior while attempting to preserve the simplicity of the initial mode. This BRDF is

$\rho \left(k_1,k_2\right)=\frac{\sqrt{\left(n_u+1\right)\left(n_v+1\right)}}{8\pi }\frac{\left(n\cdot h\right)n_u{\cos }^2\varphi +n_v\sin \varphi }{\left(h\cdot k_i\right)\max \left(\cos {\theta }_i,\cos {\theta }_o\right))}F\left(k_i\cdot \text{h}\right)\cdot \text{ (24.5)}$

Again we use Schlick’s approximation to the Fresnel equation:

$F\left(k_i\cdot \text{h}\right)=R_s+\left(1-R_s\right){\left(1-\left(k_i\cdot \text{h}\right)\right)}^5,\text{ (24.6)}$

where Rs is the material’s reflectance for the normal incidence. Because ki · h = ko · h, this form is reciprocal. We have an empirical model whose terms are chosen to enforce energy conservation and reciprocity. A full rationalization for the terms is given in the paper by Ashikhmin, listed in the chapter notes.

The specular BRDF of Equation (24.5) is useful for representing metallic surfaces where the diffuse component of reflection is very small. Figure 24.7 shows a set of metal spheres on a texture-mapped Lambertian plane. As the values of parameters nu and nv change, the appearance of the spheres shift from rough metal to almost perfect mirror, and from highly anisotropic to the more familiar Phong-like behavior.

24.5.2 Diffuse Term for the Anisotropic Phong Model

It is possible to use a Lambertian BRDF together with the anisotropic specular term; this is done for most models, but it does not necessarily conserve energy. A better approach is a simple angle-dependent form of the diffuse component which accounts for the fact that the amount of energy available for diffuse scattering varies due to the dependence of the specular term’s total reflectance on the incident angle. In particular, diffuse color of a surface disappears near the grazing angle, because the total specular reflectance is close to one. This well-known effect cannot be reproduced with a Lambertian diffuse term and is therefore missed by most reflection models.

Following a similar approach to the coupled model, we can find a form of the diffuse term that is compatible with the anisotropic Phong lobe:

$\rho d\left(k_1,k_2\right)=\frac{28Rd}{23\pi }\left(1-R_s\right)\left(1-{\left(1-\frac{\cos {\theta }_i}{2}\right)}^5\right)\left(1-{\left(\frac{\cos {\theta }_o}{2}\right)}^5\right).\text{ (24.7)}$

Here Rd is the diffuse reflectance for normal incidence, and Rs is the Phong lobe coefficient. An example using this model is shown in Figure 24.8.

24.5.3 Implementing the Model

Recall that the BRDF is a combination of diffuse and specular components:

$\rho \left(k_1,k_2\right)={\rho }_s\left(k_1,k_2\right)+\rho d\left(k_1,k_2\right).\text{ (24.8)}$

The diffuse component is given in Equation (24.7); the specular component is given in Equation (24.5). It is not necessary to call trigonometric functions to compute the exponent, so the specular BRDF can be written:

$\rho \left(k_1,k_2\right)=\frac{\sqrt{\left(n_u+1\right)\left(n_v+1\right)}}{8\pi }\left(n\cdot h\right)\frac{\left(n_u{\left(n\cdot h\right)}^2+n_v{\left(h\cdot \text{v}\right)}^2\right)/\left(1-{\left(\mathrm{hn}\right)}^2\right)}{\left(h\cdot k_i\right)\max \left(\cos {\theta }_i,\cos {\theta }_o\right)}F\left(k_i\cdot h\right)\cdot \text{ (24.9)}$

In a Monte Carlo setting, we are interested in the following proble given k1, generate samples of k2 with a distribution whose shape is similar to the cosine-weighted BRDF. Note that greatly undersampling a large value of the integrand is a serious error, while greatly oversampling a small value is acceptable in practice. The reader can verify that the densities suggested below have this property.

A suitable way to construct a pdf for sampling is to consider the distribution of half vectors that would give rise to our BRDF. Such a function is

$ph\left(h\right)=\frac{\sqrt{\left(n_u+1\right)\left(n_v+1\right)}}{2\pi }\left(\mathrm{nh}\right)n_u{\cos }^2\varphi +n_v{\sin }^2\varphi ,\text{ (24.10)}$

where the constants are chosen to ensure it is a valid pdf.

We can just use the probability density function ph(h) of Equation (24.10) to generate a random h. However, to evaluate the rendering equation, we need both a reflected vector ko and a probability density function p(ko). It is important to note that if you generate h according to ph (h) and then transform to the resulting ko:

$k_o=-k_i+2\left(k_i\cdot h\right)h,\text{ (24.11)}$

the density of the resulting ko is not ph(ko). This is because of the difference in measures in h and ko. So the actual density p(ko) is

$p\left(k_o\right)=\frac{ph\left(h\right)}{4\left(k_{i}h\right)}.\text{ (24.12)}$

Note that in an implementation where the BRDF is known to be this model, the estimate of the rendering equation is quite simple as many terms cancel out.

It is possible to generate an h vector whose corresponding vector ko will point inside the surface, i.e., cosθo < 0. The weight of such a sample should be set to zero. This situation corresponds to the specular lobe going below the horizon and is the main source of energy loss in the model. Clearly, this problem becomes progressively less severe as nu, nv become larger.

The only thing left now is to describe how to generate h vectors with the pdf of Equation (24.10). We will start by generating h with its spherical angles in the range $\left(\theta ,\varphi \right)\in \left[0,\frac{\pi }{2}\right]\times \left[0,\frac{\pi }{2}\right]$. Note that this is only the first quadrant of the hemisphere. Given two random numbers (ξ1, ξ2) uniformly distributed in [0, 1], we can choose

$\varphi -\arctan \left(\sqrt{\frac{n_u+1}{n_v+1}}\tan \left(\frac{\pi {\xi }_1}{2}\right)\right),\text{ (24.13)}$

and then use this value of ϕ to obtain θ according to

$\cos \theta ={\left(1-{\xi }_2\right)}^{1/\left(n_u{\cos }^2\varphi +n^v{\sin }^2\varphi +1\right)}.\text{ (24.14)}$

To sample the entire hemisphere, we use the standard manipulation where ξ1 is mapped to one of four possible functions depending on whether it is in [0,0.25), [0.25,0.5), [0.5,0.75), or [0.75,1.0). For example, for ξ1 ∈ [0.25,0.5), find ϕ(1 − 4(0.5 − ξ1)) via Equation (24.13), and then “flip” it about the ϕ = π/2 axis. This ensures full coverage and stratification.

For the diffuse term, use a simpler approach and generate samples according to a cosine distribution. This is sufficiently close to the complete diffuse BRDF to substantially reduce variance of the Monte Carlo estimation.

Frequently Asked Questions

• My images look too smooth, even with a complex BRDF. What am I do ing wrong?

  BRDFs only capture subpixel detail that is too small to be resolved by the eye. Most real surfaces also have some small variations, such as the wrinkles in skin, that can be seen. If you want true realism, some sort of texture or displacement map is needed.

• How do I integrate the BRDF with texture mapping?

  Texture mapping can be used to control any parameter on a surface. So any kinds of colors or control parameters used by a BRDF should be programmable.

• I have very pretty code except for my material class. What am I doing wrong?

  You are probably doing nothing wrong. Material classes tend to be the ugly thing in everybody’s programs. If you find a nice way to deal with it, please let me know! My own code uses a shader architecture (Hanrahan & Lawson, 1990) which makes the material include much of the rendering algorithm.

Notes

There are many BRDF models described in the literature, and only a few of them have been described here. Others include (Cook & Torrance, 1982; He et al., 1992; G. J. Ward, 1992; Oren & Nayar, 1994; Schlick, 1994a; Lafortune, Foo, Torrance, & Greenberg, 1997; Stam, 1999; Ashikhmin, Premože, & Shirley, 2000; Ershov, Kolchin, & Myszkowski, 2001; Matusik, Pfister, Brand, & McMillan, 2003; Lawrence, Rusinkiewicz, & Ramamoorthi, 2004; Stark, Arvo, & Smits, 2005). The desired characteristics of BRDF models is discussed in Making Shaders More Physically Plausible (R. R. Lewis, 1994).

Exercises

1. Suppose that instead of the Lambertian BRDF we used a BRDF of the form C cosa θi. What must C be to conserve energy?
2. The BRDF in Exercise 1 is not reciprocal. Can you modify it to be reciprocal?
3. Something like a highway sign is a retroreflector. This means that the BRDF is large when ki and ko are near each other. Make a model inspired by the Phong model that captures retroreflection behavior while being reciprocal and conserving energy.