4.3 Surface Reflection

When light is incident on a surface, the surface scatters the light, reflecting some of it back into the environment. There are two main effects that need to be described to model this reflection: the spectral distribution of the reflected light and its directional distribution. For example, the skin of a lemon mostly absorbs light in the blue wavelengths but reflects most of the light in the red and green wavelengths. Therefore, when it is illuminated with white light, its color is yellow. It has much the same color no matter what direction it is being observed from, although for some directions a highlight—a brighter area that is more white than yellow—is visible. In contrast, the light reflected from a point in a mirror depends almost entirely on the viewing direction. At a fixed point on the mirror, as the viewing angle changes, the object that is reflected in the mirror changes accordingly.

Reflection from translucent surfaces is more complex; a variety of materials ranging from skin and leaves to wax and liquids exhibit subsurface light transport, where light that enters the surface at one point exits it some distance away. (Consider, for example, how shining a flashlight in one’s mouth makes one’s cheeks light up, as light that enters the inside of the cheeks passes through the skin and exits the face.)

There are two abstractions for describing these mechanisms for light reflection: the BRDF and the BSSRDF, described in Sections 4.3.1 and 4.3.2, respectively. The BRDF describes surface reflection at a point neglecting the effect of subsurface light transport. For materials where this transport mechanism does not have a significant effect, this simplification introduces little error and makes the implementation of rendering algorithms much more efficient. The BSSRDF generalizes the BRDF and describes the more general setting of light reflection from translucent materials.

4.3.1 The BRDF and the BTDF

The bidirectional reflectance distribution function (BRDF) gives a formalism for describing reflection from a surface. Consider the setting in Figure 4.10: we would like to know how much radiance is leaving the surface in the direction omega Subscript normal o toward the viewer, upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline right-parenthesis , as a result of incident radiance along the direction omega Subscript normal i Baseline comma upper L Subscript normal i Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal i Baseline right-parenthesis . (When considering light scattering at a surface location, pbrt uses the convention that omega Subscript normal i refers to the direction from which the quantity of interest (radiance in this case) arrives, rather than the direction from which the Integrator reached the surface.)

Figure 4.10: The BRDF. The bidirectional reflectance distribution function is a 4D function over pairs of directions omega Subscript normal i and omega Subscript normal o that describes how much incident light along omega Subscript normal i is scattered from the surface in the direction  omega Subscript normal o .

If the direction omega Subscript normal i is considered as a differential cone of directions, the differential irradiance at normal p Subscript  is

normal d upper E Subscript Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal i Baseline right-parenthesis equals upper L Subscript normal i Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal i Baseline right-parenthesis cosine theta Subscript normal i Baseline normal d omega Subscript normal i Baseline period
(4.10)

A differential amount of radiance will be reflected in the direction omega Subscript normal o due to this irradiance. Because of the linearity assumption from geometric optics, the reflected differential radiance is proportional to the irradiance

normal d upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline right-parenthesis proportional-to normal d upper E Subscript Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal i Baseline right-parenthesis period

The constant of proportionality defines the surface’s BRDF f Subscript normal r for the particular pair of directions omega Subscript normal i and  omega Subscript normal o :

f Subscript normal r Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline comma omega Subscript normal i Baseline right-parenthesis equals StartFraction normal d upper L Subscript normal o Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline right-parenthesis Over normal d upper E Subscript Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal i Baseline right-parenthesis EndFraction equals StartFraction normal d upper L Subscript normal o Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline right-parenthesis Over upper L Subscript normal i Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal i Baseline right-parenthesis cosine theta Subscript normal i Baseline normal d omega Subscript normal i Baseline EndFraction period
(4.11)

The spectral BRDF is defined by using spectral radiance in place of radiance.

Physically based BRDFs have two important qualities:

  1. Reciprocity: For all pairs of directions omega Subscript normal i and omega Subscript normal o , f Subscript normal r Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal i Baseline comma omega Subscript normal o Baseline right-parenthesis equals f Subscript normal r Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline comma omega Subscript normal i Baseline right-parenthesis .
  2. Energy conservation: The total energy of light reflected is less than or equal to the energy of incident light. For all directions omega Subscript normal o ,
    integral Underscript script upper H squared left-parenthesis bold n Subscript Baseline right-parenthesis Endscripts f Subscript normal r Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline comma omega prime Subscript Baseline right-parenthesis cosine theta prime normal d omega Subscript Baseline Superscript prime Baseline less-than-or-equal-to 1 period

Note that the value of the BRDF for a pair of directions omega Subscript normal i and omega Subscript normal o is not necessarily less than 1; it is only its integral that has this normalization constraint.

Two quantities that are based on the BRDF will occasionally be useful. First, the hemispherical-directional reflectance is a 2D function that gives the total reflection in a given direction due to constant illumination over the hemisphere, or, equivalently, the total reflection over the hemisphere due to light from a given direction. It is defined as

rho Subscript normal h normal d Baseline left-parenthesis omega Subscript normal o Baseline right-parenthesis equals integral Underscript script upper H squared left-parenthesis bold n Subscript Baseline right-parenthesis Endscripts f Subscript normal r Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline comma omega Subscript normal i Baseline right-parenthesis StartAbsoluteValue cosine theta Subscript normal i Baseline EndAbsoluteValue normal d omega Subscript normal i Baseline period
(4.12)

The hemispherical-hemispherical reflectance of a BRDF, denoted by rho Subscript normal h normal h , gives the fraction of incident light reflected by a surface when the incident light is the same from all directions. It is

rho Subscript normal h normal h Baseline equals StartFraction 1 Over pi EndFraction integral Underscript script upper H squared left-parenthesis bold n Subscript Baseline right-parenthesis Endscripts integral Underscript script upper H squared left-parenthesis bold n Subscript Baseline right-parenthesis Endscripts f Subscript normal r Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline comma omega Subscript normal i Baseline right-parenthesis StartAbsoluteValue cosine theta Subscript normal o Baseline cosine theta Subscript normal i Baseline EndAbsoluteValue normal d omega Subscript normal o Baseline normal d omega Subscript normal i Baseline period
(4.13)

A surface’s bidirectional transmittance distribution function (BTDF), which describes the distribution of transmitted light, can be defined in a manner similar to that for the BRDF. The BTDF is generally denoted by f Subscript normal t Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline comma omega Subscript normal i Baseline right-parenthesis , where omega Subscript normal i and omega Subscript normal o are in opposite hemispheres around  normal p Subscript . Remarkably, the BTDF does not obey reciprocity as defined above; we will discuss this issue in detail in Section 9.5.2.

For convenience in equations, we will denote the BRDF and BTDF when considered together as f Subscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline comma omega Subscript normal i Baseline right-parenthesis ; we will call this the bidirectional scattering distribution function (BSDF). Chapter 9 is entirely devoted to describing a variety of BSDFs that are useful for rendering.

Using the definition of the BSDF, we have

normal d upper L Subscript normal o Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline right-parenthesis equals f Subscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline comma omega Subscript normal i Baseline right-parenthesis upper L Subscript normal i Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal i Baseline right-parenthesis StartAbsoluteValue cosine theta Subscript normal i Baseline EndAbsoluteValue normal d omega Subscript normal i Baseline period

Here an absolute value has been added to the cosine theta Subscript normal i factor. This is done because surface normals in pbrt are not reoriented to lie on the same side of the surface as omega Subscript normal i (many other rendering systems do this, although we find it more useful to leave them in their natural orientation as given by the Shape). Doing so makes it easier to consistently apply conventions like “the surface normal is assumed to point outside the surface” elsewhere in the system. Thus, applying the absolute value to cosine theta factors like these ensures that the desired quantity is calculated.

We can integrate this equation over the sphere of incident directions around normal p Subscript to compute the outgoing radiance in direction omega Subscript normal o due to the incident illumination at normal p Subscript from all directions:

upper L Subscript normal o Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline right-parenthesis equals integral Underscript script upper S squared Endscripts f Subscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal o Baseline comma omega Subscript normal i Baseline right-parenthesis upper L Subscript normal i Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal i Baseline right-parenthesis StartAbsoluteValue cosine theta Subscript normal i Baseline EndAbsoluteValue normal d omega Subscript normal i Baseline period
(4.14)

This is a fundamental equation in rendering; it describes how an incident distribution of light at a point is transformed into an outgoing distribution, based on the scattering properties of the surface. It is often called the scattering equation when the sphere script upper S squared is the domain (as it is here), or the reflection equation when just the upper hemisphere script upper H squared left-parenthesis bold n Subscript Baseline right-parenthesis is being integrated over. One of the key tasks of the integration routines in Chapters 13 through 15 is to evaluate this integral at points on surfaces in the scene.

4.3.2 The BSSRDF

The bidirectional scattering surface reflectance distribution function (BSSRDF) is the formalism that describes scattering from materials that exhibit subsurface light transport. It is a distribution function upper S left-parenthesis normal p Subscript normal o Baseline comma omega Subscript normal o Baseline comma normal p Subscript normal i Baseline comma omega Subscript normal i Baseline right-parenthesis that describes the ratio of exitant differential radiance at point normal p Subscript normal o in direction omega Subscript normal o to the incident differential flux at normal p Subscript normal i from direction omega Subscript normal i (Figure 4.11):

upper S left-parenthesis normal p Subscript normal o Baseline comma omega Subscript normal o Baseline comma normal p Subscript normal i Baseline comma omega Subscript normal i Baseline right-parenthesis equals StartFraction normal d upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript normal o Baseline comma omega Subscript normal o Baseline right-parenthesis Over normal d normal upper Phi left-parenthesis normal p Subscript normal i Baseline comma omega Subscript normal i Baseline right-parenthesis EndFraction period
(4.15)

Figure 4.11: The bidirectional scattering surface reflectance distribution function generalizes the BSDF to account for light that exits the surface at a point other than where it enters. It is often more difficult to evaluate than the BSDF, although subsurface light transport makes a substantial contribution to the appearance of many real-world objects.

The generalization of the scattering equation for the BSSRDF requires integration over surface area and incoming direction, turning the 2D scattering Equation (4.14) into a 4D integral.

upper L Subscript normal o Baseline left-parenthesis normal p Subscript normal o Baseline comma omega Subscript normal o Baseline right-parenthesis equals integral Underscript upper A Endscripts integral Underscript script upper H squared left-parenthesis bold n Subscript Baseline right-parenthesis Endscripts upper S left-parenthesis normal p Subscript normal o Baseline comma omega Subscript normal o Baseline comma normal p Subscript normal i Baseline comma omega Subscript normal i Baseline right-parenthesis upper L Subscript normal i Baseline left-parenthesis normal p Subscript normal i Baseline comma omega Subscript normal i Baseline right-parenthesis StartAbsoluteValue cosine theta Subscript normal i Baseline EndAbsoluteValue normal d omega Subscript normal i Baseline normal d upper A Subscript Baseline period
(4.16)

With two more dimensions to integrate over, it is more complex to account for in rendering algorithms than Equation (4.14) is. However, as the distance between points normal p Subscript normal i and normal p Subscript normal o increases, the value of upper S generally diminishes. This fact can be a substantial help in implementations of subsurface scattering algorithms.

Light transport beneath a surface is described by the same principles as volume light transport in participating media and is described by the equation of transfer, which is introduced in Section 14.1. Subsurface scattering is thus based on the same effects as light scattering in clouds and smoke—just at a smaller scale.