11.1 Volume Scattering Processes
There are three main processes that affect the distribution of radiance in an environment with participating media:
- Absorption: the reduction in radiance due to the conversion of light to another form of energy, such as heat
- Emission: radiance that is added to the environment from luminous particles
- Scattering: radiance heading in one direction that is scattered to other directions due to collisions with particles
The characteristics of all of these properties may be homogeneous or inhomogeneous. Homogeneous properties are constant throughout some region of space, while inhomogeneous properties vary throughout space. Figure 11.1 shows a simple example of volume scattering, where a spotlight shining through a participating medium illuminates particles in the medium and casts a volumetric shadow.
Consider thick black smoke from a fire: the smoke obscures the objects behind it because its particles absorb light traveling from the object to the viewer. The thicker the smoke, the more light is absorbed. Figure 11.2 shows this effect with a spatial distribution of absorption that was created with an accurate physical simulation of smoke formation. Note the shadow on the ground: the participating medium has also absorbed light between the light source to the ground plane, casting a shadow.
Absorption is described by the medium’s absorption cross section, , which is the probability density that light is absorbed per unit distance traveled in the medium. In general, the absorption cross section may vary with both position and direction , although it is normally just a function of position. It is usually also a spectrally varying quantity. The units of are reciprocal distance (). This means that can take on any positive value; it is not required to be between 0 and 1, for instance.
Figure 11.3 shows the effect of absorption along a very short segment of a ray. Some amount of radiance is arriving at point , and we’d like to find the exitant radiance after absorption in the differential volume. This change in radiance along the differential ray length is described by the differential equation
which says that the differential reduction in radiance along the beam is a linear function of its initial radiance.
This differential equation can be solved to give the integral equation describing the total fraction of light absorbed for a ray. If we assume that the ray travels a distance in direction through the medium starting at point , the remaining portion of the original radiance is given by
While absorption reduces the amount of radiance along a ray as it passes through a medium, emission increases it, due to chemical, thermal, or nuclear processes that convert energy into visible light. Figure 11.4 shows emission in a differential volume, where we denote emitted radiance added to a ray per unit distance at a point in direction by .
Figure 11.5 shows the effect of emission with the smoke data set. In that figure the absorption coefficient is much lower than in Figure 11.2, giving a very different appearance.
The differential equation that gives the change in radiance due to emission is
This equation incorporates the assumption that the emitted light is not dependent on the incoming light . This is always true under the linear optics assumptions that pbrt is based on.
11.1.3 Out-Scattering and Attenuation
The third basic light interaction in participating media is scattering. As a ray passes through a medium, it may collide with particles and be scattered in different directions. This has two effects on the total radiance that the beam carries. It reduces the radiance exiting a differential region of the beam because some of it is deflected to different directions. This effect is called out-scattering (Figure 11.6) and is the topic of this section. However, radiance from other rays may be scattered into the path of the current ray; this in-scattering process is the subject of the next section.
The probability of an out-scattering event occurring per unit distance is given by the scattering coefficient, . As with absorption, the reduction in radiance along a differential length due to out-scattering is given by
The total reduction in radiance due to absorption and out-scattering is given by the sum . This combined effect of absorption and out-scattering is called attenuation or extinction. For convenience the sum of these two coefficients is denoted by the attenuation coefficient :
Two values related to the attenuation coefficient will be useful in the following. The first is the albedo, which is defined as
The albedo is always between 0 and 1; it describes the probability of scattering (versus absorption) at a scattering event. The second is the mean free path, , which gives the average distance that a ray travels in the medium before interacting with a particle.
Given the attenuation coefficient , the differential equation describing overall attenuation,
can be solved to find the beam transmittance, which gives the fraction of radiance that is transmitted between two points:
where is the distance between and , is the normalized direction vector between them, and denotes the beam transmittance between and . Note that the transmittance is always between 0 and 1. Thus, if exitant radiance from a point on a surface in a given direction is given by , after accounting for extinction, the incident radiance at another point in direction is
This idea is illustrated in Figure 11.7.
Two useful properties of beam transmittance are that transmittance from a point to itself is 1, , and in a vacuum and so for all . Furthermore, if the attenuation coefficient satisfies the directional symmetry or does not vary with direction and only varies as function of position (this is generally the case), then the transmittance between two points is the same in both directions:
This property follows directly from Equation (11.1).
Another important property, true in all media, is that transmittance is multiplicative along points on a ray:
for all points between and (Figure 11.8). This property is useful for volume scattering implementations, since it makes it possible to incrementally compute transmittance at multiple points along a ray: transmittance from the origin to a point can be computed by taking the product of transmittance to a previous point and the transmittance of the segment between the previous and the current point .
The negated exponent in the definition of in Equation (11.1) is called the optical thickness between the two points. It is denoted by the symbol :
In a homogeneous medium, is a constant, so the integral that defines is trivially evaluated, giving Beer’s law:
While out-scattering reduces radiance along a ray due to scattering in different directions, in-scattering accounts for increased radiance due to scattering from other directions (Figure 11.9).
Figure 11.10 shows the effect of in-scattering with the smoke data set. Note that the smoke appears much thicker than when absorption or emission was the dominant volumetric effect.
Assuming that the separation between particles is at least a few times the lengths of their radii, it is possible to ignore inter-particle interactions when describing scattering at a particular location. Under this assumption, the phase function describes the angular distribution of scattered radiation at a point; it is the volumetric analog to the BSDF. The BSDF analogy is not exact, however. For example, phase functions have a normalization constraint: for all , the condition
must hold. This constraint means that phase functions actually define probability distributions for scattering in a particular direction.
The total added radiance per unit distance due to in-scattering is given by the source term :
It accounts for both volume emission and in-scattering:
The in-scattering portion of the source term is the product of the scattering probability per unit distance, , and the amount of added radiance at a point, which is given by the spherical integral of the product of incident radiance and the phase function. Note that the source term is very similar to the scattering equation, Equation (5.9); the main difference is that there is no cosine term since the phase function operates on radiance rather than differential irradiance.