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.

Figure 11.1: Spotlight through Fog. Light scattering from particles in the medium back toward the camera makes the spotlight’s illumination visible even in pixels where there are no visible surfaces that reflect it. The sphere blocks light, casting a volumetric shadow in the region beneath it.

11.1.1 Absorption

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.

Figure 11.2: If a participating medium primarily absorbs light passing through it, it will have a dark and smoky appearance, as shown here. (Smoke simulation data courtesy of Duc Nguyen and Ron Fedkiw.)

Absorption is described by the medium’s absorption cross section, sigma Subscript normal a , 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 normal p Subscript and direction omega Subscript , although it is normally just a function of position. It is usually also a spectrally varying quantity. The units of sigma Subscript normal a are reciprocal distance ( normal m Superscript negative 1 ). This means that sigma Subscript normal a 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 upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma minus omega Subscript Baseline right-parenthesis is arriving at point normal p Subscript , and we’d like to find the exitant radiance upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis after absorption in the differential volume. This change in radiance along the differential ray length normal d t is described by the differential equation

upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis minus upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma minus omega Subscript Baseline right-parenthesis equals normal d upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis equals minus sigma Subscript normal a Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma minus omega Subscript Baseline right-parenthesis normal d t comma

which says that the differential reduction in radiance along the beam is a linear function of its initial radiance.

Figure 11.3: Absorption reduces the amount of radiance along a ray through a participating medium. Consider a ray carrying incident radiance at a point normal p Subscript from direction minus omega Subscript . If the ray passes through a differential cylinder filled with absorbing particles, the change in radiance due to absorption by those particles is normal d upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis equals minus sigma Subscript normal a Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma minus omega Subscript Baseline right-parenthesis normal d t .

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 d in direction omega Subscript through the medium starting at point normal p Subscript , the remaining portion of the original radiance is given by

normal e Superscript minus integral Subscript 0 Superscript d Baseline sigma Super Subscript normal a Superscript left-parenthesis normal p Super Subscript Superscript plus t omega Super Subscript Superscript comma omega Super Subscript Superscript right-parenthesis normal d t Baseline period

11.1.2 Emission

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 normal p Subscript in direction omega Subscript by upper L Subscript normal e Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis .

Figure 11.4: The volume emission function upper L Subscript normal e Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis gives the change in radiance along a ray as it passes through a differential volume of emissive particles. The change in radiance per differential distance is normal d upper L Subscript Superscript Baseline equals upper L Subscript normal e Superscript Baseline normal d t .

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.

Figure 11.5: A Participating Medium Where the Dominant Volumetric Effect Is Emission. Although the medium still absorbs light, still casting a shadow on the ground and obscuring the wall behind it, emission in the volume increases radiance along rays passing through it, making the cloud brighter than the wall behind it.

The differential equation that gives the change in radiance due to emission is

normal d upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis equals upper L Subscript normal e Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis normal d t period

This equation incorporates the assumption that the emitted light upper L Subscript normal e Superscript is not dependent on the incoming light upper L Subscript normal i Superscript . 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.

Figure 11.6: Like absorption, out-scattering also reduces the radiance along a ray. Light that hits particles may be scattered in another direction such that the radiance exiting the region in the original direction is reduced.

The probability of an out-scattering event occurring per unit distance is given by the scattering coefficient, sigma Subscript normal s . As with absorption, the reduction in radiance along a differential length normal d t due to out-scattering is given by

normal d upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis equals minus sigma Subscript normal s Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma minus omega Subscript Baseline right-parenthesis normal d t period

The total reduction in radiance due to absorption and out-scattering is given by the sum sigma Subscript normal a Baseline plus sigma Subscript normal s . 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  sigma Subscript normal t :

sigma Subscript normal t Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis equals sigma Subscript normal a Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis plus sigma Subscript normal s Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis period

Two values related to the attenuation coefficient will be useful in the following. The first is the albedo, which is defined as

rho equals StartFraction sigma Subscript normal s Baseline Over sigma Subscript normal t Baseline EndFraction period

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, 1 slash sigma Subscript normal t , which gives the average distance that a ray travels in the medium before interacting with a particle.

Given the attenuation coefficient sigma Subscript normal t , the differential equation describing overall attenuation,

StartFraction normal d upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis Over normal d t EndFraction equals minus sigma Subscript normal t Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma minus omega Subscript Baseline right-parenthesis comma

can be solved to find the beam transmittance, which gives the fraction of radiance that is transmitted between two points:

upper T Subscript r Baseline left-parenthesis normal p Subscript Baseline right-arrow normal p prime right-parenthesis equals normal e Superscript minus integral Subscript 0 Superscript d Baseline sigma Super Subscript normal t Superscript left-parenthesis normal p Super Subscript Superscript plus t omega Super Subscript Superscript comma omega Super Subscript Superscript right-parenthesis normal d t
(11.1)

where d equals double-vertical-bar normal p Subscript Baseline minus normal p prime double-vertical-bar is the distance between normal p Subscript and normal p prime , omega Subscript is the normalized direction vector between them, and upper T Subscript r denotes the beam transmittance between normal p Subscript and normal p prime . Note that the transmittance is always between 0 and 1. Thus, if exitant radiance from a point normal p Subscript on a surface in a given direction omega Subscript is given by upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis , after accounting for extinction, the incident radiance at another point normal p prime in direction minus omega Subscript  is

upper T Subscript r Baseline left-parenthesis normal p Subscript Baseline right-arrow normal p Superscript prime Baseline right-parenthesis upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis period

This idea is illustrated in Figure 11.7.

Figure 11.7: The beam transmittance upper T Subscript r Baseline left-parenthesis normal p Subscript Baseline right-arrow normal p prime right-parenthesis gives the fraction of light transmitted from one point to another, accounting for absorption and out-scattering, but ignoring emission and in-scattering. Given exitant radiance at a point normal p Subscript in direction omega Subscript (e.g., reflected radiance from a surface), the radiance visible at another point normal p prime along the ray is upper T Subscript r Baseline left-parenthesis normal p Subscript Baseline right-arrow normal p Superscript prime Baseline right-parenthesis upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis .

Two useful properties of beam transmittance are that transmittance from a point to itself is 1, upper T Subscript r Baseline left-parenthesis normal p Subscript Baseline right-arrow normal p Subscript Baseline right-parenthesis equals 1 , and in a vacuum sigma Subscript normal t Baseline equals 0 and so upper T Subscript r Baseline left-parenthesis normal p Subscript Baseline right-arrow normal p prime right-parenthesis equals 1 for all normal p prime . Furthermore, if the attenuation coefficient satisfies the directional symmetry sigma Subscript normal t Baseline left-parenthesis omega Subscript Baseline right-parenthesis equals sigma Subscript normal t Baseline left-parenthesis minus omega Subscript Baseline right-parenthesis or does not vary with direction omega Subscript and only varies as function of position (this is generally the case), then the transmittance between two points is the same in both directions:

upper T Subscript r Baseline left-parenthesis normal p Subscript Baseline right-arrow normal p Superscript prime Baseline right-parenthesis equals upper T Subscript r Baseline left-parenthesis normal p prime right-arrow normal p Subscript Baseline right-parenthesis period

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:

upper T Subscript r Baseline left-parenthesis normal p Subscript Baseline right-arrow normal p Superscript double-prime Baseline right-parenthesis equals upper T Subscript r Baseline left-parenthesis normal p Subscript Baseline right-arrow normal p Superscript prime Baseline right-parenthesis upper T Subscript r Baseline left-parenthesis normal p prime right-arrow normal p Superscript double-prime Baseline right-parenthesis comma
(11.2)

for all points normal p prime between normal p Subscript and normal p double-prime (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 upper T Subscript r Baseline left-parenthesis normal o right-arrow normal p Subscript Baseline right-parenthesis can be computed by taking the product of transmittance to a previous point upper T Subscript r Baseline left-parenthesis normal o right-arrow normal p prime right-parenthesis and the transmittance of the segment between the previous and the current point upper T Subscript r Baseline left-parenthesis normal p prime right-arrow normal p Subscript Baseline right-parenthesis .

Figure 11.8: A useful property of beam transmittance is that it is multiplicative: the transmittance between points normal p Subscript and normal p double-prime on a ray like the one shown here is equal to the transmittance from normal p Subscript to normal p prime times the transmittance from normal p prime to normal p double-prime for all points normal p prime between normal p Subscript and  normal p double-prime .

The negated exponent in the definition of upper T Subscript r in Equation (11.1) is called the optical thickness between the two points. It is denoted by the symbol tau :

tau left-parenthesis normal p Subscript Baseline right-arrow normal p Superscript prime Baseline right-parenthesis equals integral Subscript 0 Superscript d Baseline sigma Subscript normal t Baseline left-parenthesis normal p Subscript Baseline plus t omega Subscript Baseline comma minus omega Subscript Baseline right-parenthesis normal d t period

In a homogeneous medium, sigma Subscript normal t is a constant, so the integral that defines tau is trivially evaluated, giving Beer’s law:

upper T Subscript r Baseline left-parenthesis normal p Subscript Baseline right-arrow normal p Superscript prime Baseline right-parenthesis equals normal e Superscript minus sigma Super Subscript normal t Superscript d Baseline period
(11.3)

11.1.4 In-scattering

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 11.9: In-scattering accounts for the increase in radiance along a ray due to scattering of light from other directions. Radiance from outside the differential volume is scattered along the direction of the ray and added to the incoming radiance.

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.

Figure 11.10: In-Scattering with the Smoke Data Set. Note the substantially different appearance compared to the other two smoke images.

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 p left-parenthesis omega Subscript Baseline comma omega prime Subscript Baseline right-parenthesis 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 omega Subscript , the condition

integral Underscript script upper S squared Endscripts p left-parenthesis omega Subscript Baseline comma omega prime Subscript Baseline right-parenthesis normal d omega Subscript Baseline Superscript prime Baseline equals 1
(11.4)

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  upper L Subscript normal s Superscript :

normal d upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis equals upper L Subscript normal s Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis normal d t period

It accounts for both volume emission and in-scattering:

upper L Subscript normal s Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis equals upper L Subscript normal e Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis plus sigma Subscript normal s Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis integral Underscript script upper S squared Endscripts p left-parenthesis normal p Subscript Baseline comma omega Subscript normal i Baseline comma omega Subscript Baseline right-parenthesis upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript normal i Baseline right-parenthesis normal d omega Subscript normal i Baseline period

The in-scattering portion of the source term is the product of the scattering probability per unit distance, sigma Subscript normal s , 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.