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 （吸收）
Emission—energy that is added to the environment from luminous particles （发光）
Scattering—how light heading in one direction is scattered to other directions due
to collisions(碰撞) with particles （散射）
Absorption
（一条光线经过 dt 长度的 participating medium，就会损失 （σa(p, ω) Li(p, −ω) dt） radiance ）
Consider thick(厚) black smoke froma fire: the smoke obscures(遮掩) the objects behind it because
its particles absorb light traveling from the object to the viewer. The thicker(厚) the smoke,
the more light is absorbed. Figure 11.2 shows this effect with a volume density 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.
Absorption is described by the medium’s absorption cross section, σa, 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 p and direction ω, although it
is normally just a function of position. It is usually also a spectrally varying quantity. The
units of σa are reciprocal distance (m−1). This means that σa can take on any positive
value; it is not required to be between zero and one, for instance.
Figure 11.3 shows the effect of absorption along a very short portion of a ray. Some
amount of radiance Li(p, −ω) is arriving at point p, and we’d like to find the exitant
radiance Lo(p, ω) after absorption in the differential volume. This change in radiance
along the differential ray length dt 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.
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 p from direction −ω. If the ray passes through a
differential(微分) cylinder filled with absorbing particles, the change in radiance due to absorption by those
particles is dLo(p, ω)=−σa(p, ω) Li(p, −ω) dt .
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 ω through the medium starting at point p, the total absorption is given by
Emission
（一条光线经过 dt 长度的 participating medium, 会增加 （Lve(p, ω) dt ） radiance ）
While absorption reduces the amount of radiance along a ray as it passes through a medium,
emission increases it, due to chemical(化学反应), thermal(热量), or nuclear(原子) 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 p in direction ω by
Lve(p, ω). Figure 11.5 shows the effect of emission in the smoke data set. In that figure
the absorption coefficient is much lower than in Figure 11.2, giving a very different
appearance.
Figure 11.4: The volume emission function Lve(p, ω) 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 dL = Lvedt .
The differential equation that gives the change in radiance due to emission is
Note that this equation is based on the assumption that the emitted light Lve is not dependent
on the incoming light Li. This is always true under the linear optics assumptions
that pbrt is based on.
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.
Out-Scattering
（一条光线经过 dt 长度的 participating medium, 可能会与 particles 碰撞，会导致 光线偏离了，
所以会导致 损失 （σs(p, ω) Li(p, −ω) dt ） radiance ）
The third basic light interaction in participating media is scattering. As a beam passes
through a medium, it may collide with particles in the medium 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(偏离) 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, σs. As with the attenuation coefficient, the reduction in radiance
along a differential length dt due to out-scattering is given by
Attenuation or Extinction
（这里主要是 考虑 absorption 和 out-scattering 一共造成多少损失，这里得到的总结主要是看图17，
当在一个点 P 反射出 Lo(p, ω) ，经过 participating medium之后 在 P‘ 得到的 Tr (p→p')Lo(p, ω)，
Tr 表示的就是 透射率，表示有百分之多少的光束是可以透射 participating medium）
The total reduction in radiance due to absorption and out-scattering is given by the sum
σa + σs. 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 σt:
Given the attenuation coefficient σt, the differential equation describing overall attenuation,
can be solved to find the beam(光束) transmittance, which gives the fraction of radiance that is
transmitted between two points on a ray:
where d is the distance between p and p', ω is the normalized direction vector between
them, and Tr denotes the beam transmittance between p and p'. Note that the transmittance
is always between zero and one. Thus, if exitant radiance froma point p on a surface
in a given direction ω is given by Lo(p, ω), after accounting for extinction, the incident
radiance at another point p' in direction −ω is
This idea is illustrated in Figure 11.7.
Figure 11.7: The beam transmittance Tr (p→p') 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 p in direction ω (e.g., reflected radiance from a surface), the radiance
visible at another point p' along the ray is Tr (p→p')Lo(p, ω).
Two useful properties of beam transmittance are that transmittance from a point to itself
is one (Tr(p→p) = 1), and in a vacuum Tr(p→p’) = 1 for all p‘. Another important
property, true in all media, is that transmittance is multiplicative along points on a ray:
for all points p’ between p and p‘’ (Figure 11.8). This property is important for volume
scattering implementations, since it makes it possible to incrementally compute
transmittance at many points along a ray by computing the product of each previously
computed transmittance with the transmittance for its next segment.
Figure 11.8: A useful property of beam transmittance is that it is multiplicative: the transmittance
between points p and p‘’ on a ray like the one shown here is equal to the transmittance from p to p‘
times(乘上) the transmittance from p‘ to p‘’ for all points p’ between p and p’‘.
The negated exponent in Tr is called the optical thickness between the two points. It is
denoted by the symbol τ :
（Tr 公式里面的 指数 表示的是 两个点之间的 厚度）
In a homogeneous medium, σt is a constant, so the τ integral is trivially evaluated and
yields Beer’s law:
In-Scattering
（很多光线经过 dt 长度的 participating medium, 可能会与 particles 碰撞，会导致 光线偏离了，光线有概率会 偏离到 wo 的方向，所以会导致 增加 （S(p, ω) dt） radiance ）
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.
Assuming that the separation between particles is at least a few times the lengths of their
radii, it is possible to ignore interparticle(粒子之间) interactions when describing scattering at a
particular location. Under this assumption, the phase function p(ω→ω’) 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 ω, the condition
must hold. This constraint means that phase functions actually define probability distributions
for scattering in a particular direction. （phase functions 其实就是一个 散射 到某一个方向 的 概率分布函数，理解起来类似 BRDF）
The total added radiance per unit distance due to in-scattering is given by the source
term S:
It accounts for both volume emission and in-scattering:
（公式中积分里面，个人理解就是，积分这个点的 所有入射光线，看看所有的入射光线 贡献多少 radiance 到这个点上的）
The in-scattering portion of the source term is the product of the scattering probability
per unit distance, σs, 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 (5.8); the main
difference is that there is no cosine term since the phase function operates on radiance
rather than differential irradiance.