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Participating media may absorb, emit and/or scatter light.
The simplest participating medium only absorbs light. That means that light
passing through the medium is attenuated depending on the density of the
medium. Max [Max95] derives the following equation for a light
absorbing participating medium

(1) 
which gives the light intensity at distance . is the light
intensity at where the ray enters the volume. denotes the
extinction coefficient at position t in the medium, which gives the fraction
of light that is absorbed rather than let through.
A good example for lightemitting
media are hot particles in a flame. The amount of light which is emitted
along a ray can be described by

(2) 
where is called the source term. is the amount of light which
enters the medium. The integrated emission along the ray is simply added to
the light that entered the medium from the outside.
Real particles both absorb and emit light, so
the equations for absorption and emission have to be
combined. Max derives an equation which gives the intensity at the eye

(3) 
Because participating media consist of small particles, light is not just
reflected or refracted by the medium, but scattered. That means that at
arbitrary points, light is scattered in different directions. The way light is
scattered is defined by socalled phasefunctions. In order to understand
phasefunctions, another term has to be defined. The particle albedo of a
participating medium gives the fraction of the
extinction which represents scattering rather than absorption. Clouds or snow,
for example, have a very high albedo and therefore appear very bright. Soot, in
contrast, has a very low albedo and therefore it appears very dark.
Phase functions describe the way light is scattered by a participating medium.
They return the fraction of light which is scattered from the lightsource into
the eye. Two different classes of phase functions can be distinguished 
isotropic and anisotropic phase functions. In an isotropic medium, light is
scattered uniformely in all directions, whereas in an anisotropic medium,
scattering depends on the angle between the incident and outgoing direction
of light. I.e. certain kinds of fog tend to scatter more light back to the
lightsource than in the forward direction. This phenomenon is called
backwardscattering.
In a medium with low albedo and low density it is unlikely that a ray of light
is scattered more than once before leaving the medium.
Therefore it is sufficient to consider only light that is scattered from the
light source directly into the eye.
In the simplest approach it is assumed that light reaches
the particles from a distant lightsource (or lightsources) and is not blocked
by objects or absorbed by the participating medium. Max gives a general
shading rule for this approach:

(4) 
where is the incoming light reaching flowing in direction
.
is the BRDF (bidirectional reflection
distribution function) which describes which fraction of the light coming in
from direction to point is reflected in the direction of
. A rule especially suited for volume rendering is

(5) 
where is the particle albedo,
the extinction coefficient and is the
phase function describing the directionality of the scattering.
The term can simply be added to the source term

(6) 
where is the direct emission at position and the
inscattered light at position . If the source term is defined this way,
equation 3 can be used to handle direct
emission as well as scattering.
The above approach is quite simple, but does not account for shadows.
Clouds, for example, often appear darker on the side which is opposite to the sun,
because the clouds itself absorb light and shadow themselves from the sun.
In order to handle shaded scattering, equation
3 has to be refined. Max [Max95]
presents a solution, where a shadowfeeler is sent to the lightsource for each
point along the primary ray. Then, the amount of incoming light at each of
these points along the primary ray is diminished using the absorption value along
the shadow feeler.
To render even more accurate images, multiple scattering effects have to be
taken into account. This means, that light is scattered more than once
before it reaches the eye. In participating media with high albedo, like clouds,
the influence of multiple scattering cannot be ignored. Modelling multiple
scattering is a very demanding task  the problem is comparable to the radiosity
problem, but instead of surfaces which can receive light from all other
surfaces, volume elements receive light from all other volume elements.
In order to calculate multiple scattering effects,
different methods have been presented to calculate approximate
solutions [RT87],
[KV84], [Max95], [Sta95].
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Up: Fundamentals of Volume Tracing
Previous: Transfer Functions
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