Participating media

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

$$I(s)=I_0\exp\left(-\int_0^s{\tau(t)dt}\right)$$ (1)

which gives the light intensity $I(s)$ at distance $s$. $I_0$ is the light intensity at $s=0$ where the ray enters the volume. $\tau(t)$ 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 light-emitting media are hot particles in a flame. The amount of light which is emitted along a ray can be described by

$$I\left(s\right)=I_0+\int_0^s{g\left(t\right)dt}$$ (2)

where $g(t)$ is called the source term. $I_0$ 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

$$I\left(D\right)= I_0\exp{\left(-\int_0^D{\tau\left(t\ri... ...t(s\right)\exp{\left(-\int_s^D\tau\left(t\right)dt\right)}ds}$$ (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 so-called phase-functions. In order to understand phase-functions, 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 backward-scattering. 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:

$$S(X,\omega)=r(X,\omega,\omega')i(X,\omega')$$ (4)

where $i(X,\omega')$ is the incoming light reaching $X$ flowing in direction $\omega'$. $r(X,\omega,\omega')$ is the BRDF (bidirectional reflection distribution function) which describes which fraction of the light coming in from direction $\omega'$ to point $X$ is reflected in the direction of $\omega$. A rule especially suited for volume rendering is

$$r(X,\omega,\omega')=a(X)\tau(X)p(\omega,\omega')$$ (5)

where $a(X)$ is the particle albedo, $\tau(X)$ the extinction coefficient and $p$ is the phase function describing the directionality of the scattering. The term $S(X,\omega)$ can simply be added to the source term $g$

$$g(X) = e(X) + S(x)$$ (6)

where $e(X)$ is the direct emission at position $X$ and $S(X)$ the in-scattered light at position $X$. 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 shadow-feeler is sent to the lightsource for each point $X$ 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].