Graphics Reference
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3.3.5 Solving the LTE: Radiosity for Volumes
A solution to the light transport equation can be obtained by integrating the equa-
tion for a fixed direction over a fixed interval. In CG rendering, the fixed direction
is normally that of the viewpoint, as the goal is ultimately to compute the color
of a pixel. The integral of the LTE in the view direction is known as the volume
rendering equation , being a kind of volume extension to the rendering equation.
The term is confusing though, because “volume rendering” methods may have
nothing to do with the volume rendering equation. However, numerical meth-
ods are necessary to solve the equation, as it generally has no analytical solution.
Some of the same numerical techniques used in volume rendering can be used in
solving (approximating) the volume rendering equation.
For example, the finite-element method can be applied to the volume render-
ing equation in a manner analogous to the radiosity method for surface rendering.
In the surface-based radiosity method, each surface is split into small patches of
constant radiosity. Under the assumption of purely diffuse reflection, light trans-
fer between patches is modeled as a linear expression. The interaction between
all patches becomes a system of linear equations, thereby discretizing the (sur-
face) rendering equation. The “volumetric radiosity method” works analogously:
the entire environment is split into voxels, under the additional assumption that
all scattering is isotropic. Light transport between voxels is linear, so the volume
rendering equation becomes a system of linear equations.
Holly Rushmeier and Kenneth E. Torrance presented the first general solution
of the LTE for CG rendering in a paper entitled “The Zonal Method for Calculat-
ing Light Intensities in the Presence of a Participating Medium” [Rushmeier and
Torrance 87]. The method described in the paper extends the ordinary (surface-
to-surface) radiosity method to include volume elements. The volume containing
the participating medium is split into voxels (zones), and form factors (analogous
to surface patch form factors) representing light transfer between the voxels are
computed. A notable aspect of the method is that the volume elements are in-
corporated into the surface-to-surface radiosity form factors: the geometric form
factors include surface-to-volume and volume-to-surface as well as volume-to-
volume interaction. As a result, the entire radiative transfer reduces to a single
system of linear equations.
The solution of the equations consists of the constant radiosity of the surface
patches, and the volumes (the volume “radiosity” is the radiant power emitted and
scattered by the volume). Like the surface radiosity method, a reconstruction step
is needed to produce a visually acceptable rendered image. This step also requires
integrating over the rays to the eye in order to determine the collected in-scattered
and attenuated light.
In Rushmeier and Torrance's method, this integration is
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