The radiosity method is based on the numerical solution of the shading
equation by the finite element method.
It subdivides the surfaces into small elemental surface patches.
Supposing these patches are small, their intensity distribution over the
surface can be approximated by a constant value which depends on the surface
and the direction of the emission.
We can get rid of this directional dependency if only diffuse surfaces are
allowed, since diffuse surfaces generate the same intensity in all
directions. This is exactly the initial assumption of the simplest radiosity
model, so we are also going to consider this limited case first.
Let the energy leaving a unit area of surface i in a unit time in all
directions be 
,
and assume that the light density is homogeneous over the surface. This
light density plays a crucial role in this model and is also called the
radiosity of surface i.
The dependence of the intensity on
can be expressed by the following argument:
,
while the flux in the solid angle
is
if
is the angle between the surface normal and the direction concerned.
since 
.
Consider the energy transfer of a single surface on a given wavelength.
The total energy leaving the surface ( 
) can be divided into its
own emission and the diffuse reflection of the radiance coming from other
surfaces (figure 1.1).
The emission term is 
if 
is the emission density which is also assumed to be constant on the surface.
The diffuse reflection is the multiplication of the diffuse coefficient
and that part of the energy of other surfaces which actually reaches surface
i. Let 
be a factor, called the form factor, which determines that fraction
of the total energy leaving surface j which actually reaches surface
i.
Considering all the surfaces, their contributions should be integrated, which leads to the following formula of the radiosity of surface i:
Before analyzing this formula any further, some time will be devoted to the meaning and the properties of the form factors.
The fundamental law of photometry (equation 
)
expresses the energy transfer between two differential surfaces if they are
visible from one another. Replacing the intensity by the radiosity using
equation 1.1, we get:
If 
is not visible from 
, that is, another surface is obscuring it
from or it is visible only from the ``inner side'' of the surface, the
energy flux is obviously zero. These two cases can be handled similarly if
an indicator variable 
is introduced:
Since our goal is to calculate the energy transferred from one finite
surface
( 
) to
another ( 
)
in unit time, both surfaces are divided into infinitesimal elements and
their energy transfer is summed or integrated, thus:
By definition, the form factor
is a fraction of this energy and the total energy leaving surface j
( 
):
It is important to note that the expression of
is symmetrical with the exchange of i and j, which is known as the
reciprocity relationship:
We can now return to the basic radiosity equation. Taking advantage of the homogeneous property of the surface patches, the integral can be replaced by a finite sum:
Applying the reciprocity relationship, the term
can be replaced by
:
Dividing by the area of surface i, we get:
This equation can be written for all surfaces, yielding a linear equation
where the unknown components are the surface radiosities ( ):
or in matrix form, having introduced matrix 
:
( 
stands for the unit matrix).
The meaning of 
is the fraction of the energy reaching the very same
surface. Since in practical applications the elemental surface patches are
planar polygons, is 0.
Both the number of unknown variables and the number of equations are
equal to the number of surfaces (N). The solution of this linear
equation is, at least theoretically, straightforward (we shall consider its
numerical aspects and difficulties later). The calculated
radiosities represent the light density of the surface on a given
wavelength. Recalling Grassman's laws, to generate color pictures at least
three independent wavelengths should be selected (say red, green and blue),
and the color information will come from the results of the three different
calculations.
Thus, to sum up, the basic steps of the radiosity method are these:
form factor calculation.
)
on the representative wavelengths, or in the simplified case on the wavelength of red, green and
blue colors. Solve the linear equation for each representative wavelength,
yielding 
, 
... 
.
Constant color of surfaces results in the annoying effect of faceted objects, since the eye psychologically accentuates the discontinuities of the color distribution. To create the appearance of smooth surfaces, the tricks of Gouraud shading can be applied to replace the jumps of color by linear changes. In contrast to Gouraud shading as used in incremental methods, in this case vertex colors are not available to form a set of knot points for interpolation. These vertex colors, however, can be approximated by averaging the colors of adjacent polygons (see figure 1.2).
Note that the first two steps of the radiosity method are independent of the actual view, and the form factor calculation depends only on the geometry of the surface elements. In camera animation, or when the scene is viewed from different perspectives, only the third step has to be repeated; the computationally expensive form factor calculation and the solution of the linear equation should be carried out only once for a whole sequence. In addition to this, the same form factor matrix can be used for sequences, when the lightsources have time varying characteristics.
Szirmay-Kalos Laszlo
