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:
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:
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.