The surfaces of the scene to be rendered are each divided up into one or more smaller surfaces (patches). A view factor is computed for each pair of patches. View factors (also known as form factors) are coefficients describing how well the patches can see each other. Patches that are far away from each other, or oriented at oblique angles relative to one another, will have smaller view factors. If other patches are in the way, the view factor will be reduced or zero, depending on whether the occlusion is partial or total.
The view factors are used as coefficients in a linearized form of the rendering equation, which yields a linear system of equations. Solving this system yields the radiosity, or brightness, of each patch, taking into account diffuse inter-reflections and soft shadows.
Progressive radiosity solves the system iteratively in such a way that after each iteration, we have intermediate radiosity values for the patch. These intermediate values correspond to bounce levels. That is, after one iteration, we know how the scene looks after one light bounce, after two passes, two bounces, and so forth. Progressive radiosity is useful for getting an interactive preview of the scene. Also, the user can stop the iterations once the image looks good enough, rather than wait for the computation to numerically converge.
As the algorithm iterates, light can be seen to flow into the scene, as multiple bounces are computed. Individual patches are visible as squares on the walls and floor. Another common method for solving the radiosity equation is "shooting radiosity," which iteratively solves the radiosity equation by "shooting" light from the patch with the most error at each step. After the first pass, only those patches which are in direct line of sight of a light-emitting patch will be illuminated. After the second pass, more patches will become illuminated as the light begins to bounce around the scene. The scene continues to grow brighter and eventually reaches a steady state.
198 3.7.1 Mathematical formulation
The basic radiosity method has its basis in the theory of thermal radiation, since radiosity relies on computing the amount of light energy transferred among surfaces. In order to simplify computations, the method assumes that all scattering is perfectly diffuse. Surfaces are typically discretized into quadrilateral or triangular elements over which a piecewise polynomial function is defined.
After this breakdown, the amount of light energy transfer can be computed by using the known reflectivity of the reflecting patch, combined with the view factor of the two patches. This dimensionless quantity is computed from the geometric orientation of two patches, and can be thought of as the fraction of the total possible emitting area of the first patch which is covered by the second patch.
More correctly, radiosity B is the energy per unit area leaving the patch surface per discrete time interval and is the combination of emitted and reflected energy:
where:
1. B(x)i dAi is the total energy leaving a small area dAi around a point x.
2. E(x)i dAi is the emitted energy.
3. ρ(x) is the reflectivity of the point, giving reflected energy per unit area by multiplying by the incident energy per unit area (the total energy which arrives from other patches).
4. S denotes that the integration variable x' runs over all the surfaces in the scene 5. r is the distance between x and x'
6. θx and θx' are the angles between the line joining x and x' and vectors normal to the surface at x and x' respectively.
7. Vis(x,x' ) is a visibility function, defined to be 1 if the two points x and x' are visible from each other, and 0 if they are not.
The geometrical form factor (or "projected solid angle") Fij.
Fij can be obtained by projecting the element Aj onto a the surface of a unit hemisphere, and then projecting that in turn onto a unit circle around the point of interest in the plane of Ai. The form factor is then equal to the proportion of the unit circle covered by this projection.
Form factors obey the reciprocity relation AiFij = AjFji
199 If the surfaces are approximated by a finite number of planar patches, each of which is taken to have a constant radiosity Bi and reflectivity ρi, the above equation gives the discrete radiosity equation,
where Fij is the geometrical view factor for the radiation leaving j and hitting patch i.
This equation can then be applied to each patch. The equation is monochromatic, so colour radiosity rendering requires calculation for each of the required colours.
3.7.1.1 Solution methods
The equation can formally be solved as matrix equation, to give the vector solution:
This gives the full "infinite bounce" solution for B directly. However, the number of calculations to compute the matrix solution scales according to n3, where n is the number of patches. This becomes prohibitive for realistically large values of n.
Instead, the equation can more readily be solved iteratively, by repeatedly applying the single-bounce update formula above. Formally, this is a solution of the matrix equation by Jacobi iteration. The reflectivities ρi are less than 1, this scheme converges quickly, typically requiring only a handful of iterations to produce a reasonable solution. Other standard iterative methods for matrix equation solutions can also be used, for example the Gauss–Seidel method (jefferys, 1988), where updated values for each patch are used in the calculation as soon as they are computed, rather than all being updated synchronously at the end of each sweep. The solution can also be tweaked to iterate over each of the sending elements in turn in its main outermost loop for each update, rather than each of the receiving patches. This is known as the shooting variant of the algorithm, as opposed to the gathering variant. Using the view factor reciprocity, Ai Fij = AjFji, the update equation can also be re-written in terms of the view factor Fji seen by each sending patch Aj:
This is sometimes known as the "power" formulation, since it is now the total transmitted power of each element that is being updated, rather than its radiosity.