Radiosity (computer Graphics) - Mathematical Formulation

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:

• B(x)_i dA_i is the total energy leaving a small area dA_i around a point x.
• E(x)_i dA_i is the emitted energy.
• 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).
• S denotes that the integration variable x' runs over all the surfaces in the scene
• r is the distance between x and x'
• θ_x and θ_x' are the angles between the line joining x and x' and vectors normal to the surface at x and x' respectively.
• 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.

If the surfaces are approximated by a finite number of planar patches, each of which is taken to have a constant radiosity B_i and reflectivity ρ_i, the above equation gives the discrete radiosity equation,

where F_ij 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 color radiosity rendering requires calculation for each of the required colors.