Rendering equation revisited: how to avoid explicit visibility computations

The rendering integral equation was introduced by Kajiya [16] to model the equilibrium of radiant energy exchange in a 3D scene with non partecipating medium and surfaces exhibiting a variety of local illumination effects including diffuse, glossy and specular reflectance. This equation has been extensively applied in the area of photorealistic rendering, specially in its radiosity form. In this paper we first derive a new form of such an integral equation and then we apply to the new equation the radiosity solution technique. There are two main novelties with respect to the previously known radiosity treatments. First of all we avoid any explicit global visibility computation by simulating visibility as a superimposition of independent effects at the level of the integral equation. The calculations do not need to carry around any explicit visibility information; nor such visibility information is kept in expensive auxiliary data structures. Secondly, by use of integral geometric transformations, we reduce the six-dimensional integrals representing coefficients of the radiosity matrix to two-dimensional ones. We show that such integrals can then be reduced to a form for which a result in [25] gives an high precision numerical approximation algorithm. The technique presented works for any piecewise polynomial functional basis adopted in generalized radiosity discretization and places mild conditions on the other functions of the model.