# The Rendering Equation

In 1986 Kajiya introduced an appropriate equation from thermodynamics into the formula that generally today serves as the starting point for all lighting models [103]. The emitted radiance L of a point on the surface at a place x in a direction in space J is given by

L(x, J) = Le(x, J)+ fr (x, J,Jr) L(x’,lu’) G(x, x) dA’, (9.5)

Js

where we have to integrate over all surface points x’ of all surfaces dA’ of the viewed scene S, which also can be seen as an integration over the hemisphere surrounding the object. The term J’ denotes the direction vector that points from x’ to x. For the solution of the integral we furthermore assume energy equilibrium, i. e., a condition in which the energy input and output are in an equilibrium state for the entire scene and also for each subspace of the scene.

The term Le(x, J) in Eqn. (9.5) is the radiance of the surface at place x in the direction J. If the examined surface shines – meaning it is a light source – this will be noted here, otherwise Le = 0 applies.

Chapter 9 The function fr (x, ш, ш’) is the so-called bidirectional reflection distribution Rendering function (BRDF) of the material at the position x. It indicates which part of the

radiance in point x is emitted in the direction Ш, if it is illuminated from the direction ш’. In this function the material properties of the surface at the point x are contained. If this, for example, is a mirror, then radiance is only emitted in the direction in which the reflection vector v points due to the vector ш’. The function fr thus turns into the delta function fr = 5(ш — v). If the surface consists of a purely diffuse material, the radiance is scattered independently of ш equally in each direction. It thus applies for fr = const. The BRDF was approximated in the local lighting model by the ambient, diffuse, and specular light terms.

The function L(x’, U’) describes the outgoing radiance of the observed point x’ of another object in the scene in the direction of x, and is thus responsible for the global energy exchange.

The geometry term G(x, x’) finally considers whether the points x and x’ are blocked by an object lying between them, and, if not, how the associated sur­faces are positioned. If the surfaces face each other, a larger part of the light is transferred from one surface to the other surface, as if they were tilted to each other. Finally, also the distance is taken into consideration, which likewise af­fects the brightness.

The problem with Eqn. (9.5) is the term L(x’, U’); it is found on both sides of the equation, once under the integral. Hence, we are dealing with an inte­gral equation, which usually cannot be solved analytically and thus must be approximated numerically.

Updated: October 4, 2015 — 7:24 am