In equation 15 the unknown function I appears on both sides. Substituting the right side's I by , which is obviously I according to the equation, we get:
Repeating this step n times, the original equation can be expanded into a Neumann series:
If integral operator is contractive, that is if
then , thus
The contractive property of comes from the fact that a reflection or refraction always decreases the energy.
The terms of this infinite Neumann series have intuitive meaning as well: comes from the emission, comes from a single reflection, from two reflections, etc.
In order to study the structure of , let us consider the case of i=2:
To evaluate the integrand at point , the following algorithm must be executed:
This algorithm can easily be generalised for arbitrary number of reflections. In order to compute the contribution of the nth reflection of the light to , the following function should be integrated: A ray is emanated recursively from at direction then from the found surface at , etc. until . The emission intensity at the end of the walk is read and multiplied by the BRDFs and the cosine terms of the stages of the walk.
These walks provide the value of the integrand at ``point'' . The integral is estimated as the average of these values. Directions should be selected to minimise the error of the approximation.
Thus we can conclude that the solution of the rendering equation requires the evaluation of high-dimensional ingegrals.