The global illumination problem is formulated by the rendering equations. This Fredholm type integral equation can be solved by the classical radiosity or finite-element method. Since this method requires a meshing, the complexity of the scene is limited and the approximation of surfaces can produce accumulated errors. On the other hand specular reflections or refractions cannot be handled by the classical radiosity at all. A promising alternative for modelling all the possible light-object interactions, is the application of hybrid algorithms which combine the radiosity and ray-tracing methods. These algorithms are typically based on particle tracing, where particles perform a random walk through the scene. In order to estimate the colour of the pixels according to the contributions of the particles, the Monte Carlo integration methods are usually applied. The variance of the estimation can be reduced by using the importance sampling technique.
The next section reviews the derivation of the rendering equation, in the third section the random walk method is introduced, the fourth and fifth sections discuss the Monte Carlo and quasi-Monte Carlo integration techniques, the sixth one reviews the related work and finally some hybrid rendering methods are introduced.