Scenes with Analytic Solutions

One of the benefits of having benchmarking scenes and images is that it can help verify that an algorithm is computing correct solutions. One technique that can help with this task is to use environments that have easily computed analytic solutions. The simplest analytic test case is to compute the direct transfer from an emitting polygon to a receiving plane. This can be compared at each point with the analytic differential area to area form factor. As analytic unoccluded direct illumination is often built into global illumination algorithms, it does not always test anything useful.

Another simple type of test environment for global illumination is the furnace. The name for this class comes from a property of a furnace in thermal equilibrium. In this situation, at every point and direction the emission at each wavelength exactly cancels out the absorption at that wavelength, and the interior has a completely uniform appearance, causing all geometric features to disappear. We can create a version of this for global illumination with an enclosed environment where every surface has the same diffuse reflectivity $\rho$ and emitted radiance L_e. Now the solution at any point is

$$L(x) = \frac{1}{1 - \rho} L_e.$$ (1)

The simplest proof of this involves simulating a naive path tracer. At each bounce, L_e at the current point is weighted by the current ray weight, and the current ray weight is scaled by $\rho$. This gives the contribution for any infinite path as

$$L_{\rm path} = \sum_{d=0}^{\infty} \rho^d L_e.$$ (2)

A slight modification on this is to include any object that absorbs and emits no energy into the scene (an perfect, but not necessarily specular, reflector). This object simply changes the direction of a path based on its BRDF (perfect importance sampling). This bounce can be ignored for the purpose of counting the depth, as it has no effect on the accumulated weight or returned luminance. Therefore the above path tracing proof still holds.

Although furnaces allow analytic solutions for complex environments, the path tracing proof shows a flaw for using them to verify a brute force path tracer. Since any ray in any direction contributes the same amount to the calculation, the path tracer may be distributing rays incorrectly without that showing up as an error in the solution. These environments do work very well for testing finite element approaches.

The property that allows the basic furnace to be evaluated analytically is that every point has the same incident radiance distribution at every bounce, therefore the solution simply involves calculating the infinite sum. This property can be generalized by looking at environments partitioned into n subsets P_i where all points in P_i have exactly the same view of the world on bounce d. This creates a simple finite element systems of low dimension. These are easiest to find when n = 2, such as a pair of infinite planes or concentric spheres each with different BRDF's and emissions.