I: Exact vs approximate methods

As noted above, we cannot solve the equation exactly - there is always some error, although it can be made very small. In some rendering methods, the desired error is specified in advance by the user and it determines the accuracy of the calculations (f.e. GI sample density, or GI rays, or number of photons etc.). A disadvantage of these methods is that the user must wait for the whole calculation process to complete before the result can be used. Another disadvantage is that it may take a lot of trials and errors to find settings that produce adequate quality in a given time frame. However, the big advantage of these methods is that they can be very efficient within the specified accuracy bounds, because the algorithm can concentrate on solving difficult parts of the rendering equation separately (e.g. splitting the image into independent regions, performing several calculation phases etc.), and then combining the result.



In other methods, the image is calculated progressively - in the beginning the error is large, but gets smaller as the algorithm performs additional calculations. At any one point of time, we have the partial result for the whole image. So, we can terminate the calculation and use the intermediate result.
Exact (unbiased or brute-force) methods.

Advantages:

* Produce very accurate results.
* The only artifact these methods produce is noise.
* Renderers using exact methods typically have only few controls for specifying image quality.
* Typically require very little additional memory.



Disadvantages:

* Unbiased methods are not adaptive and so are extremely slow for a noiseless image.
* Some effects cannot be computed at all by an exact method (for example, caustics from a point light seen through a perfect mirror).
* It may be difficult to impose a quality requirement on these methods.
* Exact methods typically operate directly on the final image; the GI solution cannot be saved and re-used in any way.



Examples:

* Path tracing (brute-force GI in some rendereres).
* Bi-directional path tracing.
* Metropolis light transport.

Approximate (biased) methods:

Advantages:

* Adaptive, so typically those are a lot faster than exact methods.
* Can compute some effects that are impossible for an exact method (e.g. caustics from a point light seen through a perfect mirror).
* Quality requirements may be set and the solution can be refined until those requirements are met.
* For some approximate methods, the GI solution can be saved and re-used.



Disadvantages:

* Results may not be entirely accurate (e.g. may be blurry) although typically the error can be made as small as necessary.
* Artifacts are possible (e.g. light leaks under thin walls etc).
* More settings for quality control.
* Some approximate methods may require (a lot of) additional memory.



Examples:

* Photon mapping.
* Irradiance caching.
* Radiosity.
* Light cache in V-Ray.



Hybrid methods: exact methods used for some effects, approximate methods for others.

Advantages:

* Combine both speed and quality.



Disadvantages:

* May be more complicated to set up.



Examples:

* Final gathering with Min/Max radius 0/0 + photon mapping in mental ray.
* brute force GI + photon mapping or light cache in V-Ray.
* Light tracer with Min/Max rate 0/0 + radiosity in 3ds Max.
* Some methods can be asymptotically unbiased - that is, they start with some bias initially, but it is gradually decreased as the calculation progresses