Programming Assignment 2 - Monte Carlo Techniques
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Due date: 8/06/2004.

The goal of this exercise is to get some hands-on experience with Monte
Carlo integration in the context of global illumination.

More specifically, your assignment is to implement a Monte Carlo based
"final gather" post-process, which follows the hierarchical radiosity
global pass that you implemented in the first programming assignment.

As you have noticed, the global illumination solution produced by the
global pass stage of hierarchical radiosity is not suitable for direct
display of high quality images. The radiosity function is approximated
in a piecewise-constant manner and there are various artifacts around
shadow boundaries.

One way to generate a high quality shaded image after the global
distribution of radiosity has been computed is by tracing rays from
the camera via each pixel. After identifying the surface visible by
each eye ray, an integration is performed in order to gather light
from the rest of the environment to the visible point. This is not
possible in ordinary ray tracing, since we have no information about
the radiance on other surfaces in the scene, but it is possible to do
after a radiosity pass, since we have an approximation of the radiance
at each and every point in the scene.

The problem is that you would like to use a limited budget of rays to
shade each point, so the question is how to perform the integration
efficiently. Here are three ways for you to implement and compare:

1. The naive way - sample uniformly the hemisphere about the normal at
the point of interest. Minor improvements to this strategy would be to
use stratification of the hemisphere, or alternatively generate the
rays according to a cosine distribution, rather than a uniform one.

2. Simple importance sampling - a crude form of importance sampling will
integrate the primary light sources, separately from the other surfaces
in the scene. You need to decide which fraction of your ray budget will
be spent in each of these two integrals, then compute each of them
separately and add the results.

3. A more sophisticated importance sampling scheme will attempt to make
smarter use of the information about the irradiance at each point of
interest by looking at the links contributing to the element containing
the point, and all of its ancestors in the hierarchy. From each of these
links we can obtain a fairly good estimate of the relative importance
of the source on the other end of the link to the receiving point.
Design a sampling strategy that uses this information to get cleaner
pictures faster than in the two previous methods.

Bonus:

4. Diffuse scenes are just so boring... Make a floor, or some other
surfaces in your scene(s) reflect the rest of the environment by recursively
tracing reflected rays if they hit a reflective surface. This effectively
turns your program into a hybrid radiosity/ray tracing renderer. It isn't
physically correct, but makes nicer pictures.