Virtually all modern GI renderers are based on the rendering
equation introduced by James T. Kajiya in his 1986 paper "The
Rendering Equation". This equation describes how light is propagated
throughout a scene. In his paper, Kajiya also proposed a method for computing
an image based on the rendering equation using a Monte Carlo method called
path tracing.
It should be noted that the equation has been known long before that in engineering
and has been used for computing radiative heat transfer in different environments.
However, Kajiya was the first to apply this equation to computer graphics.
It should also be noted that the rendering equation is only "an approximation
of Maxwell's equation for electromagnetics". It does not attempt to model
all optical phenomena. It is only based on geometric optics and therefore
cannot simulate things like diffraction, interference or polarization. However,
it can be easily modified to account for wavelength-dependent effects like
dispersion.
Another, more philosophical point to make, is that the rendering equation
is derived from a mathematical model of how light
behaves. While it is a very good model for the purposes of computer graphics,
it does not describe exactly how light behaves in the real world.
For example, the rendering equation assumes that light rays are infinitesimally
thin and that the speed of light is infinite - neither of these assumptions
is true in the real physical world.
Because the rendering equation is based on geometric optics, raytracing is
a very convenient way to solve the rendering equation. Indeed, most renderers
that solve the rendering equation are based on raytracing.
Different formulations of the rendering equation are possible, but the one
proposed by Kajiya looks like this:
where:
L(x, x1) is related to the light passing from point x1 to point x;
g(x, x1) is a geometry (or visibility term);
e(x, x1) is the intensity of emitted light from point x1 towards point
x;
r(x, x1, x2) is related to the light scattered from point x2 to point x
through point x1;
S is the union of all surfaces in the scene and x, x1 and x2 are points
from S.
What the equation means: the light arriving at a given point x in
the scene from another point x1 is the sum of the light emitted from
all other points x2 towards x1 and reflected towards x:
Except for very simple cases, the rendering equation cannot be solved exactly
in a finite amount of time on a computer. However, we can get as close as
we want to the real solution - given enough time. The search for global illumination
algorithms has been a quest for finding solutions that are reasonably close,
for a reasonable amount of time.
The rendering equation is only one. Different renderers only apply different
methods for solving it. If any two renderers solve this equation accurately
enough, then they should generate the same image for the same scene. This
is very well in theory, but in practice renderers often truncate or alter
parts of the rendering equation, which may lead to different results.
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.
V-Ray supports a number of different methods for solving the GI equation -
exact, approximate, shooting and gathering. Some methods are more suitable
for some specific types of scenes.
Exact methods
V-Ray supports two exact methods for calculating the rendering equation:
brute force GI and progressive path tracing. The difference between the two is that
brute force GI works with traditional image construction algorithms (bucket rendering)
and is adaptive, whereas path tracing refines the whole image at once and
does not perform any adaptation.
Approximate methods
All other methods used V-Ray (irradiance map, light cache, photon map) are
approximate methods.
Shooting methods
The photon map is the only shooting method in V-Ray. Caustics can also be
computed with photon mapping, in combination with a gathering method.
Gathering methods
All other methods in V-Ray (brute force GI, irradiance map, light cache) are gathering
methods.
Hybrid methods
V-Ray can use different GI engines for primary and secondary bounces, which
allows you to combine exact and approximate, shooting and gathering algorithms,
depending on what is your goal. Some of the possible combinations are demonstrated
on the GI examples page.
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