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In-Depth Information
possible light paths, or else take a great many samples to get a good estimate (i.e.,
a low-noise image).
What's the difference between a path tracer and a conventional ray tracer?
Well, the path-tracing result tends to be noisy, as we mentioned earlier: We're
making Monte Carlo estimates of the radiance at each point, and there's vari-
ance in these estimates. A basic ray tracer that only computes recursive rays when
there's a mirror reflection uses a very
low
-variance estimate of the diffusely scat-
tered indirect light: It estimates it as zero! That makes the basic ray-traced image
darker than it should be, but uncluttered with noise. On the other hand, by taking
more samples in the path tracer, we can reduce the noise a lot. For a small ray-
casting budget (i.e., you can only afford to cast a certain number of rays in your
scene), the simple ray-tracer result is wrong but nice looking; the path-traced result
is generally “right on average,” but noisy. As the ray-casting budget increases, the
ray-traced result does not really improve (except for deeper levels of reflection),
while the path-traced result gets less and less noise, and correctly includes dif-
fusely scattered indirect light.
Finally, we've treated “measurement” as part of the wrapper for a path tracer,
but we could instead include it in the thing being computed, so rather than esti-
mating
L
, we could estimate
L
multiplied by the measurement function
M
. When
we do so, the thing we're integrating, expanded out recursively, is a product of
some number of scattering functions and cosines, an
L
e
at the end, and
M
at the
beginning. (If we're integrating with respect to area rather than solid angle, there
will also be some change-of-variables factors.) There's a symmetry in this formu-
lation: We could swap the roles of
M
and
L
e
, and imagine rendering a scene in
which the eye was emitting light according to
M
, and it was being measured at the
light sources using
L
e
as the measurement function. The integral we'd write down
for estimating light transport in this “swapped” scene would be exactly the same
as in the original. This provides some theoretical justification for the “trace rays
from the eye” approach rather than tracing photons from the lights, the way nature
does it: The integrals we're estimating are the same.
When we consider alternative ways to sample from the space of light paths, we
may find one sampling strategy that is effective for one class of paths and another
that works well for a different class. It's difficult to know in advance which will be
useful in any particular scene. Veach developed
multiple importance sampling
as a way to use many different sampling strategies in evaluating a single inte-
gral, by weighting samples from the different strategies differently. He describes
a motivating example: a single glossy surface reflecting an area light source (e.g.,
think of a slightly rippled ocean reflecting moonlight). In this case, there are only
two interesting kinds of paths:
LE
and
LDE
; if the lights are outside the visible part
of the scene, then there are only
LDE
paths. There are no paths of length greater
than two in the scene. This makes the analysis particularly easy. We'll look at only
the
LDE
paths.
Suppose that we try to estimate the light arriving at some pixel,
P
.One
approach to sampling paths is that we trace a ray through
P
, meeting the glossy
surface at a point
x
. Then we sample from the BRDF at
x
and trace a second ray
that may hit the area light source, in which case the sample contributes some radi-
ance, or that may miss the light altogether, in which case the sample contributes
