Graphics Reference
In-Depth Information
Chapter 30
Carlo Integration
In preparation for studying rendering techniques, we now discuss Monte Carlo
integration. We start with a rapid review of ideas from discrete probability theory,
and then generalize to continua like the real line or the unit sphere. We apply these
notions to describe how to generate random samples from various sets. We then
introduce Monte Carlo integration, treating all the basic ideas through the integra-
tion of a function on an interval
[
a
,
b
]
, where the ideas are easiest to understand.
We then show how these ideas apply to integration on a hemisphere or sphere, and
hence how they are used to find reflected radiance via the reflectance equation, for
instance.
We'll start with a high-level overview of the use of randomization in numer-
ical integration. Sometimes we need to integrate functions where computing
antiderivatives is impossible or impractical. For instance, the integrand in the
reflectance equation might not be described by an algebraic equation at all. In
these cases, numerical methods often are the only workable solution. Numerical
methods fall into two categories: deterministic and randomized. Here's a quick
comparison of the two.
A typical deterministic method (see Figure 30.1) for integrating a function
f
over an interval
[
a
,
b
]
is to take
n
+
1 equally spaced points in the interval
[
a
,
b
]
,
t
0
=
a
,
t
1
=
a
+
b
−
a
n
y
t
0
1
t
0
1
5
t
1
t
1
y
f
(
x
)
1
2
,
f
( )
2
2
t
i
+
t
i
+
1
2
of each of the
intervals these points define, sum up the results, and multiply by
b
−
n
, the width
of each interval. For sufficiently continuous functions
f
,as
n
increases this will
converge to the correct value. Similar methods for surface integrals, which divide
,
...
,
t
n
=
b
, evaluate
f
at the midpoint
x
a
5
t
0
t
1
t
n
2
1
b
5
t
n
Figure
30.1:
Integration
using
equal partitioning.
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