Graphics Reference
In-Depth Information
k
t
k
,
∞
(
1
+
t
)
α
=
(31.8)
k
=
0
where
k
=
α·
(
α−
1
)
·...·
(
α−
k
+
1
)
(31.9)
k
!
is defined for any
real
number
α
and for
k
=
0, 1,
...
. We'll be applying this to
t
=
1
50
, so that evaluating Equation 31.8
will give us the value of the solution in Equation 31.7.
To do so requires summing an infinite series, however. The great insight is
the realization that the sum of an infinite series can be
estimated
by looking at
individual elements of the series.
1
37
the case
α
=
2.1
and
t
=
−
50
, so that 1
−
We now lay the foundations for all the Monte Carlo approaches to rendering,
starting with a few simple applications of probability theory.
31.5.1.1 Finite Series
Suppose that we have a
finite
series
A
=
a
1
+
a
2
+
a
3
+
...
+
a
20
,
(31.10)
and we want to estimate the sum,
A
. We can do the following: Pick a random inte-
ger
i
between 1 and 20 (with probability 1
20 of picking each possible number),
and let
X
=
20
a
i
. Then
X
is a random variable. Its expected value is the weighted
average of its values, with weights being the probabilities, that is,
/
E
[
X
]=(
1
/
20
)(
20
a
1
)+(
1
/
20
)(
20
a
2
)+
...
+(
1
/
20
)(
20
a
20
)
(31.11)
=
a
1
+
a
2
+
...
+
a
20
(31.12)
=
A
.
(31.13)
We've got a random variable whose expected value is the sum we're seeking!
By actually taking samples of this random variable and averaging them, we can
approximate the sum.
Inline Exercise 31.1:
Suppose that all 20 numbers
a
1
,
a
2
,
are equal. What's
the variance of the random variable
X
? How many samples of
X
do you need
to take to get a good estimate of
A
in this case?
...
In general, the variance of
X
is related to how much the terms in the sequence
vary: If all the terms are identical, then
X
has no variance, for instance. It's also
related to the way we
chose
the terms, which happens to have been uniform, but
we'll use nonuniform samples in other examples later. When we apply these ideas
to rendering, we will end up sampling among various paths along which light can
travel; the value being computed will be the light transport along the path. Since
some paths carry a lot of light (e.g., a direct path from a light source to your eye)
and some carry very little, a large variance is present; to make estimates accurate
