:
I
(
i; j
)
1
N
N
X
k
=1
w
(
x
k
; y
k
)
L
(
i
+ 0
:
5 +
x
k
; j
+ 0
:
5 +
y
k
)
p
(
x
k
; y
k
)
(3)
This assumes a co ordinate system where a pixel (
i; j
) has unit area and is
centered at (
i
+ 0
:
5
; j
+ 0
:
5) as suggested by Heckbert[9].
But what is a goo d
p
? Does the analysis from the last section imply that
we should choose
p
=
w
? We are applying the same
p
to a large number
of integrals with dierent
L
, and saying that there is an average
< L
2
>
over a large number of pixels and images is not unreasonable because the
distribution of intensities is fairly uniform if taken over many viewing frames.
In other words, the bright spots will not favor particular places on the lm.
Once a
w
is chosen for ltering, implementation is straightforward pro-
vided that
w
is strictly positive (as it must b e if negative pixel colors are disal-
lowed). Points can b e chosen uniformly from [0
;
1]
2
and then a warping trans-
formation can b e applied to distribute the p oints according to
w
[32, 28 , 16].
For several practical and theoretical reasons[27 ] we have chosen the width
2 weighting function that is non-zero on (
x; y
)
2
[
?
1
;
1]
2
:
w
(
x; y
) = (1
? j
x
j
) (1
? j
y
j
) (4)
We generate random p oints with density equal to
w
by applying a transfor-
mation to a uniform random pair (
r
1
; r
2
)
2
[0
;
1]
2
. The transformed sample
point is just (
t
(
r
1
)
; t
(
r
2
), where the transformation function
t
is:
t
(
u
) =
8
<
:
?
1 +
p
2
u
if
u <
0
:
5
1
?
q
2(1
?
u
) if
u
0
:
5
4 The Two Forms of the Rendering Equation
In this section we describe how a distribution ray tracer uses Monte Carlo
integration to solve the rendering equation. This amounts to designing a
probability density function
p
. This is a solved, but tricky, problem for diuse
and ideal specular surfaces. However, it is not only an unsolved problem for
imperfect sp ecular surfaces, we show that the simple diuse and sp ecular
strategy do es not easily generalize to the imperfect specular case for certain
BRDF.
4
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