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D j =
Q R j
v i ) V ( P , Q ) (
v i · n P )(
v i · n Q )
f s ( P ,
v i ,
) E j ( Q ,
dQ
(32.13)
v
Q
P
2
v i ) V ( P , Q j ) (
v i ·
n P )(
v i ·
n ( Q j ))
Area ( R j ) f s ( P ,
v i ,
) E j ( Q j , E ,
,
(32.14)
v
Q j
P
2
where Q j is a single point chosen uniformly with respect to area on the region R j
that constitutes the j th source.
The second integral, representing the scattering of indirect light, can also be
split into two parts, by decomposing the function
v i
f s ( P ,
v i ,
) in the inte-
v
grand into a sum,
)= f s ( P ,
)+ f s ( P ,
f s ( P ,
) ,
v i ,
v
v i ,
v
v i ,
v
(32.15)
where f s represents the impulses like mirror reflection (and later, Snell's law
transmission), and f s is the nonimpulse part of the scattering distribution (i.e., f s is
a real-valued function rather than a distribution). Each impulse can be represented
by (1) a direction (the direction
v i such that
v i either reflects or transmits to
v
at P ), and (2) an impulse magnitude 0
k
1, by which the incoming radiance
in direction
.We'll
index these by the letter m (where m = 1 is reflection and m = 2 is transmission).
Thus, we can write
refl. indir. light =
v i is multiplied to get the outgoing radiance in direction
v
) L r ( P ,
f s ( P ,
v i ,
v i )
v i ·
n P d
v i ,
(32.16)
v
v i S + ( n P )
=
m
v m ) +
k m L r ( P ,
(32.17)
f s ( P ,
) L r ( P ,
v i ,
v i )
v i ·
n P d
.
(32.18)
v
v i
v i S + ( n P )
diffusely reflected indirect light
Finally, we can again estimate that last integral—the diffusely reflected indi-
rect light—by a single-sample Monte Carlo estimate: We pick a direction
v i
according to some probability density on the hemisphere (or the whole sphere,
when we're considering refraction as well as reflection), and estimate the integral
with
1
density (
v i ) f s ( P ,
) L r ( P ,
diff. refl. indir. light =
v i ,
v i )
v i ·
n P .
(32.19)
v
Note that while the BRDF doesn't literally make sense for an impulse-
like mirror reflection, the computation we perform to compute mirror-reflected
radiance has a form remarkably similar to that of Equation 32.19. We wrote it
(Equation 32.17) in the form
k 1 L ( P ,
v 1 ) ,
(32.20)
where
v 2 was the transmitted direction). The coeffi-
cient k 1 plays the same role as the coefficient
1
density (
v 1 was the reflection of
v i (
v i ) f s ( P ,
v i ,
v
)
| v i ·
n P |
(32.21)
 
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