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value averages out in the integral.
In other words, long BRDF lobes must be
appropriately thin.
The energy conservation requirement turns out to be a significant challenge in
the development of useful BRDF models. Many BRDF formulas in common use
do not conserve energy. Although such models can produce visually acceptable
results for local reflection, they can cause errors in global illumination computa-
tion that grow exponentially through the course of multiple reflections.
1.2.4 Calculation of Reflection
If a surface is illuminated only by a small or sufficiently distant light source, es-
sentially all the incoming radiance comes from the direction
ω of the light source.
In this case, the reflected radiance in an outgoing direction
ω
is the irradiance
from the source multiplied by the BRDF value:
· ω ) Φ
4
, ω , ω )(
L r (
x
, ω )=
f r (
x
n
r 2 ,
(1.10)
π
as follows from Equation (1.4), where
is the flux (intensity) of the source.
If indirect illumination is considered, the reflectance computation involves
integrating the incoming radiance against the BRDF in all directions ( Figure 1.6 ) .
The irradiance caused by incoming radiance L i (
Φ
, ω )
x
incident on a surface point
ω is given by L i (
, ω )(
· ω )
x from direction
x
n
. The total outgoing radiance in a
direction
ω
is thus
, ω , ω )(
· ω )
, ω )
ω .
(
, ω )=
(
(
L r
x
f r
x
n
L i
x
d
(1.11)
Ω
The conservation of energy requirement assures that the outgoing power does not
exceed the collected incoming power. Equation (1.11) is the fundamental expres-
L r ( x,ω )
L i ( x,ω' )
Hemisphere Ω
x
Figure 1.6
Calculation of reflection including indirect illumination must consider light from all direc-
tions on the hemisphere.
 
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