Game Development Reference
In-Depth Information
dA
N
x
N
u i
ω r
u i
2
x
x
ω i
x
Figure 15.6 Geometry associated with the surface formula of the rendering equation
An alternative formula, which can be called a surface formula, can be associated
to this directional formula of the rendering equation. In fact, the luminance incident on
a surface at a point x is the luminance emitted or reflected towards x by other surfaces
of the scene that are visible from x . We can thus estimate the total energy coming from
all x points that form the surfaces of the scene. Hence the equivalent formula of the
rendering equation:
cos θ r
f r ( x , ω i , ω r , λ ) L i ( x , ω i , λ ) cos θ i
2 dA
L r ( x , ω r , λ )
=
L e ( x , ω r , λ )
+
(15.8)
x
x
x
This expression gives rise to a purely geometrical term, estimated between the two
points x and x , function of the solid angle supported by dA
in x :
cos θ r cos θ i
g ( x , x )
=
i cos θ i =
(15.9)
2
x
x
In order to take into account the occlusion phenomena in the scene, we introduce the
visibility function v between the points x and x , equal to one if they are directly visible,
else zero. The final surface expression we get is:
f r ( x , ω i , ω r , λ ) L i ( x , ω i , λ ) g ( x , x ) v ( x , x ) dA
L r ( x , ω r , λ )
=
L e ( x , ω r , λ )
+
(15.10)
x
These two versions of the rendering equation are equivalent, but the first is more
suitable for an image-oriented approach, as we know the useful directions thanks to
the sensor's position. Whereas the second version is naturally suitable for the scene
approach because it requires reviewing all the existing possibilities of receiving the
energy, i.e. all the surfaces of the scene. The choice of approach for simulating the
lighting will thus depend on the space in which we want to calculate this lighting.
The calculation of lighting in the space of an image is the method most commonly
used in virtual reality. Several reasons can be provided to explain this choice. On one