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
)

=

dω
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

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