Game Development Reference

In-Depth Information

N

ω
i
u
i

ω
r

d
ω
i

x

Figure 15.5
Geometry associated with the directional formula of the rendering equation

This residual function then can be effectively modelled by comparing with the

Blinn specular component.

δ
(
x
,
ω
r
,
ω
i
)

=

s
(
x
)

·

H

(15.6)

Here,
H
represents the bisector between
ω
r
and
ω
i
and
s
represents the term we want

to approximate. This residual function has several advantages. On one hand, its cal-

culation is very quick as a single scalar product is required and on the other hand the

term s that we want to approximate is a three-component vector and thus requires

only one 2D texture for storing it.

Modelling of the BRDF thus corresponds to the calculation, based on measured

samples, of a set of 2D textures encoding different parameters of the model. Thanks

to this formulation, this method can be integrated very easily in a method for real time

rendering.

15.2.2 Modelling the lighting

The final rendering of a scene in virtual reality consists of formally solving the rendering

equation (Kajiya, 1986):

L
r
(
x
,
ω
r
,
λ
)

=

L
e
(
x
,
ω
r
,
λ
)

+

f
r
(
x
,
ω
i
,
ω
r
,
λ
)
L
i
(
ω
i
,
λ
) cos
θ
i
dω
i

(15.7)

i

The equation of the rendering defines the luminance at a point as the sum of the actual

emission on this point and the reflection as per the BRDF of the incident luminance

on the upper hemisphere
i
=

H
2

+

. This expression considers that the surfaces are

opaque, but the same formula can be obtained with transparent surfaces by replacing

the BRDF with the BTDF and the upper hemisphere of integration with the lower

hemisphere. In addition, this is not the most general equation possible. It assumes

the light reflected by the surfaces at a point in time. Theoretically, it would have to

use the BSSRDF to be more general (internal scattering) and integrate on the sphere

(Veach, 1997).

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