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viewing direction. We propose to use the well-known Phong model [20]. However,
note that this model is not based on physical laws, but comes from the computer
graphics community. Although empirical, it is widely used thanks to its simplicity,
and because it is appropriate for various types of materials, whether they are rough
or smooth. Note that other models could be considered such as the Blinn-Phong [3]
as reported in [5].
According to the Phong model (see Figure 5.1), the intensity
I
(
x
) at point
x
writes
as follows
I
(
x
)=
K
s
cos
k
α
+
K
d
cos
θ
+
K
a
.
(5.10)
This relation is composed of a diffuse, a specular and an ambient component and
assumes a point light source. The scalar
K
s
describes the specular component of the
lighting;
K
d
describes the part of the diffuse term which depends on the
albedo
in
P
;
K
a
is the intensity of ambient lighting in
P
. Note that
K
s
,
K
d
and
K
a
depend on
P
.
θ
is the angle between the normal to the surface
n
in
P
and the direction of the light
source
L
;
is the angle between
R
(which is
L
mirrored about
n
) and the viewing
direction
V
.
R
can be seen as the direction due to a pure specular object, where
k
allows to model the width of the specular lobe around
R
, this scalar varies as the
inverse of the roughness of the material.
In the remainder of this chapter, the unit vectors
i
α
,
j
and
k
correspond to the axis
of the camera frame (see Figure 5.1).
Considering that
R
,
V
and
L
are normalized, we can rewrite (5.10) as
I
(
x
)=
K
s
u
1
k
+
K
d
u
2
+
K
a
(5.11)
where
u
1
=
R
V
and
u
2
=
n
L
. Note that these vectors are easy to compute, since
we have
x
V
=
−
(5.12)
x
R
= 2
u
2
n
−
L
(5.13)
with
x
=(
x
,
y
,
1). In the general case, we consider the following dependencies
⎧
⎨
V
=
V
x
(
t
)
n
=
n
x
(
t
)
t
,
L
=
L
x
(
t
)
t
(5.14)
⎩
,
R
=
R
x
(
t
)
t
.
,
From the definition of the interaction matrix given in (5.4), its computation requires
to write the total derivative of (5.11)
I
=
kK
s
u
k
−
1
1
u
1
+
K
d
u
2
.
(5.15)
However, it is also possible to compute
I
as
I
=
I
x
+
I
t
=
I
L
x
v
+
I
t
∇
∇
(5.16)
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