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denotes the strength of specular relative to diffuse reflection, while the parameter
m
governs the width of the specular reflection. Setting
m
→∞
yields an ideal mirror.
The angle
θ
r
can be expressed in terms of the angles
θ
i
,
θ
e
, and
α
by
cos
θ
r
=
cos
α.
(3.11)
It is important to note that the Phong BRDF is not motivated by the physical
processes governing the reflection of light at real surface materials. As a conse-
quence, the Helmholtz reciprocity condition (
3.7
) is not fulfilled by the Phong
model.
A physically motivated BRDF model for rough surfaces is introduced by Tor-
rance and Sparrow (
1967
). It is described in the overview by Meister (
2000
)
that for many rough surfaces the specular reflection component does not reach
its peak exactly in the direction corresponding to
θ
e
=
2 cos
θ
i
cos
θ
e
−
θ
i
, as would be expected
from an ideal mirror, but that the maximum occurs for higher values of the emis-
sion angle
θ
e
. Torrance and Sparrow (
1967
) assume that the surface is composed
of a large number of microfacets which reflect the incident light in a mirror-like
manner. According to Meister (
2000
), the corresponding BRDF is given by the
weighted sum of a Lambertian component and a specular component
f
spec
TS
given
by
w
2
δ
2
)
F(θ
i
,θ
e
,φ,n,k)G(θ
i
,θ
e
,φ)
exp
(
−
f
spec
TS
(θ
i
,θ
e
,φ,n,k,w,δ)
=
.
(3.12)
cos
θ
i
cos
θ
e
In (
3.12
), the angle
φ
denotes the azimuth difference
(φ
e
−
φ
i
)
and
δ
the an-
gle of the surface normal of the microfacet with respect to the macroscopic
surface normal. The refraction index of the surface is assumed to be com-
plex and amounts to
n
ik
with
k
as the attenuation coefficient (cf. also
Sect.
3.4.2
). The expression
F(θ
i
,θ
e
,φ,n,k)
is the Fresnel reflectance (Hapke
1993
) and
G(θ
i
,θ
e
,φ)
+
∈[
]
the 'geometrical attenuation factor', which takes
into account shadows and occlusions. Meister (
2000
) points out that it is com-
monly possible to set
G(θ
i
,θ
e
,φ)
0
,
1
=
1, and he gives the analytical expres-
sion
min
1
,
2 cos
δ
cos
θ
e
cos
θ
i
,
2 cos
δ
cos
θ
i
cos
θ
i
G(θ
i
,θ
e
,φ)
=
(3.13)
as derived by Nayar et al. (
1991
) with
θ
i
as the local illumination angle of the sur-
face facet. It is furthermore stated that cos 2
θ
i
=
sin
θ
i
sin
θ
e
cos
φ
.
The model by Torrance and Sparrow (
1967
) that the orientations
δ
of the normals of
the surface facets have a Gaussian distribution proportional to exp
(
−
w
2
δ
2
)
, where
w
denotes a width parameter. An analytical expression for
δ
as a function of
θ
i
,
θ
e
,
and
φ
is given by Meister (
2000
).
Simple BRDF models like the Lambertian or the Phong BRDF are generally of
limited accuracy when used in the context of three-dimensional surface reconstruc-
tion. However, sometimes determination of the parameters of physically motivated
BRDFs such as the Torrance-Sparrow model is not possible or requires consider-
able experimental efforts. Hence, in the application scenario of industrial quality in-
spection described in detail in Chap.
6
(d'Angelo and Wöhler
2005a
,
2005b
,
2008
;
cos
θ
i
cos
θ
e
−
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