Graphics Reference
In-Depth Information
In (
3.52
),
I
c
denotes the intensity averaged over all orientations of the polarising
filter and
I
v
the intensity amplitude. The polarisation degree
D
p
is defined as
I
max
−
I
min
I
max
+
I
v
I
c
.
D
p
=
I
min
=
(3.53)
The rotation angle of the polarising filter for which maximum intensity
I
max
is ob-
served corresponds to the polarisation angle
Φ
. To determine the polarisation im-
age, at least three pixel-synchronous images of the surface are acquired. For each
pixel, (
3.52
) is fitted to the measured pixel pixel grey values (where Rahmann,
1999
suggests a computationally efficient linear optimisation approach), which yields the
parameters
I
c
,
I
v
, and
Φ
.
Atkinson and Hancock (
2005a
) reconstruct the orientation of dielectric surfaces
based on the Fresnel equations. In this scenario, the refraction index
n
i
of the exter-
nal medium, e.g. air, can be approximated as
n
i
=
1, while the refraction index of
the dielectric material is denoted by
n
t
. The angle
θ
i
is the incidence angle as de-
fined in Sect.
3.2.2.1
and
θ
t
the angle between the surface normal and the direction
of the light inside the material. The angles
θ
i
and
θ
t
are interrelated by Snellius's
refraction law according to
n
i
sin
θ
i
=
n
t
sin
θ
t
.
(3.54)
The Fresnel reflection coefficients are defined as
n
i
cos
θ
i
−
n
t
cos
θ
t
n
i
cos
θ
i
+
n
t
cos
θ
t
F
⊥
=
(3.55)
n
t
cos
θ
i
−
n
i
cos
θ
t
F
=
,
(3.56)
n
t
cos
θ
i
+
n
i
cos
θ
t
where (
3.55
) and (
3.56
) yield the fraction of the amplitude of the electric field of the
incident light and that of the light polarised orthogonal and parallel, respectively,
with respect to the plane defined by the surface normal
n
and the illumination direc-
tion
s
. Since the Fresnel coefficients denote amplitudes, the corresponding intensity
ratios are given by
F
2
⊥
and
F
2
.
Based on (
3.53
) together with
F
2
⊥
and
F
2
computed according to (
3.55
), Atkin-
son and Hancock (
2005b
) derive a relation between the incidence angle
θ
i
and the
polarisation degree
D
spec
for specular reflection, which yields for
n
i
=
1:
p
2sin
2
θ
i
cos
θ
i
n
t
F
2
F
2
sin
2
θ
i
−
⊥
−
D
spec
(θ
i
)
=
=
.
(3.57)
p
F
2
F
2
n
t
sin
2
θ
i
−
n
t
sin
2
θ
i
+
2sin
4
θ
i
⊥
+
−
Equation (
3.57
), however, is not valid for diffuse polarisation, which is due to
internal scattering of incident light penetrating the surface. Atkinson and Han-
cock (
2005b
) state that the light becomes partially polarised after entering into the
medium, random internal scattering processes then lead to a depolarisation of the
light, and then the light becomes partially polarised again once refraction into the
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