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wrinkled surfaces, where using (
3.20
) leads to an unsatisfactory result, it can be re-
placed by the departure from integrability error term (
3.25
) as discussed in detail in
Sect.
3.2.3
.
In our scenario, the incident light is unpolarised. For smooth metallic sur-
faces the light remains unpolarised after reflection at the surface. Rough metal-
lic surfaces, however, partially polarise the reflected light, as shown e.g. by Wolff
(
1991
). When observed through a linear polarisation filter, the reflected light has
a transmitted radiance that oscillates sinusoidally as a function of the orienta-
tion of the polarisation filter between a maximum
I
max
and a minimum
I
min
.The
polarisation angle
Φ
0
◦
,
180
◦
]
∈[
denotes the orientation under which the maxi-
mum transmitted radiance
I
max
is observed. The polarisation degree is defined by
D
=
(I
max
−
I
min
)/(I
max
+
∈[
]
(cf. Sect.
3.4.2
for details). Like the re-
flectance of the surface, both polarisation angle and degree depend on the surface
normal
n
, the illumination direction
s
, and the viewing direction
v
. No sufficiently
accurate physical model exists so far which is able to describe the polarisation be-
haviour of light scattered from a rough metallic surface. We therefore determine
the functions
R
Φ
(
n
,
s
,
v
)
and
R
D
(
n
,
s
,
v
)
, describing the polarisation angle and de-
gree of the material, respectively, for the phase angle
α
between the vectors
s
and
v
over a wide range of illumination and viewing configurations. To obtain analytically
tractable relations rather than discrete measurements, phenomenological models are
fitted to the obtained measurements (cf. Sect.
3.4.2
).
To integrate the polarisation angle and degree data into the three-dimensional sur-
face reconstruction framework, we define two error terms
e
Φ
and
e
D
which denote
the deviations between the measured values and those computed using the corre-
sponding phenomenological model, respectively:
I
min
)
0
,
1
L
Φ
(l)
R
Φ
θ
(l)
(u, v), θ
e
(u, v), α
(l)
2
e
Φ
=
uv
−
(5.18)
i
u,v
l
=
1
L
D
(l)
R
D
θ
(l)
(u, v), θ
e
(u, v), α
(l)
2
.
e
D
=
uv
−
(5.19)
i
u,v
l
=
1
Based on the feature-specific error terms
e
I
,
e
Φ
, and
e
D
, a combined error term
e
is
defined which takes into account both reflectance and polarisation properties:
e
=
e
s
+
λe
I
+
μe
Φ
+
νe
D
.
(5.20)
Minimising error term (
5.20
) yields the surface gradients
p
uv
and
q
uv
that opti-
mally correspond to the observed reflectance and polarisation properties, where
the Lagrange parameters
λ
,
μ
, and
ν
denote the relative weights of the individual
reflectance-specific and polarisation-specific error terms. With the discrete approx-
imations
∂p
∂x
}
uv
=[
∂p
∂y
}
uv
=[
{
p
u
+
1
,v
−
p
u
−
1
,v
]
/
2 and
{
p
u,v
+
1
−
p
u,v
−
1
]
/
2forthe
second derivatives of the surface
z
uv
and
p
uv
as the local average over the four
nearest neighbours of pixel
(u, v)
we obtain an iterative update rule for the surface
gradients by setting the derivatives of the error term
e
with respect to
p
and
q
to
zero, resulting in
¯
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