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air occurs. Analogous to the case of specular reflection, Snellius's law and the Fres-
nel equations yield an equation for the degree of polarisation
D
dif
p
of light diffusely
scattered by the surface in terms of the emission angle
θ
e
(again,
n
i
=
1 is assumed):
n
t
)
2
sin
2
θ
e
1
(n
t
−
D
diff
p
=
4 cos
θ
e
n
t
.
(3.58)
n
t
)
2
sin
2
θ
e
+
1
sin
2
θ
e
2
n
t
2
+
−
(n
+
−
Rahmann (
1999
) exploits the fact that, according to the Fresnel equations, light
which is specularly reflected by a surface can be decomposed into two components
oriented parallel and perpendicular, respectively, relative to the 'reflection plane'
spanned by the illumination vector
s
and the viewing direction
v
. For unpolarised
incident light, the first component corresponds to the fraction of light which remains
unpolarised after reflection from the surface. Rahmann (
1999
) concludes that the
orientation of the polarised component, which is given by the polarisation angle
Φ
,
denotes the direction perpendicular to the reflection plane.
However, this approach can only be applied if it is possible to distinguish spec-
ular from diffuse reflection. While this distinction is usually possible for dielectric
surfaces, it will be shown in Sect.
3.4.2
that the reflectance behaviour of metallic
surfaces is more complex and that they display several specular reflection compo-
nents which strongly overlap with the diffuse reflection component.
It is favourable to employ polarisation imaging for the reconstruction of specu-
larly reflecting surfaces based on polarisation images, as shown by Rahmann and
Canterakis (
2001
), who utilise for shape recovery the projection of the surface nor-
mals directly provided by the polarisation angle. Based on the measured polarisa-
tion angle information, iso-depth curves are obtained, the absolute depth of which,
however, is unknown. These level curves provide correspondences between the dif-
ferent images, which are in turn used for triangulation, thus yielding absolute depth
values. Rahmann and Canterakis (
2001
) demonstrate that a three-dimensional sur-
face reconstruction is possible based on three polarisation images. By formulat-
ing the surface reconstruction problem as a minimisation of an error functional (cf.
Sect.
3.2.2.2
) describing the mean-squared difference between the observed and the
modelled polarisation angle, Rahmann and Canterakis (
2001
) show that two polari-
sation images are sufficient for a three-dimensional reconstruction of asymmetrical
objects, while the reconstruction of simpler, e.g. spherical, surfaces requires at least
three polarisation images.
Atkinson and Hancock (
2005a
) derive the polarisation degree and the polarisa-
tion angle from a set of 36 images acquired under different orientations of the linear
polarisation filter mounted in front of the lens. They systematically examine the
accuracy of the value of
θ
i
obtained based on (
3.53
) and (
3.58
). The azimuthal com-
ponent of the surface orientation corresponds (up to an 180
◦
ambiguity) to the mea-
sured polarisation angle. Furthermore, they utilise the result to estimate the BRDF of
the surface material at zero phase angle (parallel illumination vector
s
and viewing
direction
v
).
This method to estimate the surface normals based on the Fresnel equations is
extended by Atkinson and Hancock (
2005b
) to a multiple-view framework in which
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