Intensity-Based and Polarisation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 164

2

n t ( 1 k 2 )

|ˆ | = +

the complex refraction index by 1, which is generally true for

visible wavelengths, and accordingly obtain the relation

n

2 n tan θ i sin θ i
tan 2 θ i sin 2 θ i +|ˆ

D met = (3.59)

p

2

n |

for the polarisation degree in terms of the incidence angle. Equation ( 3.59 )isvalid
for smooth, specularly reflecting metallic surfaces. Provided that the complex re-
fraction index of the material is known, Morel et al. ( 2005 ) compute the surface
gradients and determine the depth map of the surface by integration of the gradient
field. The surface is illuminated diffusely by a hemispherical dome. The described
method is used in the context of quality inspection of very smooth polished metallic
surfaces.
According to polarised light scattering measurements performed by Germer et
al. ( 2000 ), it is questionable, however, if a simple polarisation model like ( 3.59 ) can
be applied to metallic surfaces. The steel surface samples examined by Germer et
al. ( 2000 ) were polished with various polishing emulsions, and some were etched
in a sulphuric acid solution. In the course of their ellipsometric measurements, they
determine the BRDF of the surface, the degree of circular polarisation, the overall
degree of polarisation, and the polarisation angle of the light reflected from the sur-
face for a constant incidence and emission angle of θ i = θ e =

60 ◦ over a range of
incident polarisation states and azimuthal scattering angles. The surface roughness
is determined by means of atomic force microscopy. Germer et al. ( 2000 ) com-
pare their polarimetric measurements with theoretical predictions of the scattering
behaviour of surfaces exhibiting a roughness on microscopic scales ('microrough-
ness') and a variable subsurface permittivity. For most values of the azimuthal scat-
tering angle, the measurements are neither fully consistent with the microroughness
model nor with the subsurface scattering model, which suggests a combination of
both mechanisms.
The experimental results of Germer et al. ( 2000 )giveanimpressionofthedif-
ficulties encountered when attempting to apply polarisation models to metallic sur-
faces. This is especially true for rough metallic surfaces. As a consequence, for
the raw forged iron surfaces regarded in the industrial quality inspection scenar-
ios described in Sect. 6.3 , it was found to be favourable to determine empirically
the reflectance and polarisation properties instead of relying on physical models
(d'Angelo and Wöhler, 2005a , 2006 , 2008 ).
The measurement procedure employed by d'Angelo and Wöhler ( 2005a , 2006 )
for determining the polarisation properties of the surface is similar to the method
described by Atkinson and Hancock ( 2005a ). A flat sample part is attached to a
goniometer, which allows a rotation of the sample around two orthogonal axes.
The corresponding goniometer angles γ 1 and γ 2 can be adjusted at an accuracy
of a few arcseconds. As illustrated in Fig. 3.5 , adjusting γ 1 is equivalent to rotat-
ing the surface normal n around an axis perpendicular to the plane spanned by the
vectors s and v , while adjusting γ 2 causes a rotation of n around an axis lying in
that plane. The phase angle α between s and v is independent of γ 1 and γ 2 , since
the centre of rotation lies on the sample surface, and is assumed to be constant