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An advantage of the described local approach is that the three-dimensional recon-
struction result is not affected by additional constraints such as smoothness of the
surface but directly yields the surface gradient vector for each image pixel. A draw-
back, however, is the fact that due to the inherent nonlinearity of the problem, exis-
tence and uniqueness of a solution for p uv and q uv are not guaranteed for both the
albedo-dependent and the albedo-independent case. However, in the experiments
presented in Sect. 5.3.4 and Chap. 6 we show that in practically relevant scenarios
a reasonable solution for the surface gradient field and the resulting depth z uv is
obtained, even in the presence of noise.
5.3.2 Estimation of the Surface Albedo
For the specular surfaces regarded for the experimental evaluations based on syn-
thetic data (cf. Sect. 5.3.4 ) and on real-world objects in the context of industrial
quality inspection (cf. Chap. 6 ), the three-component reflectance function according
to ( 3.14 ) is used by d'Angelo and Wöhler ( 2005a , 2005b , 2005c , 2008 ), correspond-
ing to the reflectance function
cos θ i +
cos α) m n
N
R I i e ,α) = ρ
σ n · ( 2 cos θ i cos θ e
(5.22)
n
=
1
with N
2. The term in round brackets corresponds to cos θ r with θ r as the an-
gle between the direction v from the surface to the camera and the direction of
mirror-like reflection (cf. ( 3.11 )). For θ r > 90 only the diffuse component is con-
sidered. This reflectance function consists of a Lambertian component, a specular
lobe, and a specular spike (Nayar et al., 1991 ). For a typical rough metallic surface,
the measured reflectance function is shown in Fig. 3.4 , where the material-specific
parameters according to ( 5.22 )aregivenby σ 1 =
=
3 . 85, m 1 =
2 . 61, σ 2 =
9 . 61, and
m 2 =
15 . 8. The specular lobe is described by σ 1 and m 1 , and the specular spike by
σ 2 and m 2 , respectively.
One possible way to determine a uniform surface albedo ρ is its estimation based
on the specular reflections in the images used for three-dimensional reconstruction,
which appear as regions of maximum intensity I (l)
spec as long as the reflectance be-
haviour is strongly specular, i.e. at least one of the parameters σ n is much larger
than 1. Note that the pixel grey values of these regions must not be oversaturated.
For these surface points we have θ r =
0 and θ (l)
i
= α (l) / 2. Relying on the previously
determined parameters σ n ,( 5.22 ) yields
cos α (l)
2
1
L
N
σ n α (l)
1
L
I (l)
ρ =
spec ·
+
.
(5.23)
l
=
1
n
=
1
In principle, a single image is already sufficient to determine the value of ρ as long
as it contains specular reflections. Note that in ( 5.23 ) the dependence of the param-
eters of the reflectance function on the phase angle α is explicitly included.
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