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Ta b l e 5 . 3 Evaluation results obtained based on the synthetically generated surface shown in
Fig. 5.15 a. The error values for z are given in pixels
Utilised information
RMSE (without DfD)
RMSE (with DfD)
z
p
q
z
p
q
Reflectance
11 . 6
0.620
0.514
9.62
0.551
0.514
Pol. angle
17 . 0
0.956
0.141
6.62
0.342
0.069
Pol. degree
4 . 72
0.138
0.514
4.73
0.135
0.514
Pol. angle and degree
1 . 83
0.121
0.057
1.71
0.119
0.056
Reflectance and pol. angle
12 . 0
0.528
0.099
2.52
0.280
0.055
Reflectance and pol. degree
10 . 9
0.575
0.514
8.46
0.418
0.517
Reflectance and polarisation
(angle and degree)
10 . 2
0.277
0.072
0.91
0.091
0.050
angles of ψ ( 1 )
30
and ψ ( 2 )
30 , respectively. This setting results in iden-
=−
=+
tical phase angles α ( 1 )
75 for the two light sources. A set of 500 random
points is extracted from the ground truth surface, to which Gaussian noise is added
as described below prior to using them as sparse depth data for three-dimensional
reconstruction.
The reflectance functions of the rough metallic surface measured according to
Sects. 3.2.1 and 3.4.2 were used to render the synthetic images shown in Fig. 5.16 c.
Gaussian noise is applied with a standard deviation of 5
α ( 2 )
=
=
10 4
×
for the intensity I ,
10 2 ,1 for the polarisation angle Φ ,
and 0 . 4 pixel for the depth values ( z between 0 and 6 pixels). The weights for the
error terms according to ( 5.32 )aresetto λ
where the maximum grey value is about 6
×
=
=
=
=
450, μ
40, ν
100, and χ
1. The
surface gradients are initialised with zero values.
Figure 5.16 shows the reconstruction results on noisy synthetic images, where
the plots (d)-(f), (j), and (k) were obtained by applying SfPR alone, while the plots
(g)-(i) depict the results obtained based on integration of sparse depth data into the
SfPR framework. The respective reconstruction errors are given in Table 5.4 .Itis
apparent that the shape from shading reconstruction using a single light source fails
to reconstruct the surface (Fig. 5.16 d), while the surface shape can be reconstructed
approximately using a single intensity and polarisation angle image (Fig. 5.16 f).
To reach similar reconstruction accuracy without polarisation information, illumi-
nation from two different directions is required (Fig. 5.16 e). Table 5.4 illustrates
that using intensity and polarisation degree in the three-dimensional reconstruction
process leads to poor accuracy both for the global and the local approach, while
using intensity and polarisation angle yields a high accuracy which does not fur-
ther increase when the polarisation degree is additionally used. The reason for this
behaviour is the fact that intensity and polarisation degree contain somewhat redun-
dant information, since both display a maximum in or near the specular direction
( θ r =
0 ) and decrease in a qualitatively similar lobe-shaped manner for increasing
values of θ r . The dependence on surface orientation, however, is much stronger for
the intensity than for the polarisation degree, while the measurement error tends
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