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model, an additional disparity error is computed. If a point is situated in an image
region with repetitive structures, the disparities of all possible correspondences are
determined and compared to the model, which leads to the matrix
E
d
of disparity
errors. These two matrices are merged into the total error matrix
E
t
according to
E
t
=
E
SSD
+
λ
e
E
d
,
(1.136)
involving the weight parameter
λ
e
. The refined correspondence analysis is per-
formed based on the matrix
E
t
, using the same constraints (uniqueness, uniqueness
and ordering, minimum weighted matching) as for the initial stereo analysis. The in-
fluence of the deviation between the three-dimensional points and the model on the
resulting three-dimensional point cloud increases with the increasing value of
λ
e
.It
is shown in Sect.
1.6.4
, however, that the three-dimensional reconstruction result is
not strongly sensitive with respect to the chosen value of
λ
e
and that a significant
improvement of the initial stereo analysis is achieved for a variety of scenes using a
unique value of
λ
e
.
Since correspondence analysis based on the matrix
E
t
involves the computation
of disparity differences between three-dimensional points and the model, and since
our experimental evaluation requires an assignment of three-dimensional points to
the model or a foreground object based on a disparity difference threshold, it is
necessary to examine to which extent inaccuracies of the estimated parameters
a
s
,
b
s
, and
c
s
of a model plane in scene space translate into inaccurate model disparities.
For the errors
a
d
,
b
d
, and
c
d
of the model plane parameters in disparity space,
the law of error propagation yields the following relations:
∂a
d
∂c
s
∂a
d
∂a
s
la
s
c
s
l
c
s
a
d
=
c
s
+
a
s
=
c
s
+
a
s
(1.137)
∂b
d
∂c
s
∂b
d
∂b
s
lb
s
c
s
l
c
s
b
d
=
c
s
+
b
s
=
c
s
+
b
s
(1.138)
∂c
d
∂c
s
lf
c
s
c
d
=
c
s
=
c
s
.
(1.139)
For an approximate quantitative error analysis a camera constant of
f
=
1350 pixels
and a baseline of
l
0
.
1 m (cf. Sect.
1.6.4
) are assumed. The average distance to the
repetitive structures approximately corresponds to the value of
c
s
, which amounts
to about 10 m for the outdoor scene and 1 m for the indoor scenes regarded in
Sect.
1.6.4
. Under the fairly pessimistic assumption that the value of
c
s
can be esti-
mated at an accuracy of 5 %, the resulting error of
c
d
according to (
1.139
) amounts
to 0
.
7 pixel for the outdoor scene and 6
.
8 pixels for the indoor scenes.
The extension of the object in the image is denoted by
g
and typically corre-
sponds to 100-300 pixels, such that we may set
g
=
200 pixels. The maximum
disparity error
d
max
due to an inaccuracy
a
d
of the model parameter
a
d
in dis-
parity space then corresponds to
d
max
=
=
g
·
a
d
. For simplicity, it is assumed that
for the true model plane we have
a
s
=
0, i.e. the model plane is parallel to the
image plane. If we (again pessimistically) assume that
a
s
=
b
s
=
0
.
6, corresponding to
an angular error of more than 30
◦
in the case of the frontoparallel plane, we obtain
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