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a three-dimensional point
C
1
x
(m
i
. For each pixel the surface normal
C
1
n
(m
i
is
known as a result of the SfPRD method. The three-dimensional point cloud is
transformed into the rectified coordinate system
S
2
of camera 2 according to
S
2
x
(m)
i
C
1
T
C
1
x
(m)
i
,
S
2
=
(5.34)
where
S
C
1
T
denotes the transformation (a rotation and a translation) from coordi-
nate system
C
1
into coordinate system
S
2
. The same transformation is performed
for the surface normals
C
1
n
(m)
i
C
1
s
, resulting in the
and the illumination vector
vectors
S
2
n
(m
i
and
S
2
s
.
2.
Render a synthetic image for rectified camera 2
: Based on the known reflectance
function, a synthetic image
R
(m
I
(
S
2
u,
S
2
v)
is rendered, which represents the pixel
grey values expected for the rectified coordinate system
S
2
.
3.
Determine disparity corrections
: Deviations between the estimated and the
true surface profile are now revealed by a position-dependent lateral offset
d
(m
j
(
S
2
u
(m
j
,
S
2
v
(m
j
)
between the rendered and the observed image of rec-
tified camera 2. In each iteration cycle
m
, the blockmatching stereo algo-
rithm re-determines the pixels
(
S
2
u
(m)
j
,
S
2
v
(m)
j
)
for which correspondences be-
tween the rendered and the observed image in the rectified coordinate system
S
2
can be established. Due to the chosen standard geometry, a depth error
of a pixel in the image of camera 1 translates into an offset along the corre-
sponding epipolar line, i.e. image row, in the rectified image of camera 2. The
value of
d
(m)
j
(
S
2
u
(m)
j
,
S
2
v
(m)
j
)
corresponds to the disparity error of the pixel at
(
S
2
u
(m)
j
,
S
2
v
(m)
j
)
in the rectified image of camera 2. We determine the offset based
on the same correlation-based blockmatching approach as utilised for the initial
stereo image analysis.
4.
Compute corrected three-dimensional points:
The positions
(
S
2
u
(m
j
,
S
2
v
(m
j
)
and
disparities
d
(m
j
of all pixels for which the blockmatching technique is able to
determine a value
d
(m)
j
are updated according to
S
2
u
(m),
corr
j
S
2
u
(m)
j
d
(m)
j
=
−
S
2
v
(m),
corr
j
S
2
v
(m)
j
=
(5.35)
d
(m),
corr
j
d
(m)
j
d
(m)
j
=
+
.
The corrected three-dimensional point cloud
S
2
r
(m
+
1
)
j
is obtained from the cor-
rected pixel positions
(
S
2
u
(m),
corr
j
,
S
2
v
(m),
corr
j
)
and disparities
d
(m),
corr
j
determined
according to (
5.35
), relying on the basic equations of stereo analysis in standard
epipolar geometry (Horn,
1986
). Transformed into the coordinate system of cam-
era 1, the corrected three-dimensional points are denoted by
C
1
r
(m
+
1
)
j
. Finally,
1.
5. Iterate steps 1-4 until the average and the standard deviation of the disparity
corrections
d
(m)
j
the iteration cycle index
m
is incremented:
m
←
m
+
are of the order of 1 pixel. Note that the disparities
d
(m)
j
are
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