(AB3 - 1)? — min. (3)
with A — design matrix
P = weight matrix
1 = observation vector
In a combined adjustment of radiometric and geometric data,
(Weisensee, 2000), a 3D point from any kind of measuring
device gives rise to an error equation of a geometric pseudo-
measurement
1 1 .
Vzp 7 3. o ag (X p. Yp): Zjj - Zp (4)
i=0 j=0
expanding the condition to the solution to
AB HA Pech) min. (5)
3. RESULTS
Depending on the respective measurement task an adequate
accuracy for a 3D point has to be kept. The resolution of a
geometric description of a surface also depends on this task. For
the experiments made here a representation of object surfaces in
a grid of 1 mm size has been chosen.
As different sensors acquire a surface independently from
different positions and in separate systems, the relation between
those systems has to be re-established by some means. Here, a
global registration has been determined using control points on
the objects. These control points have been established in the
photogrammetric bundle adjustment and do not represent a
reference system of higher accuracy. From this follows that no
statement can be made on absolute accuracy of surface
acquisition. Instead, different systems are to be registered
relatively, giving the opportunity to judge the results on a
relative basis.
For this, approaches which minimise the square difference
between irregularly spaced point clouds by a spatial
transformation are readily at hand, e.g. given by (Besl & Mc
Kay, 1992), and are already integrated in the processing of
stripe projector data, (GOM, 2002).
In special cases where a surface can be modelled by a 2 1/2 D
approach, the mathematical model of registration can be
simplified because only the square difference in z-direction is to
be minimised. This model has been illustrated by (Heipke et al.,
2002) and is used here.
The following table shows the offset values and their standard
deviations between the surfaces reconstructed by
photogrammetric methods and the stripe projecting system. For
the first test surface also the differences to CAD-data are given.
3.1 Combined adjustment
After applying registration offsets in a first iteration several
computations have been performed with different weights for
3D points and intensity measuments respectively. The following
cases have been considered. First, the overall sum of all weights
of 3D points has been choosen equally to the weights of all
image data, i. e. 100/100. Then, the weight of 3D points is
reduced in three steps. In Table 2 the resulting mean differences
with their standard deviations and the minimum and maximum
differences between the computation with weights from
100/100 to 1/100 are shown for the test surface. Table 3 shows
the same results for the concrete tile.
factor ' Z mean standard minimum maximum
[mm] deviation
[mm]
100/100 -0.0021 0.0023 -4.6235 2.7354
50/100 -0.0018 0.0025 -4.6471 3.5491
10/100 -0.0015 0.0026 -4.6961 4.4104
1/100 -0.0016 0.0027 -4.6769 4.8409
Table 2. Results — test surface
factor Z mean standard minimum maximum
[mm] deviation
[mm]
100/100 -0.4152 0.0066 -10.8729 14.5787
50/100 -0.4373 0.0073 -11.1835 16.8064
10/100 -0.4725 0.0081 -11.5396 19.1585
1/100 -0.4791 0.0085 -11.4268 19.5025
test surface concrete surface
correction | standard | correction | standard
[mm] deviation [mm] deviation
[mm] [mm]
Fast Vision 30.0095 0.0024 -0.0453 0.0014
Correlation +0.0309 0.0014 -0.3348 0.0097
CAD-data -0.2085 0.0004 = —
Table 1. Registration offsets
Table 3. Results — concrete surface
Figure 6. Colour coded model of concrete surface
—108-—
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