International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
Table 2: Quality analysis 1: estimation of the virtual cameras
observing area: left lower corner (3,3, —17) [em]
observing area: right upper corner (21, 23, —11) [em]
number of points used for estimation 845
distance between points 2 [em]
Gapproxvc (cameraC1) 0.04 [pel]
Table 3: Coordinates of the projection centre of camera Cl for
the strict model and the approximation VC
[Projection center | Xi [em] | X» [em] | Xs [em] J
Strict model 3.08 4.53 63.46
Approximation VC 3.07 4.5] 85.98
Quality analysis:
l. A priori quality DLT:
residuals as backprojection errors in image space
2. Quality DLT:
residuals in object space for new points — "
3. Quality point matching algorithm:
comparison of the reconstructed points (before final es-
timation) using the strict and the approximated model
4. Quality point matching algorithm:
comparison of the reconstructed points (after final esti-
mation) using the strict and the approximated model
5.4 Prediction of 3D points using the virtual camera
5.4.1 Estimation of the virtual cameras To define the seg-
mentation of the object space a priori quality test have to be calcu-
lated (see (Wolff and Fórstner, 2001)). These a priori tests show,
that the determination of only one virtual camera (VC) for the
whole object space is enough. For the position of the four cam-
eras see Fig.4.
To investigate the quality of the determined virtual cameras (Qual-
ity analysis 1), we project the object points which were used for
the estimation of P into the image space and get the image points
x’. The estimated DLT (11 independent parameters) yields resid-
uals x — x’ being systematic errors. To get an a priori quality of
the projective model we give the r. m. s. error
$i Ta xy
Gapprox = 2n — 11
where n is the number of points used.
Tab. 2 gives the entities and results of estimating the virtual cam-
eras of camera C1. The number of points used for the estimation
need not to be as high as in this case. Tab. 3 gives the coordinates
of the camera projection center for the three different orienta-
tions. The multi media geometry influence mostly the hight of
an object point, which is here the X3 coordinate of the projec-
tion center. Therefor the projection center of the two orientations
differ mostly in the hight.
5.4. Results of the point matching using the approximation
As mentioned above, the algorithm should be calculated for dif-
ferent starting images, to guarantee that also the points, which
are not extracted in the starting image, can be found. Here we
use four cameras, every camera could see the whole object scene.
Together with the constrain, that at least three corresponding im-
age points of an object point are needed, it is enough to have two
different starting images. Therefor and because of the constraints,
610
that the image points of an object point should be seen in at least
three image points, we got the constraint for our clustering algo-
rithm: a group of at least three define an object point.
First, we want to examine if the constraint for a object point, that
at least three close points in a group define an object point, is
sufficient. Fig. 3 shows the hypotheses of two matched image
points by there corresponding object points (seen from the side).
The distribution of the 3D points shows a very dense part, where
the sediment surface is supposed to be. All the other points might
be wrong hypotheses and should be deleted by the clustering al-
gorithm. Fig. 4 shows the results after the clustering. All points
which differ significantly from the surface are eliminated (Fig. 4
a). Fig. 4 b) shows the distribution of the object points on the
surface, which are evenly distributed.
i
Figure 3: Hypothesis of 3D point matchings before clustering. A
group of at least three points define an object point.
92
Figure 4: Results after clustering the point hypotheses. The right
figure shows the point cloud from the side, the left figure shows
is from above together with the positions of the cameras.
Using the strict model gave 156 reconstructed 3D points, the use
of the virtual camera VC found 161 points. For the quality anal-
ysis 3. we have to compare the two sets of points. Therefor a
threshold e is defined, so that a point X, is defined as equal to a
referent point X; if X, — X;| « e. The number of points found
as equal in dependency of the threshold is shown in Fig. 5.
The main influence of the approximation refers to the hight ofthe
object points. The r. m. s. error of the Xs coordinate of the
reconstructed object points X. — (X1, X», X3) is
Ox; =
where n is the number of points used. The error of the approxi-
mation is given in table 5 in comparison to the referent data before
calculating the final bundle adjustment.
5.5 Final 3D determination of the predicted points using the
strict model
After the matching process, including an approximated determi-
nation of the object point, we calculate a final bundle adjustment
for the strict model and for the approximation VC. The clusters
resulting from the clustering algorithm contain that points, which
were found as corresponding points. To compare this clusters
Inte
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