International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
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Figure 1: Sketches of the five image configuration. Upper row - configurations with strong image geometry: "Tetra,
'Airl^ and "Air2'. Lower row - configurations with weak image geometry: "Street1' and 'Street2'. The arrows point in
direction of the cuboid compression, the limit of which is represented by the small bar. The camera geometry for "Tetra,
'Street1? and. "Street2! is based on the Nikon DCS460; i.e. 3000 x 2000 pizels (9 jum) with principal distance of 3500 pixel.
The camera geometry for ’Air1’ and ’Air2” is based on the aerial normal angle case; i.e. image size 23x23 cn? , assumed
to be scanned with 15 pm, and principal distance 300 mm.
is computed by the five methods mentioned above. From
the computed tensor projection matrices are extracted fol-
lowing [Hartley 1997], which are then used to determine
all no points into object space by spatial intersection. This
results in a projective, not a Euclidian, reconstruction of
all no points. Using the known Euclidian object coor-
dinates of the k sample points a transformation M be-
tween the projective reconstruction and the true Euclidian
space can be computed. With M the other no — k points
are also transformed from the projective reconstruction to
the Euclidian space and the mean and maximum differ-
ences (termed ground errors) between the transformed and
known Euclidian positions are determined.
These steps are repeated for 1000 samples and the overall
mean of the mean and maximum ground errors are stored.
Then the cuboid is compressed in a certain direction and
the whole process is repeated for the thinner cuboid. The
overall means of the mean and maximum ground errors are
then plotted against the compression rate of the cuboid;
see figure 2 (left part) which shows the plot for the "Tetra'-
configuration and 8 point correspondences.
For each image configuration a threshold for the mean
ground error was set. If this threshold is hurt by the mean
ground error of a certain sample, the number of failures
for the respective pair (image configuration and cuboid
compression) is increased by 1. After all 1000 samples the
percentage of failures are also plotted against the cuboid
compression; see figure 2 (right part).
Due to the space limitation only the plot for the image con-
figuration ’Tetra’ and 8 corresponding points is included
in this article. For a detailed description of these synthetic
experiments with all respective plots see [Ress] 2003]. The
results found there can be summarized in the following
way:
e The ground errors obtained for the different compu-
tation methods tend to be the same
for a particular image configuration, if the num-
ber of point correspondences increases, and
for a particular number of point correspondence,
if the stability of the image geometry increases.
Therefore, from 10 points onwards for Tetra’, Air?’
and 'Air2", and from 25 points onwards for '"Streetl
and 'Street2', all computation methods return prac-
tically the same result; i.c. then the simple direct lin-
ear solution ('"UCA") is equivalent to the rigorous con-
strained computation (CR!) minimizing reprojection
error.
e For small numbers of point correspondences, € 8 for
the stable configurations "Tetra', 'Airl? and 'Air2' and
for the unstable configurations 'Streetl' and ‘Street?’
in general, the unconstrained minimization of repro-
jection error generally performed better than any al-
gebraic (constrained or unconstrained) minimization.
Therefore, the benefit of minimizing only reprojection
error, is larger than of considering only the constraints.
e Concerning the impact of the minimum thickness of
the cuboid on the various computation methods, we
can say that
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