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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part BS. Istanbul 2004
As expected, the bigger the target size the more accurately the
target normal can be determined (Table 1). Whereas the results
of the first three target classes are satisfactory, the target
normals of the smallest target are undoubtedly not as good.
These targets contain only very limited elliptical information
(Figure 9) and therefore it is quite impressive how accurately
the target normal can still be determined.
Average Target | Average Angle |Std. Error of Angle
Diameter [pixel] | Error [degrees] [degrees]
13.2 0.43 0.25
8.6 0.57 0.27
7 1 1.09 0.48
4.2 3.06 1.20
Table 1: — Target normal statistic separated into the four target
groups
Figure 9: Target image of the smallest target group including
the computed ellipse
VM is often used in surface inspections, where points on the
surface are required. However, with the usage of targets the
computed 3D coordinates are always positioned above the
sought-after surface. With the knowledge of the target normal
the corresponding point on the surface can be computed
directly. This is a clear advantage for practical applications.
Using a typical target thickness of 0.11 mm it can be estimated
an angle error of 5 degrees results in a horizontal offset of only
10 pm (vertical offset less than | um). This shows clearly that
the achievable accuracy of the target normal even for small
targets is good enough to compensate for target thickness.
4. THE ELLIPSE CENTRE ECCENTRICITY AND ITS
DISTORTION EFFECTS ON THE BUNDLE
ADJUSTMENT
Earlier investigations (Dold, 1996; Ahn et al., 1997) have
studied the impact of the eccentricity error on a bundle
adjustment. It was reported that in a free network adjustment
with or without simultaneous camera calibration, the
eccentricity error caused by moderately sized image targets is
almost fully compensated for by changes in the exterior
orientation parameters (and the principal distance) without
affecting the other estimated parameters (Ahn et al., 1999).
Network simulations performed by the authors have shown
good agreement with earlier findings, especially when
employing test fields with little variation in the target normals.
However, test fields with a significant range of target
orientations and with medium to big-sized targets can show
significant distortions within the triangulated object point
coordinates. Such a simulated test field result is presented in the
following.
The selected test field represents a sinus-shaped surface with
the horizontal extent of 5 by 3 m (Figure 10). 16 images were
artificially generated using a typical VM camera (c of 20 mm,
resolution of 1500x1000 pixels). This resulted in an average
target diameter of approximately 22 pixels within the images.
Figure 10: 3D view of sinus-shaped surface including target
normals.
To investigate the eccentricity error, a free network adjustment
using intensity-weighted target centroids was performed.
Afterwards, the computed object coordinates were transformed
into the original (error-free) coordinate reference frame. The
resulting positional discrepancies are illustrated in Figure 11
and listed in Table 2.
Average | Maximal
[mm] [mm]
Discrepancies (transformation) 0.25 0.38
Object point sigma (bundle adjustment) 0.02 0.03
Table 2: Numerical results of bundle adjustment and the
discrepancies within the original object coordinates.
Figure 11: Discrepancy vectors between adjusted object
coordinates and original coordinates.
From the results, interesting conclusions can be drawn. First,
the eccentricity error can systematically distort the object point
coordinates, this being visible in Figure 11. The amount of
