231 
In order to verify’ weather or not the flatness of the 
wall is critical, the grid has been surveyed by means of 
two theodolite stations (fig. 3) and the final 3D-terrain 
co-ordinates of the grid points have been obtained 
with an accuracy of 1/10 of mm (fig.4). 1 With the real 
co-ordinates we can correct for the dX and dY relief 
displacements of the grid points. 
Fig. 3 - The theodolite measurement 
The largest value of the deviation from plane is 
dZmax=l/2mm giving a relief displacement of 
dXmax=l/3mm. In figure 5 are depicted as circles the 
deviations from the medium plane. 
Fig. 4 - The relief displacement correction 
Then an homografic transformation performed. 
. a,X + a 7 Y + a, 
x + dx = — - - 
a 1 X + a s Y +1 (9) 
y + dy = a * X+a ’ r+a > 
a^X + a $ Y + 1 
The residuals of the transformations are the vector 
components of the distortion. The distortion is the 
difference from the real lens to the pinhole model. The 
homographic transformation is a particular case of the 
DLT (1) where the object space reduces to an object 
plane. It has the advantage in comparison with (1) or 
the self-calibration procedure, to reduce the 
parameters from 11 to 8, and therefore to avoid or 
reduce the problems arising with the correlation 
among the coefficients. 
On the left of fig. 6 the residual vectors are depicted, 
overlaid to the grid, for testl. The principal point is 
then computed as intersection of the straight lines 
passing through the vectors giving weigh proportional 
to the lenght of their module. In this manner the 
vectors can be split in two components, the radial and 
the tangential ones. The radial components are then 
arranged accordingly to the distance from the principal 
point in the graph on the right of the fig. 6. The radial 
components can be interpolated by the curve (8) where 
the distance ro is not arbitrary but found by least 
squares interpolation. Fig. (7) shows the tangential 
components. 
Fig. 8 displays the three estimated characteristic 
distortion curves and the calibrated Rollei curve taken 
from the calibration certificate. After the 
transformation (4), consisting in adding an arbitrary 
linear term to the characteristic distortion, the 
calibrated distortion curves have been obtained, 
allowing the comparison with the certificate curve (fig. 
9). 
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Survey 
Results 
* The survey was performed using the relative orientation by 
coplanarity (Fangi, 1998). The obtained model co-ordinates 
have been transformed into the ground system by a similarity 
transformation in space (7 parameters). As reference the 
theoretical co-ordinates have been used, having the depth 
equal to 0. The residuals of the transformation represent the 
deviation of the real co-ordinates from the theoretical ones 
(fig. 5). 
Fig. 5 - The deviations from plane of the grid