D 
Y=-—-C (3) 
g c 
Putting (3) into (2) the final correction equations result: 
deine n ai 
—'C, t€ 
3 y 
dx' (ex) = a ex (4) 
D 
In analogy the correction of image coordinates y' yield to 
y 7 y'&dy (ey) * dy (ez) 
y^ey 
  
and hence dy'(ey)=- D and 
a ‘Cr +ey 
dy'(ez) = d ez (5) 
D 
In order to compute the exterior orientations of the images 
within the panorama the mathematical model of spatial 
similarity transformation is used. By default it is based on 
object coordinates: 
x X 
e, [7 A-R-|Y (6) 
y Z 
The presented image corrections based on excentricity and the 
formulation of object coordinates as angular directions the 
following equation results: 
x'+dx' (ey) + dx' (ex) sin(@) 
Cr = À-R-| cos(a) (7) 
y'+dy' (ey) + dy' (ez) tan() 
and the modified collinearity equation can be derived as 
xX +dx' (ey) + dx' (ex)y = 
x Ail -sin(œ, ) + 51, : cos(a) * 03, -tan( B) 
  
m. 
: 3, - sin(@,) + 753, * COS(O; ) + 733, -tan(B;) 
1 1 : 1 : (8) 
V'u +dy' (ey);, + dy' (ez)x = 
Fy, -Sin(@y) + ry, * COS(@ ) + 732, -tan(B,) 
hs, -Sin(at) * 755, : cos(a ) * 73, -tan(B,) 
  
=C; 
These equations consists of all parameters that have to be 
estimated: 
=  excentricity parameters ex, ey and ez 
= direction, tilt angle and roll of image (7) of the rotation 
matrix R(e,œ,K) 
= direction and slope to object poin (k) (a and p) 
  
x target point 
   
  
    
image plane A er diei) 
image plane 
/ 
perspective centre 4 / 
ei) 
© tripod axis 
Fig. 8: Excentricity ex and ey 
3.3 Practical process for panorama images generation 
For the practical realisation of geometric exact panoramas the 
determination of the excentricity parameters are of major 
importance. Only for the case that the excentricity vanishes to 
zero, exact panoramas can be calculated from single frame 
images. Therefore the "imaging system" camera/tripod is 
calibrated in advance and justified according to the following 
steps. 
Firstly, circular targets with known diameter are distributed in 
object space that is imaged as a panorama by the camera to be 
calibrated. A least-squares adjustment based on the observation 
equations (8) yields the excentricity quantities. Subsequently 
the camera positioin on the tripod is corrected. In the following 
an arbitrary number of panoramas can be aquired by this 
system. 
The panoramas themselves can be computed from the single 
images by resampling based on (8) in addition with a 
radiometric matching of the overlapping zone (Fig. 9). Tie 
points are measured in order to calculate the local orientation 
parameters of the images (similar to relative orientation). The 
resulting quality can be assessed by the remaining residuals of 
tie points. A number of experiments has shown that the stability 
of the mechanical adjustment of the tripod/camera system is 
better than 1mm, even if the camera has been dismounted 
several times. 
Fig. 9: Panorama, calculated using orientation data 
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