ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
Finally the orientation parameters ( X ;K;) are introduced into 
the well known collinearity equations to transform the point 
from the ground system to the sensor system. 
x X =X 
y, | AR(2»,9.x), Y, -Y, (6) 
-f 2-1 
By dividing equations one and two by equation three in (6) 
above, the unknown scale À, cancels out. These equations are 
the collinearity equations as used for frame sensors. 
ha, (X, =X Jr AY mo tr, (Z, zs) 
ur (7) 
i RX RX yer, ZZ) 
  
n, GE, -X)tra Q, = Yn, , 72.) 
BUG X rr, (Y ern (ZZ) 
  
Nm. (8) 
The mathematical model used is very flexible. In areas without 
sufficient measurements of tie points, e.g. lakes or forests, the 
orientation parameters are primarily or exclusively determined 
by the GPS/IMU measurements. On the other hand, if 
GPS/IMU data cannot be used or is not available, the 
orientation can be determined by tie point measurements only. 
In the latter case the definition of the orientation fixes interval 
and the tie point distribution must be chosen very carefully to 
avoid known singularities of the mathematical model, as they 
are described in the analysis of Müller (1991). 
The figure below illustrates graphically the relation between the 
orientation parameters, the orientation fixes and the GPS/IMU 
observations. 
  
Orientation 
parameter 
Orientation fixes 
Interpolation / \ 
correction p 
dr d 
   
  
   
  
True | 
orientation 
Orientation for each line 
from GPS/IMU 
FN 
La 
  
  
  
  
Figure 7: Example of one orientation parameter over time. 
Typically, the GPS/IMU sensors will have a systematic offset to 
the actual sensor head. This systematic offset between the true 
orientation and the GPS/IMU observations is compensated and 
computed by additional parameters within the self-calibration 
process of the triangulation. 
After the adjustment, the orientation of the GPS/IMU is updated 
by piecewise interpolation from the orientation fixes, which 
were computed in the bundle adjustment. 
  
Orientation 
parameter 
      
Time 
  
  
  
Figure 8: Precise orientation for each line of Level 0. 
3.2 Spacing of Orientation Fixes 
The spacing of the orientation fixes has a strong impact on the 
determinability of all parameters. Therefore it must be defined 
carefully. 
   
   
  
a) This example shows a very large spacing. The solid black 
lines indicate the three observations in forward, nadir and 
backward scenes. The dashed purple lines indicate the effect of 
the rays in the adjustment. Although the point is observed in 
three scenes the geometric effect is reduced to two rays. If the 
fixes were even further apart from each other the effect of the 
three observations could approach a single ray point. This is 
due to the mathematical model. The effect of an observation 
depends on the distance to the fixes. If the spacing is defined to 
be too wide, the adjustment becomes singular. 
  
b) This example shows a medium spacing. As two observations, 
nadir and backward, fall into one interval, the geometry is not 
truly comparable with a three ray point. The weight of the 
dashed purple lines should indicate the influence of each ray on 
the point determination. 
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