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Figure 2 Derivation of fast polynomial mapping functions 
reason for that is the fact that the SPOT sensor is linear 
and thus the perspective relations are valid only within an 
image line. It was often stated that the epipolar geometry 
does not exist for SPOT images and that resampling to 
epipolar images requires a DTM. However, strictly 
speaking the epipolar geometry does not even exist for 
frame cameras (which is why most bundle adjustment 
programs use additional parameters). So, the aim of our 
investigations was to check to what extent by using 
Kratky's PMFs an epipolar geometry could be 
established. 
To check that, the following approach was used. A height 
error AZ (two versions; AZ = 50 m and 100 m) was 
added and subtracted to the known heights of the 136 
points. For each point, these two erroneous heights and 
the image to image PMFs were used to transform the 
pixel coordinates of the left image in two points in the 
right image. They defined a straight line which passed 
through the known correct pixel coordinates of the point. 
The question that had to be answered was whether by 
arbitrarily changing the height, the projection by using 
PMFs of the left point in the right image would fall on 
this straight line, i.e. whether this straight line was the 
epipolar line (Figure 3). Thus, the known height was 
sequentially incremented by 25 m in positive and 
negative direction (leading to object points like P, in 
Figure 3), and the projection of the left point in the right 
image and its distance from the straight line were 
computed. This distance is a measure of deviation from 
straight epipolar lines. The results for all 136 points are 
listed in Table 2. The results are identical for both 
versions of AZ, and for positive and negative increments. 
As it can be seen from the table a deviation of 0.25 pixels 
is reached only with a height error of over 7 km! Since 
such errors are impossible, even more for matching 
361 
which requires good approximations in order to be 
successful, straight epipolar lines can be assumed. 
Table 2 Deviations from a straight epipolar line for 
  
  
  
  
  
  
different height errors 
Threshold of | Mean Z error to Standard 
: deviation of Z 
distance to the reach the error to reach 
straight line threshold threshold 
[pixel] [km] 
[km] 
0.25 7.44 0.15 
0.5 10.53 021 
1 14.90 0.30 
2 21.08 0.42 
  
  
  
  
  
The above knowledge was used to modify the Multiphoto 
Geometrically Constrained Matching (Baltsavias, 1992) 
for automatic DTM generation. The points to be matched 
were selected in one of the two images (reference image). 
For each point, by using a height approximation and an 
error AZ as above, the epipolar line in the other image 
was determined. If only approximations for the pixel 
coordinates exist, then a height approximation can be 
derived by the image to image PMFs from the pixel 
coordinates of the point in the reference image and the x 
pixel coordinate in the other image. Weighted geometric 
constraints force the matching to search for a 
corresponding point only along the epipolar line. This 
reduction of the search space from 2-D to 1-D increases 
the success rate and reliability of the matching results.