X Y. X-X 
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follows for images, sensors and new points (Oy accordingly): 
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3.3 Parallel Section in Space 
With the previously derived orientation parameters, we are now 
able to do object reconstruction with large amounts of points 
which were not included in the parallel block adjustment. A 
linear forward section for central projection is shown in (Kraus, 
1997; Moré, 2000). We have transferred it to the parallel case, 
using the following equations: 
x =(474)' (472), (5) 
with 
and 
The above is a very fast way of computing three dimensional 
coordinates from an arbitrary number of images, since this 
adjustment is linear. Therefore neither approximate values are 
required, nor will there be an iteration. 
The computation of errors, however, is not reliable with this 
method. A closer look at the error matrix Oy, = (4 Tq)! reveals, 
that the observations do not take part in this matrix, hence all 
image points seem to have the same error. Other ways have 
been and are being explored to reveal blunders in the 
observations. A rather simple approach is to use the correction 
matrix v = AX-L for the detection of errors. A more 
sophisticated, yet time consuming approach, is the computation 
of the shortest distance of the new point to the line of sight of 
each image. Since all lines of sight should merge in one point, 
these distances ought to be rather small and equal to each other. 
Analysis of the distances can help detecting blunders and the 
originating image. 
As mentioned above, this is currently a topic of research with a 
growing variety of setup situations and point measurements. 
4. RESULTS 
The suggested methods have been applied to several projects, 
involving the tilting sample stage and some calibration objects. 
As expected, the best results could be achieved with the latest 
cascade pyramid. As an example, a typical tilting series will be 
stated, using 13 images with up to 38 measured control points, 
each. All images shared one sensor, where the radial lens 
distortion and affine transformation were kept unknown as 
additional parameters. Thus, the whole adjustment system had 
70 unknowns and 966 observations. Some results are shown in 
table 1. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
sensor magnification magnification error 
[pixel / nm] [pixel / nm] 
XL30 FEG 94.559 0.116 
image Xo [nm] my [nm] 9 [3 mo [7 
number Yo [nm] myo [nm] 0 [7 ms [9] 
x ['] my [*] 
1 2.621 0.004 0.261 0.22 
1.941 0.004 351.038 0.21 
88.550 0.06 
2 2.640 0.004 0.171 0.21 
1.882 0.004 345.935 0.19 
88.332 0.06 
3 2.460 0.004 359.838 0.22 
2.017 0.004 356.701 0.22 
88.628 0.06 
4 2.675 0.004 359.993 0.21 
1.794 0.004 341.131 0.18 
88.279 0.07 
5 2.452 0.004 359.042 0.22 
2.033 0.004 1.728 0.22 
88.694 0.06 
  
  
  
  
  
  
Table 1. The first 5 images of the tilting series. 
The above table exemplarily shows the first five images of the 
tilting series, mentioned before. The tilting sample stage was set 
to: 0°, 355°, 5°; 350°, 10°, etc. Comparison with the adjusted 
results proves the high accuracy of the positioning table. 
Furthermore it provides high stability and convenient handling 
in use. The pyramid allows precise measurement, therefore 
supporting accurate results. Nevertheless, deviations may occur 
-914- 
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