2.3 The Microscopic Calibration Pyramid 
For the calibration of the sensor and the verification of the table 
a reference object is necessary. The object has to be small 
enough to correspond to the approximate magnification of 
subsequent evaluation projects. Additionally, definite control 
points with known spatial coordinates have to be identified. The 
first obvious choice was a regular 2D grid. But we encountered 
problems with mathematical ambiguities due to the parallel 
projection model. Figure 3 illustrates these ambiguities, in 
contrast to central projection. 
planar regular grid 
    
central parallel 
projection projection 
image 
   
Figure 3. Ambiguities when projecting 2D objects with parallel 
projection. While the central perspective image is non- 
ambiguous (left), the parallel projection gives the same result, 
when the sample is tilted in either direction, or even if the image 
is scaled in one direction. 
Therefore, a two dimensional object was not found suitable, 
although such specimen are easy to acquire and offered in a 
large variety from different providers. Consequently an object 
offering spatially distributed control points (nanomarkers) was 
required. A cascade pyramid with sloping edges was developed 
with 21 nanomarkers on the bottom level, 12 markers on the 
middle and 5 markers on the top level, serving as well 
distinguishable control points. The distribution on the lower 
level is non-symmetrical to be always informed of the 
pyramid’s orientation. Figure 4 shows that the imaging 
ambiguities due to parallel projection are resolved with the use 
of such a pyramidal shaped calibration object. From the images 
we are able to derive the non-ambiguous tilting angles and 
eventually significant affine distortion parameters. 
Figure 4 also shows that due to the objects sloping edges, 
nanomarker points will only be invisible, when extremely tilted. 
The slopes being vertical, nanomarker points would easily be 
covered by higher pyramidal stages at low tilting angles. 
cascade 
  
  
central parallel 
proj schon projection 
image 
   
Figure 4. Central and parallel projection deliver a non- 
ambiguous result with three dimensional objects. 
Another advantage of the sloping edges is the ability to measure 
the spatial coordinates of the nanomarkers with an atomic force 
microscope (AFM). The sensor of an AFM consists of a tip with 
a radius of only a few nanometers. Due to the tip shape, it is 
difficult to measure vertical structures. With the calibration 
structure developed for this project, the AFM can access all the 
nanomarkers. Figure 5 finally shows an SEM image of the 
pyramid. 
    
1205 20000 20KV WD13 Sp 4 Pyramid 12 1 pm — 
Figure 5. SEM image of the calibration pyramid 
Such a pyramid approximately measures 6um in width and 
length and 2um in height. To give an idea about the real 
dimensions: if the pyramid had the size of a One Euro coin, this 
coin would correspond the diameter of a soccer field and be 9 
meters in height. This might help understanding the difficulties 
of moving and keeping the specimen in the eucentric axis. 
—212— 
  
  
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