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camera-axes, non-negligible lens distortion and multimedia 
geometry (object and sensor in media with different refrac- 
tive indices) the epipolar line will be a slightly bended line. 
Its length / can be restricted if approximate knowledge 
about the depth range in object space is available, e.g. the 
range of the illuminated test section. Adding a certain toler- 
ance width € to this epipolar line segment (due to data 
quality) the search area for the corresponding particle 
image becomes a narrow twodimensional window in image 
space. 
2. Two-camera arrangement 
With the large number of imaged particles a problem of 
ambiguities occurs here, as often two or more particles will 
be found in the search area. If the particle features like size, 
shape or color do not allow a reliable distinction of parti- 
cles, these ambiguities cannot be solved by a system based 
on only two cameras. 
For a quantification of the probability of the occurence of 
ambiguities a point P centered in object space shall be 
considered: X = b,,/2, Z = (Zmin + Zmax)/2, Y = 0 
(Figure 2, consideration in epipolar plane without loss of 
gencrality). 
Pi Zmax 
( 5 ^. Zmin 
  
b 
12 Im 
Figure 2: length of epipolar search window 
With 
  
  
  
X X 
E X = €max' 7 , x, = Zin ; > (Eq. 1) 
the length of the epipolar search window becomes 
l x" ^" fms E] 
12 = —x = C+ de 
2 ! Zin Zmax 
= b12 ) 
I C . — —M! 
Zmin Lona = 
C-by-(Z.,..—7Z ; 
= 12 ( mex min) (Eq. 2) 
Loin : Lor 
and with the average number of ambiguous particles per 
search window 
Pz, = (n- 1) "uec (Eq. 3) 
one receives the expectable number of ambiguities per ster- 
eopair 
2-eie bo (er Zi) 
F7, 2 .  (Eq.4) 
max 
max 
N, = (n? — n) . 
  
The number of ambiguities grows 
* approximately with the square of the number of parti- 
cles 
* linearly with the length of the epipolar line segment 
* linearly with the width of the epipolar search window 
With realistic suppositions for the number of particles per 
image and the dimensions of the epipolar search window in 
a reasonable camera setup the number of ambiguities to be 
expected becomes that large (see table 1), that a two- 
camera-system will not allow for a robust solution of the 
correspondence problem, if the number of targets or the 
depth range in object space cannot be controlled strictly. 
Instead algorithms based on three or more cameras rather 
than two will be discussed in the following, which allow a 
drastical reduction of the expectable number of ambigui- 
ties. 
2.1 Intersection of epipolar lines 
À consequent solution of the problem is the use of a third 
camera in a setup as shown in Figure 3 with the aim of 
reducing the search space from a line plus tolerance to the 
intersection of lines plus tolerance. 
  
  
  
  
  
Figure 3: arrangement of three CCD cameras for the method of 
intersection of epipolar lines 
This setup can be exploited as shown in Figure 4: 
  
  
  
  
  
13 (223) 
  
  
  
  
  
  
  
  
  
  
Figure 4: principle of intersection of epipolar lines