The short base by, guarantees for a small number of ambi- 
guities in the establishment of correspondences between 
image 1 and 2, while the long base by3 fulfills the require- 
ment of good depth accuracy. As shown later (Eq. 19 - 23), 
this arrangement can minimize the probability of occurence 
of ambiguities but does not take into consideration that 
ambiguities can be solved; thus it does not represent an 
ideal setup if the total number of unsolvable ambiguities is 
to be minimized. 
Like the method of intersection of epipolar lines this 
collinear arrangement has some remaining ambiguities, 
which cannot be solved. Two kinds of ambiguities can be 
distinguished: 
  
  
1 h3 
b ? 5 r 
13 123 
Figure 10: length of epipolar line segments for three-camera-setup 
1. A point R’” is accidently imaged in the search area 123 
of a ‘wrong’ candidate Q” on 1,2. 
With 
by 
la = 2-£-— , 
123 bo 
LO. (bi — b) * (Zo, 7 Zmin) 
23 7 7 7 
min max 
  
one receives 
4 (n- 1) -£?- b, 
P = Pin Por = 
a(l) 12 23 F-. (b14 — b45) 
(Eq. 16) 
29 
2. A second point Q"" is detected in the search area 154 of 
the ‘correct’ candidate: 
2: (n — 1) (£l 
Pig = 5 
4. (n — 1) £2. b, 
TRES Pr : (Eq. 17) 
With (Eq. 17), (Eq. 18) the probability of an unsolvable 
ambiguity for this camera arrangement becomes 
P, Ply Poo 
4 - (n—1) -e?-bj, 
= Eq. 18 
F:by: (bi 7 bi) ab 
and the number of remaining unsolvable ambiguities is 
. 4 - (n?- n) - £?. p?, 
eT F Bo (Gab) - (Eq. 19) 
If n, £ and b5 are given by the the number of targets, the 
image quality and the requirements of depth accuracy, the 
optimum choice of by, can be calculated; for P, > min 
the derivative (dP,) / (db,,) has to be zero: 
  
oF, ! 0 
ob; 
4. (n-VD7E Dig 1 I S 
F ar =) 
=> bi, = b,3/2 (Eq. 20) 
This shows that the ideal camera arrangement of three 
collinear cameras is a symmetric arrangement with bj5 — 
b33 = b,3/2. Like the method of intersection of epipolar 
lines the length of the epipolar lines does not have an influ- 
ence on the number of ambiguities. The efficiency of the 
method is almost as good as the method of intersection of 
epipolar lines (see table 1). 
2.3 Comparison of the methods 
The expectable numbers of remaining ambiguities for the 
methods discussed above are compiled in table 1 for real- 
istic assumptions for the number of particles (n), the depth 
range in object space (AZ) and the width of the epipolar 
search area (g) for a base bj4 = 200 mm and a camera 
constant c = 9 mm: 
Table 1: numbers of remaining ambiguities 
  
  
  
Number of cameras 2 3 3 
arrangement coll. triang. 
eaaet enne e ihe renes deni Mp de enee see deu eren ette: 
parameters : 
n-1000,£-10um,AZ-40mm | 401 40 35 
n-2000£-10um,AZ-40mm  : 1605 160 140 
n = 1000, £ = 5 jum, AZ = 40mm } 201 10 9 
n = 1000, £ = 10 um, AZ = 80mm | 3802 40 35 
^ 
  
With two cameras the expectable numbers of unsolvable 
(but detectable) ambiguities becomes that large that the 
method itself becomes questionable. The geometric 
constraint of a third camera leads to a reduction of the 
numbers of ambiguities by at least one order of magnitude. 
If the number of remaining ambiguities is still considered 
too large, a further reduction is possible in a straightfor- 
ward manner by employing a fourth camera and either 
  
    
pmo 068 PA n4 - 08 ~~)