points. For each camera used in the calculation of 
object points, a sub-matrix Q; is defined as: 
Q i 0 0 
Q = D "Tg (40) 
0.0 
where Q i is the cofactor matrix from equation (34) of 
the computed parameters for camera with index i. The 
ones in the diagonal is due to the fact that the mea- 
sured image coordinates of the used check point have 
unit weight. The resulting Qj matrix for k cameras is: 
Q 
Q = 2 = (41) 
Q 
4. RESULTS 
In order to estimate precision, the standard deviation 
o is calculated for each of the coordinate components 
of the check points. The diagonal elements q;; of the 
cofactor matrix QC. in equation (38) have been used to 
calculate o: 
s-«Yd = VE (42) 
where 69 is set to 1. 
The standard deviation have been calculated for each 
check point in the grid of figure 2. In order to get a 
value for comparison of case A, B, C and D in table 2, 
the mean deviation has been computed for each of the 
coordinate components, i.e. G,, G,, G, see table 4 and 5. 
  
y, 
| test | component | A | B | c | D | 
1 Ox 3.50 2.47 1.83 1.30 
Sy 3.78 2.66 1.88 1.30 
Oz 9.74 7.27 5.29 3.70 
2 Ox 2.32 1.70 1.40 1.01 
Oy 2.47 1.77 1.48 1.01 
Oz 6.49 5.08 4.38 3.07 
  
Table 4: Mean deviation c of the coordinate components X, Y and Z 
using 6 control points. Test 1 uses two cameras and test 2 uses four 
cameras. See table 2 for case A, B, C, and D. 
Figure 4 Shows the mean deviation in the X-compo- 
nent, 0,, for case A, B, C, and D. The mean deviations 
of the Y-component gives the same relative difference 
for case A, B, C, and D. 
40 
  
| test | component A B C D 
3 Ox 1.57 1.12 1.39 1.01 
Sy 1.66 1.13 1.47 1.01 
Oz 4.84 3.41 4.35 3.05 
4 Ox 1.46 1.05 1.37 0.98 
Oy 1.53 1.05 1.43 0.98 
Cz 4.51 3.16 4.25 2.98 
Table 5: Mean deviation c of the coordinate components X, Y and Z 
using 13 control points. Test 3 uses two cameras and test 4 uses four 
cameras. 
  
  
  
  
  
  
  
  
  
  
  
  
G 40 
case 
H A 
204 B B 
€ 
FA p 
2,04 
10 4 7 
LA 
A 
7 
LA 
0,0 - T ; 
1 3 3 4 
Fig4 Mean deviation in the X-component, oy, from the check 
point calculation in test 1, 2, 3, and 4, see table 1. 
Figure 5 shows the Z-component of the same check 
point grid as in figure 4. 
  
  
   
  
  
  
   
   
  
  
  
  
  
  
  
  
  
  
  
  
  
G 100 
case 
8,0 4 "^... 
? EB p 
C 
6,0 DÀ D HH 
40 7 
2,07 
0,03 T T 
1 2 3 4 
Fig5 Mean deviation in the Z component, ¢,, from the check 
point calculation in test 1, 2, 3, and 4. 
To get an idea of the internal reliability the correlation 
of the residuals resulting from the calculation of trans- 
form parameters can be studied. Figure 6 shows an 
average of the correlation values of the residuals for 
the four cases A, B, C, and D. Correlation is calculated 
from Oo, in equation (29) and (33) by: 
Qi; 
= a (43) 
Another way of getting an idea of the internal reliabi- 
lity is 
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