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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
Table 1: Camera pair listing. Stations numbers are from 
the Zürich data set and Figure |. Distance, object depth, 
and image coverage are average for both cameras. 
  
  
  
  
  
  
  
  
  
, S oc zt 
y 8 EU E E E 3 
«22:70 #5 Ÿ sé ai 
5 e R92: 2 ÉD 2 2% ES 
e 2 22.4 B85 BA BS EF 
| 5, 14 674 133 96 477 34 15 
2 5.13 667 53 Si SI 24 29 
3 15,14 1659 85 46 81 22 19 
4 4, 25 294 38 74 28 13 42 
S 1415 283 29 8 9i 29 11 
6 28,31 263 26 24 39 14 32 
7.436,27 260 9 2 24 14 43 
8 15,28 259 89 85 54 17 24 
9. 15,31 250 81 65 64 13 11 
10 4,26 257 57 os 33 21 30 
11 4,5 249 50 71 34 16 25 
12 14,28 245 103 77 65 16 23 
]3 14,31 245 88 $7 74 12 11 
14 4, 30 211 21 32. 27 10 44 
15 4, 29 209 23 27 28 8 44 
16 5,29 205 27 44 28 10 41 
17 5, 30 200 29 39 30 10 39 
18 24,25 176 12 22. 2! 6 53 
19 4, 24 172 26 53 26 10 42 
20 25,26 167 20 24 24 9 29 
21 4,27 166 65 94 38 21 31 
22 29,30 165 2 6 22 8 44 
25 27,28 157 14 14 26 21 33 
24 512 121 58 S7 3 24 26 
25 12.13 120 6 6. 57 13 20 
26 12,14 111 79 40 84 22 17 
27 26,28 85 22 13 2] 2] 26 
28 13,15 65 56 37 68 20 21 
29 24.26 52 31 42 25 7 15 
30 5,15 SI. 4108 88 167 31 16 
31 4, 28 49 79 108 43 34 18 
32. 4 12.15 43 S] 32.72 20 19 
38 25,27 42 28 23-27 3 8 
The measured points and internal camera parameters were 
exported from Photomodeler (EOS Systems Inc.) and load- 
ed into Matlab for processing. 
6 ADJUSTMENTS 
For each camera pair, an initial guess of the relative ori- 
entation between the cameras was calculated with the nor- 
malised 8-point algorithm and the essential matrix, as de- 
scribed in (Hartley and Zisserman, 2000). All homologous 
points were used. Given the camera relative orientation, 
initial object point estimates were calculated by forward 
intersection. 
The four algorithms were given the same starting approx- 
imations and used to solve the two-camera bundle adjust- 
ment problem. To handle the datum problem, the first cam- 
era position and orientation and one coordinate of the sec- 
ond camera were kept fixed, i.e. a dependent relative ori- 
entation (Mikhail et al., 2001). The success or failure to 
converge and the used number of iterations were recorded 
591 
= 
Algorithm 1U, 2U 
  
f Algorithm 1D, 2D 
a 
Figure 2: Algorithm behaviour for Case 2, all points. Large 
initial steps for the undamped algorithms lead to failure 
after a few iterations. Transparent cameras indicate posi- 
tion estimates during the iterations, opaque cameras indi- 
cate final camera position. Light points indicate initial ob- 
ject point approximations, dark point indicate final object 
points (converged case only). 
for each algorithm. The maximum number of iterations 
was set to 20. The termination criteria was 
17a] < ev llvill and max [gi(ws)| € e; 
from (Borlin et al., 2003, Equation (16a)) with constants 
e, = 1079,e, = 1078, roughly corresponding to termi- 
nation when the largest update was around 1 mm.' Other 
algorithmic constants were set to y = 0.1, Qmin = 1079. 
Furthermore, the process was repeated on 500 random sub- 
sets of random size from each camera pair. The number of 
points in each subset were taken from a uniform distribu- 
tion between 8 and n,,,4,, where 1,,44 is the number of 
homologous points for the specific camera pair. The same 
points and starting approximations were given to all algo- 
rithms. 
7 RESULTS 
The optimisation results using all homologous points for 
each camera pair are shown in Table 2. Based on the con- 
vergence results, the cases may be classified as follows: