anbul 2004 
2] 
2D 
ints. The 
onverges. 
and con- 
vergence 
D. 
| case 22, 
t the two 
is behind 
lation by 
baseline 
)ed algo- 
hms. In- 
aled that 
22 itera- 
ptimisa- 
case are 
fferences 
ained al- 
1ed algo- 
er failure 
substan- 
d by the 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
js 
Algorithm 1U, 2U 
Ca i: 
: T Di 444 t 
++ 3 x i 
UeARUCAM LAU 
lieb ai 
A sit i i 
i ii ii: i 
A d 
  
Algorithm 1D, 2D 
Figure 4: Algorithm behaviour for Case 6, all points. The 
damped algorithms use more iterations than the undamped. 
damped algorithms; in case 25 only 594. 
In the 12865 cases where all algorithms converged, the av- 
erage number of iterations used by each algorithm were 
4.4] (Alg. IU), 4.52 (Alg. 1D), 4.38 (Alg. 2U), and 4.52 
(Alg. 2D). 
No case of convergence towards a matrix M with determi- 
nant —1 was detected. 
8 CONCLUSIONS 
When a non-linear problem is solved as a sequence of lin- 
earised problems, the convergence properties will be af- 
fected by how good each linear approximation is of the 
original problem. Any linearisation of a continuous func- 
tion is “good” in a neighbourhood of the approximation 
Table 3: Percentage failure for each algorithm on 500 point 
subsets of each camera pair. 
  
  
  
  
  
  
  
  
Pair Failure [%] 
# Stations Alg. IU Alg. ID Alg. 2U Alg. 2D 
1 5, 14 68 6 60 4 
2 5.13 60 22 58 23 
3 13, 14 4 3 4 3 
4 4, 25 10 3 9 3 
5 14, IS 5 S 5 5 
6 28.31 7 4 7 3 
7 26, 27 3 2 3 2 
8 15,28 14 4 14 4 
9 15, 31 12 3 12 3 
10 4, 26 2 1 2 1 
11 4,5 5 3 3 3 
12 14, 28 8 3 9 3 
13 14, 31 9 4 9 4 
14 4, 30 3 1 3 ] 
15 4,29 6 2 6 2 
16 5, 29 4 3 5 3 
17 5,30 3 1 3 1 
18 24, 25 16 3 18 3 
19 4, 24 6 4 7 4 
20 25,26 3 1 3 1 
21 4,27 3 2 4 2 
2 29, 30 77 53 77 53 
23 27, 28 22 6 22 6 
24 5.12 15 8 13 9 
25 12,13 94 90 95 90 
26 12, 14 23 11 23 11 
27 26,28 14 5 14 6 
28 13,15 87 16 87 16 
29 24, 26 14 [1 14 H 
30 5.15 15 11 15 11 
31 4, 28 11 7 11 7 
32 12. 15 67 18 66 19 
33 25,27 18 7 17 7 
Average 21.5 9.9 21.2 9.8 
point. However, the size of the neighbourhood varies. If 
the problem is highly non-linear, there is a risk that the up- 
dates calculated by solving equation (3) or (5) are outside 
this neighbourhood, which may lead to slow convergence 
or failure. 
In this paper, two parameterisations of the rotation matrix 
Were compared with respect to their effect on the conver- 
503 
gence properties of the bundle adjustment algorithm. The 
hypothesis was that since the rotation matrix parameteri- 
sation is less non-linear than the Euler angle parameteri- 
sation, the linear approximations uséd by e.g. bundle ad- 
justment would be better, and the convergence properties 
would improve. 
In addition to the two parameterisations, the /ine search 
technique was optionally applied. The line search tech- 
nique is a different approach to reducing the risk of adding 
updates that are too large. Different line search techniques 
exist, but the common denominator is that a fraction œ < 1 
ofthe update is added. The fraction is determined such that 
the new parameter approximations are better than the cur- 
rent. 
The results indicate that the re-parameterisation hypothe- 
sis holds, but that the difference is probably too small to 
justify using it, except to avoid singularities. However, the 
results show that using line search, the number of conver- 
gence failures on average is reduced by 54% at the insignif- 
icant extra cost of 0.15 extra iterations. Thus, we conclude 
that a technique such as line search should be part of any 
standard implementation of bundle adjustment.