International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
considering the generated camera parameters and randomly 
modified itself within their a priori standard deviation (4). 
Starting point 
| (systems time) 
srand() Step 1 
Generation of Random Numbers às a 
pre-process for simulation Swp 2 
RANDOM NU 
random value [0.Rand MA 
uniform random values [ 
  
    
  
     
     
Gl = rand(): 
RNG2 
DOLL 
7 te 
BOX-MULLER Transformation 
  
  
| 
| input: uniform distributed (gRNGI, gRNG2) 
| output: normal distributed (nRNGI, nRNG2) 
Y 
Y 
For each image of the Input bundle for one simulation | 
Step3 | 
vi (exe Ye ApA A Á 
/ BB C. / 
# to be modified in quoted order / 
  
T 
{ 
| Randomly modified (rm) camera parameter 
| e.g. ¢ (m) = ¢ + (NRNGI * 5.) 
  
Further camera. YES 
< parameters to reer 
s 
7. be modified? |.— 
Y NO 
Calculation of image coordinates of one | 
image (standard observation equation) | 
+ 
  
  
A dans Allowance of image | 
^ Randomly 7, xS a age] 
T : i YES coordinate due to 
Pa 5 = 
«o allowance with > ; : 
T EON higher spread of 
respect to r^ 2 - x 
NS a rad.sym. dist. for. | 
ied E ——— Re — 
ye LT 
  
| Randomly allowance of image en = | 
i coordinate x.y" = f(s.) 
t 1 
j 
    
  
Figure 3. Flow chart of simulation data generation for one 
image / of one simulation process S 
Consideration of higher deviation of radial-symmetric 
distortion with respect to radial distance 
The radial-symmetric lens distortion Aj, A», A5 considers the 
highest influence of the distortion parameters. Due to its 
functional definition (5) 
202 dw 
Artz ra n -üy e A, ft th Az A ) (5) 
with r'  - radial distance 
rp = second zero-crossing 
Ar' = radial-symmetric distortion 
the standard deviation resp. its influence increases with the 
radial distance like graphically shown in Figure 4 for the 
following example (Table 1). 
  
  
  
  
  
  
  
  
  
  
  
  
  
Kodak DCS 645 M — 35mm lens 
Sensor format: 36.648 x 36.648 mm? 
ck -35.6637 sck 0.0005 
xh -0.0993 sxh 0.0007 
yh 0.4083 syh 0.0007 
Al -9.01 E-05 SA 1.56E-07 
A2 6.23E-08 sA2 6.88E-10 
A3 -1.48E-11 SA3 9.20E-13 
Bl 2.37E-06 sB1 2.23E-07 
B2 -1.46E-07 sB2 2.21E-07 
C1 1.06E-04 sCI 3.30E-06 
C2 -1.04E-05 sC2 3.17E-06 
  
  
  
  
  
Table 1. Example camera paramter 
  
radial distortion of standard deviation 
UD16 4-—————— —-————- 
{1.014 Sree 
0.012 
0.01 
3 0.002 
= 0.006 
0.004 
0.002 
  
  
  
  
  
  
  
  
0+ 
3 10 15 
radial distance [rur] 
  
  
  
Figure 4. Standard deviation of radial-symmetric lens distortion 
Applying this effect to the simulation process, an additive is 
calculated for the concerning image coordinate. In order to 
pursue the effect of the modified lens distortion with respect to 
the superior input bundle parameter, the additive's sign is 
generated of the difference of these two functional models. In 
the following this modification is called RADVAR- 
modification. 
4. SIMULATION RESULTS AND ANALYSIS 
The results and analyses are based on free-net bundle 
adjustments (free camera geometry), restrictively with three 
fixed scales placed to the coordinate systems axes. Because of 
the random generation of data sets, different blunder might 
appear due to an instable new data bundle. Modern bundle 
programs like BUNDY (own development of our institute), 
which is used for this simulation process, have integrated and 
non-changeable blunder detection algorithms. Strictly speaking 
the simulation results are based on different object geometry. 
The importance of this effect does mainly appear when scale 
points are eliminated within the calculation process, which 
causes different scales in object space. Concerning the 
following results and analyses these false-scaled bundle results 
are eliminated. In the following the expression input value 
defines the randomly modified values of the simulation process. 
4.1 Camera geometry 
4.1.1 Input values: With respect to the example of the 
Kodak DCS 645 M (Table 1) the normal distributed input 
values for principal distance and principal point result as shown 
in the diagrams (Fig. 5,6). The principal distance input values 
span from 36.66187mm to 36.66556mm. Regarding its standard 
deviation sy = 0.0005mm, 0.4% of all values (200 simulations * 
60 images per bundle) lie outside the triple standard deviation, 
equally 0.4% of all values for the y-direction of the principal 
   
Internati 
point, w 
with refc 
Due to a 
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are frame 
  
Figure 5 
  
Figure 
The dist 
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Figure 7. 
yield to : 
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dr nasi = 
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41.2" | 
at the out 
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demonstr 
InputB v 
example 
input val 
Prm) d