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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
2. MONTE-CARLO-METHOD 
The Monte-Carlo-Method (MCM) is a statistical simulation 
technique. Within the simulation process it generates a sample 
of non-interdependent variations of which the optimum will be 
chosen. The probability finding the absolute optimum increases 
with the number of simulation trials (Schmitt, 1977). Statistical 
simulation techniques are useful solving complicated linear 
systems. Furthermore the MCM can be used for solving 
problems, which focus on the evaluation of uncertainty and 
randomness of single system components, and additionally 
getting information about the whole systems behaviour, 
(Schwenke, 1999). Cox et al. (2001) divides the uncertainty 
evaluation process using MCM into two phases: Phase | 
includes as formulation phase the declaration of the probability 
density function (pdf) (1) of the input quantities (2). 
ZO = (Yn Zl (1) 
Xk Xx (2) 
The pdf's, together with the measurement model constitute the 
inputs to the calculation phase (Phase 2) for the Monte-Carlo- 
Simulation process (Cox et al., 2001). Figure 1 shows the flow 
chart for a Monte-Carlo simulation process. 
Model Y=f(X) 
Probability density | Number M of Monte 
functions g(X) | Carlo trials 
i T 
| Y a 
i M samples x,.....X | 
Y | 
of X from g(X) : | 
M model values 
Y= YY) = {fx fx} 
| M sorted model values | 
  
  
  
  
  
  
  
  
  
Ya Yım) 
  
Figure 1. Flow chart of simulation process (Cox et al., 2001) 
After specifying the functional model to be simulated, an 
appropriate probability density function g(X) needs to be 
selected. In this case of simulating photogrammetric bundles a 
univariate normal distribution, known as Gaussian distribution, 
has been chosen. Using numerical pseudo-random number 
generators uniform distributed numbers within an [0,1]-interval 
are the basis for randomly controlled simulation processes 
(Schmitt, 1977). Most programming languages support the 
generation of uniform distributed random numbers, algorithms 
like Hill-Wichmann or Kiss are applicable, too. The Box- 
Muller Algorithm (3) provides the generation of values from 
the standardized Gaussian distribution N(0,1) (Cox et al., 2001). 
If U,, U, are independent and identically continuous uniform 
distributed Ujo, random values, the variables X, and X, 
defined by 
X= 2log(U, ) cos(2zU » ) 
Mo = - 2log(U; )sin(2zU ; ) 
are then independent and identically univariate normal 
distributed No; values (Robert & Casella, 2002). All normal 
density curves (Gaussian distribution curve, Fig. 2) satisfy the 
following property. 68.3% of the observations fall within 1 
standard deviation of the mean, 95.4% within 2 and 99.794 
within 3 standard deviations of the mean for infinite random 
samples. Thus, for a normal distribution, almost all values lie 
within 3 standard deviations of the mean (Narasimhan, 1996). 
For finite-dimensional samples the normal distribution is 
replaced by the student's distribution (Graf et al. 1998). 
fo 8 += 
  
7-8 [4245 
  
  
3-6 | +364 
  
  
  
  
  
Figure. 2 Gaussian distribution curve 
3. MODEL DEFINITION AND IMPLEMENTATION 
The simulation process is based on existing and evaluated 
image bundles (/nputB) that are made for verification purposes 
of high-resolution digital cameras. Therefore the whole 
simulation process is based on the standard observation 
equations. In case of evaluating the simulation process 
considering the described extended camera model, the standard 
observation equation is extended by image-variant parameters, 
which is explicitly exposed at Hastedt et al. (2002). 
Due to many calculations caused by a high number of input 
values for one bundle adjustment, a step-by-step simulation is 
carried out. Caused by the definition of the C++ random 
number generator, all needed random numbers are first 
generated (Steps 1-2, Fig. 3). Dependent on the predefined 
number of Monte-Carlo trials (S), the process of data generation 
and calculation of the bundle adjustment will be executed S 
times. For each image / of one bundle the data generation will 
be executed as shown in Figure 3, Step 3. 
First, camera parameters to be modified are randomly changed 
within their standard deviation arisen from the input bundle. In 
order to be able to analyse single system components and their 
influence within their standard deviation, the random 
modification. of the camera parameters is selectable. One 
parameter of interior orientation will then be recalculated (4). 
Pom) = P(iy) s (nRNG, * $5] (4) 
with P{rm) = randomly modified parameter 
Piv) = parameter's input value of /nputB 
nRNG, - normal distributed random value 
Sp = parameter's standard deviation of /nputB 
Afterwards the image coordinates need to be recalculated. 
Using the standard observation equation the image coordinates 
will be generated from predefined object space to image space