The linear solution showed a rather unstable behaviour 
and very few of the outliers were detected, especially in 
the general cases. The iterative algorithm shows a much 
more stable behaviour for the aerial image data, but also 
here very few of the outliers are detected. For "small" 
errors, set III, one error is detectable but not more. 
5.2 Strategy ll: Adding good observations 
The linear solution using eight points with repeated 
solutions were used together with the LMedS criteria for 
including correct observations (fig 4). 
Direct solution, 8 points, LMedS 
  
o 
S 
e 
=] 
  
  
  
ë 
Number of data sets 
t2 
e 
  
Number of introduced errors 
fig. 4 Adding good observations,strategy (ii), linear 
solution with 8 points, LMedS, algorithm 2 
The LMedS algorithm finds the solutions and the correct 
observations even for large number of outliers, but shows 
an unstable behaviour when the numbers of outliers are 
small. This is mainly due to errors of type I, i.e., correct 
observations have been removed. The reason for this lies 
partly in the behaviour of the relative orientation. If eight 
points are picked randomly some correct observations 
might very well have large residuals even if the median is 
low. The presence of noise influences the result in a 
similar way as for small gross errors. The number of 
errors of type I increases very quickly. 
5.3 Strategy lll: Including the outliers in the model 
The linear solution using eight points with repeated 
solutions were used together with the MDL criteria for 
modelling correct observations and outliers in a cost 
function (fig 5). The results of the MDL cost function 
shows a similar behaviour as the LMedS algorithm, but 
the unfavourable behaviour for few outliers of the LMedS 
algorithm is not present. 
When applying the  six-point algorithm formula 
[Hoffman-Wellenhof, 1979].instead of the eight-point 
algorithm to the data some interesting observations can be 
made. The algorithm has a higher percentage of recovered 
orientations when data contains large fractions of outliers. 
For few outliers the algorithm makes more errors of type 
I, especially for "small" error in data set IV and a high 
noise level in data set II. 
46 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
Direct solution, 8 points, MDL 
  
o 
e 
a 
= 
  
  
  
+ 
=> 
Number of data sets 
t2 
e 
  
Number of introduced errors 
fig. 5 Modelling correct observations and outliers using 
MDL in a cost function, strategy (iii), eight points 
linear solution, algorithm 2 
Direct solution, 6 points, MDL 
  
  
  
  
HI 
100 
5 80 
s 
S 60 
= 
9 40 
X 
5 
z 20 
0 
  
Number of introduced errors 
fig. 6 Modelling correct observations and outliers using 
MDL in a cost function, six points algorithm, 
algorithm 3 
6. DISCUSSION 
The calculation of relative orientation parameters in the 
general case is a difficult task, in which the geometry of 
the observations interact with dependencies and 
correlations of the estimated parameters in a very 
complex manner. Large fractions, and indeed also small 
fractions, of erroneous observations or outliers can 
severely disturb the solution. 
The chosen algorithms in this study are not claimed to be 
the optimal ones, but the conclusions and comments are 
believed to be general. 
Several methods for detecting outliers and making the 
estimates more robust than the LS estimate have been 
presented in the photogrammetric society over the years, 
most of the methods having in common that they look at 
the residuals originating from some type of LS estimate 
using all data. Observations meeting some criteria are 
then kept while others are removed or given new weights 
and the process is iterated until no more points are 
removed. The types of weight functions and test statistics 
range from looking at the residuals [Krarup et al 1980] to 
more statistically elaborate methods like data snooping 
     
   
   
   
  
     
   
    
   
    
     
    
  
   
    
  
    
  
  
     
  
    
   
   
   
   
    
    
   
   
    
   
   
   
   
   
   
    
   
    
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