med (residuals?) -> minimum 
This method is able to cope with an amount of outliers up 
to 49.9%. In this study the method is not applied. 
2.3 M-estimation, a robust adjustment technique 
In contrast to the methods mentioned before the maximum- 
likelihood-type or M-estimation is a deterministic 
procedure. In the M-estimation one ends up at minimizing 
the sum of a function of squared residuals (Huber, 1981), 
(Forstner, 1989). That function is called weight function 
with the values w.. 
Y w, 1%, - min. (2-4) 
i 
The minimization procedure is an iteratively reweighted 
least squares adjustment. A difficult problem is the proper 
choice of the weights w;. The weight function has to fulfil 
certain mathematical properties, described in (Huber, 1981), 
to guarantee the adjustment converge to a solution. 
For the method the calculation time can be predicted. 
It is planned to use the M-estimation for the line 
reconstruction with the Danish reweighting method 
(Krarup/Kubik, 1981). The residuals should be derived 
from the coplanarity values f. Some very first experiments 
showed that the method seems to be able to cope with 
noise and outliers up to a certain amount and size. The 
required approximate values might be taken from the 
results of the RANSAC technique. 
3. ALGORITHM FOR ROBUST 3D-LINE 
RECONSTRUCTION 
In this chapter a method is explained how to reconstruct a 
3D-line from errorous observations in several images. Up 
to now the algorithm is specialized on lines as used in the 
simulation experiments. Some routines and thresholds still 
have to be tested for more general cases. A method is 
presented of combining the RANSAC method with the least 
squares adjustment. In the experiments this turned out to be 
a suitable way to cope with the noise and the outliers. 
3.1 A closed form solution for the RANSAC procedure 
The underdetermined perspective view from 2D- to 3D- 
space is an inverse problem, which possesses no direct 
analytical solution. 
A closed form solution for the 6 line parameters can be 
obtained from space geometry. No linearizations or initial 
values are needed, but knowledge is required about the 
camera constant c, the position L and the orientation R of 
two perspective centers. The presented solution requires the 
input of two different image points out of two images. 
    
   
   
   
    
  
  
  
    
  
  
    
   
   
    
    
    
    
   
   
    
   
     
   
  
   
   
   
    
     
   
  
    
   
    
   
  
    
3.1.1 Closed form solution for the line center point C 
and the line direction vector B 
  
  
  
  
  
Fig. 3: Space geometry in the closed form solution 
In a first step the four image rays p, from the perspective 
centers to the image points are constructed. The observed 
ray (X;; Y;y -G ) Of a point j in image i is transformed into 
space with the help of the rotation matrix R;: 
T = R, Yi (3-1) 
Two rays belonging to one image construct a 'projection' 
plane, which is described by its normal vector 
N; = Pu X Pız (3-2) 
The space line is the intersection of the two projection 
planes. Thus the direction vector B of the line is 
perpendicular to both normal vectors N;, N,. 
p. m (3-3) 
INxN, | 
To calculate the center point C of the line, a linear equation 
system is designed. It contains the equations of two 
projection planes in space and the second model constraint. 
The position of a projection plane in space can be described 
by its normal vector N and a point lying in the plane, the 
perspective center L. 
0=N,L,+N,L, +N, Ly, 7 d, (3-4) 
O-N,L,*N,L, *N,L, - d, 
Knowing N,, N,, L, and L, we get d, and d, out of these 
equations. Then the 3x3 linear equation system can be 
solved for the line center point C. 
constraint: B.C, + p, C, * N C, - 0 
planel: N,, C, + N,, C, + SC =d, (3-5) 
plane2: N,,C, +N, C +N, C= d,