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Figure 4. Relative orientation using non homologous points 
on straight line features. 
  
The unknown parameters may be determined from above 
equations by standard least squares adjustment, if sufficient 
measurements are available. How many measurements 
are needed to make the problem uniquely determined? Let 
us assume we measure N points in both images for each of 
the Mlines. This gives: 
(1) Number of observation equations - 
4N.M. 
(2) Number of unknowns - 
5 orientation unknowns: 
(3 + 2) M unknowns per linear feature; 
M(N-1) unknown distances S for points measured in 
left image; 
M N unknown distances S for points measured in right 
image. 
(3  Inequality condition - 
(4 N M) equations = 5 + M (5 + 2N - 1) unknowns; or 
M — > 
2N-4 
(8) 
As two points measurements per line should suffice (N=2) 
the required number of lines M follows to be infinite. Thus, 
the relative orientation problem appears to be non-solvable 
as stated above. 
In a variation to the above formulation, we may assume, 
that pairs of parallel lines are measured. We then ask, how 
many of such pairs of parallel lines are required for relative 
orientation. This variation of our argument follows from the 
requirements of many constructions in projective geometry 
to know the vanishing points of bundles of parallel lines 
(Wylie, 1970). Also, in most industrial tools, parallel edges 
are readily available and recognisable as such. 
We now assume measurements of N points made in both 
images for each of M pairs of parallel lines. 
This gives: 
(1) Number of observation equations - 
8 NM. 
(2) Number of unknowns - 
5 orientation unknowns; 
(6 + 2)M unknowns per pair of parallel lines; 
2M(N-1) unknown distances S for points measured in 
left image; 
2M N unknown distances S for points measured in 
right image. 
689 
(3) Inequality condition - 
(8 zN M) equations = 5 + M(8 + 4N - 2) unknowns; or 
5 
My OS 
E AN 6 
(9) 
For N equal to 2, it results M = 2,5; that means we need 
three or more pairs of parallel lines in object space to 
complete relative orientation. 
Again, the above can be verified by a geometric argument. 
Envisage a cube imaged from two stations. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
Figure 5. Exemplification of Relative Orientation Process 
The attitudes of the left image can be determined from the 
shape of the triangle formed by the vanishing points, while 
its relative position can be determined by the position of the 
focal point with respect to the vanishing points (H. Wylie, 
1970). It may however, be difficult to find three pairs of 
parallel edges in an object. This condition can probably be 
relaxed and further research is required on suitable minimal 
measuring arrangements for relative orientation.