P is an arbitrary point and S is a fixed point on the line, 
d is the direction vector of the 3D line and t is a scalar 
factor. The endless 3D line is totally described with the 
six dependent parameters, Xs, Ys, Zs and Xq, Ya, Za- 
Point S can be arbitrary chosen, the same is possible for 
the length of the vector d. This means that two addi- 
tional constraints are necessary to get an unique 
solution for the endless line. Person [4] and Mulawa [3] 
chose the constraint |d| = 1, which requires that the 
length of the direction vector d is unique. Secondly the 
point S is defined in such a way, that the radius vector 
to point S and direction vector d are perpendicular. 
This is realised by setting the scalar product ( d * S) to 
Zero. 
Mulawa [3] introduced a coplanarity condition in 
object space as link between the inner and outer orien- 
tation elements of the image, the image coordinates of 
an arbitrary line point and the unknown line 
parameters. This relation is expressed by equation (2) 
and shown graphically in figure (1). 
   
X 
Figure 1: shows the coplanarity constraint which com- 
bines the 3D straight line, the image measurements 
and the orientation parameters of the images. 
| eCc,dmi - 0 (2a) 
x 
m = R! 7] (2b) 
-C 
The projection centre of the camera in cartesian 
ground coordinate system is represented by C and m is 
a vector including the measured image coordinates of 
an arbitrary point transformed into a coordinate 
system parallel to the ground coordinate system. This 
transformation is done by the rotation matrix Rt. 
2.2 Four straight line parameters 
The fact that the parameters ( Xs , Ys, Zs, Xa , Ya, Za?) 
of a straight linear feature are dependent cause some 
disadvantages. Numerical problems may arise in an 
overestimated case, where a least square adjustment 
algorithm is used to calculate the line parameters. 
There exist two possible reasons. First, the size of the 
matrices used in the adjustment process get larger and 
second the a posteriori covariance matrix of the un- 
known line parameters is singular. On the other hand 
there are two advantages using the parametric straight 
line equation and the coplanarity constraint presented 
in chapter 2.1. They prevent the possibility of a geo- 
metric interpretation of the results. Second, the noise 
670 
parameters, the scalar factor t and the coordinates of 
point P are isolated from the process. 
Due to the reasons described, an alternative solution 
will be introduced using four straight line parameters. 
These parameters still allow the use of the coplanarity 
formulation as link between image measurements and 
the 3D line. This means that the parameters S and d 
have to be expressed as functions of four preliminary 
parameters A, B, C, D until the final parameters are 
defined. 
I SSA,B,C,D)-C,d( A,B,C,D),ml 
0 (3 
In a cartesian system the point S on the 3D line can be 
described by the two angles 8, $ and the distance r. à is 
the angle between the Z-axis and the radius vector to 
points S. ¢ is the angle between the X-axes and the 
projection of the radius vector to S onto the X-Y-plane 
and r is the distance from origo O to point S. 
Using 3, ¢ and r the cartesian coordinates of the point S 
are expressed by 
Xg = rsind cosó (4a) 
Yg = rsind sing (4b) 
Zg - rcosó (4c) 
To obtain the fourth line parameter, a spherical coor- 
dinate system is defined with its origo in point S and 
the three orthonormal base vectors ( eg, eq, er ). 
  
  
  
  
  
  
Figure 2 : Point S and vector d expressed by the four 
parameters 6, Q, r, y. 
The vector eg is the tangent to the meridian going 
through S. Vector eg is the tangent to the parallel circle 
through point S and vector ey is the normal vector of 
the plan (eg, eg) defined by the cross product (eg x eg).