/ 
/ 
4 ^ 
/ 
Aa 
/ 
7 
A! 7 
x 
/ 
  
  
   
   
    
  
  
  
  
  
  
   
  
   
  
  
   
  
   
   
   
  
  
   
  
  
  
  
   
  
  
  
  
  
  
   
   
    
f 
Lo 
ob:ect diane a 
Figure 1: The model 
respectively, can be determined relative to an arbi- 
trary coordinate system O'; X', Y' in 7. 
It is required to find the space position of the image 
plane 7 and the camera station S relative to the space 
coordinates X,Y, Z. This will be done by finding the 
line of intersection t of with a (called trace), as 
well as its angle of inclination w, as well as the space 
coordinates of S(Xs, Ys, Zs) will be determined. The 
principal point P’ and the focal length f can then be 
easily calculated. 
3. DETERMINATION OF THE ORIENTA- 
TION PARAMETERS 
As said before, exactly five control points are needed 
for the solution of the problem. 
3.1 Homogeneous coordinate systems 
In the object plane o, a homogeneous coordinate sys- 
tem is chosen with origin at A1, the r1- and r2-axes 
passing through A5 and A4 respectively, fig.(2), and 
the zg-axis at infinity. The homogeneous coordinates 
of a given point are the three ratios (zo : z1 : z2), not 
all zeros, which can be arbitrarily multiplied by any 
non-zero factor. From fig.(2), our control points have 
the following Homogeneous coordinates: 
A109 70) A;(0 70: 0); A300: 6; 702): A40 *0 * 4), 
where, since zg is taken — 1; b,d,c; and c9 are met- 
ric lengths, which can be calculated from the space 
coordinates, or directly measured. 
A similar homogeneous coordinate system is chosen in 
the image plane 7 as shown in fig(3). The coordinates 
2 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
  
22 
—- 
X 
Figure 2: Homogeneous and rectangular coordinates 
ina 
of the image points are: 
AM: 0:50), AL. 9.0), ASQ 6. c5), AL 0 : 4) 
and AZ(1: 1^ : 13). 
where, as before, 5b',c5,c5, d',t! and t5 are metric 
lengths. 
" 
  
  
  
9: 
Figure 3: Homogeneous and rectangular coordinates 
in 7 
3.2 Steps of solution: 
We are going to carry out the following steps: 
1. The collinear relation between the correspon- 
dence 
Aj€a — Alem i=1,23,4 
will first be determined. 
2. The vanishing line v € o and v' € 7 are found. 
3. À pair of congruent corresponding lines t € a = 
i' € 1 is determined. t should be the trace of = 
on a. 
3.3 
Q 
     
. The plai 
the cong 
In this p 
age poin 
toa 
. The coll 
terminec 
a: axis i 
. The tru 
space lie 
the vani: 
ular to t 
. Through 
T € a is 
Hence tl 
in space 
its true 
As and 
plane v. 
Transfc 
Fig.(4a,b) sh 
Ai(zo DI 
homogeneous 
tions of such 
in the form: 
8 8 
t2 
Hu og 
where aj; are 
Substituting