International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
  
  
coordinates of some feature points (L', L", C', C") in each 
image. 
In Figure 4 two planes can be seen: each containing the 
projection rays of all points of L producing the images L' and 
L^'. Some of these rays are used for the reconstruction of L by 
measuring the corresponding image point.  Fictitious 
observations state the fact that the object point — which is non- 
uniquely defined by the image ray - lies on the straight line L. 
  
Figure 4: Line reconstruction with non-homologous points 
In Figure 5 the shape to be reconstructed is a circle (C) in 
object space. On the images of this circle (C' and C") non- 
homologous points are observed through a cone of rays. 
  
Figure 5: Circle reconstruction with non-homologous points 
3.1.1 The Similarity Transformation 
The mathematical implementation of such ideas in ORIENT is 
based on the spatial similarity transformation (Figure 6). 
This transformation describes the relation of points in two 
different Cartesian coordinate systems, labelled ‘local’ and 
‘global’ coordinate systems (Eq. 1 and 2). 
X. AA 
y-»e4a- 8] y-v, (D 
ZZ 7-7 
or (-£)-4 R'(Y— Y) (2) 
where R is the spatial Rotation Matrix, 
po holds the coordinates (xo, vo. zo) of the inner 
reference point (i.c. in the local system), 
P, holds the coordinates (Xo, Yo, Z5) of the 
exterior reference point (1.e. in the global 
system), 
A is the scale 
factor that converts the 
  
  
  
  
length |(P - Po) to (=p, | 
Pp 
5 i | 
AZ * 
| 
12 
x | 
Xo 
Y 
Yo im X 
  
  
Figure 6: Similarity transformation 
When assuming that the local coordinate system is the image 
space coordinate system, equation (1) (Kraus, 1996) can be 
rewritten as follows: 
Sets "uGX — XQ) t rg Q Na) HA - Z4) 
y-7 yo |2 Acl ro (X 7 Xo) rS - Yo rs (Z - Z4) 
— Cy BX =X Ver yt.) 
(3) 
where rj are the elements of the rotation matrix, which is 
parameterised by three angles (Kraus, 1996). 
This (Eq. 3) is the well-known collinearity equation describing 
the image to object space relation in central perspective 
geometry. 
After elimination of A, the observable image coordinates can be 
expressed through: 
  
cU An - tn 729 (4) 
Xm ace 
aX =X Yn Y ~Y Vi (2-2) 
GC: nj(X A) TFT) FL Zy) 
R(X =X) + mY =X) +m (Z2-Z,) 
(5) 
  
XY. 7 
3.1.2 Observed Planes 
A local coordinate system is considered in which the plane is 
mathematically easily describable (e.g. xy-plane). The local 
coordinate system is linked to the global coordinate system 
through an arbitrarily chosen reference point Py and its rotation 
parameters (see Figure 7). 
The observations are the zero-z-coordinates of the points lying 
in the xy-plane. 
Xo. Yo and Zy can have any value in the global coordinate 
system. In this particular case only zo of the interior reference 
point Py is of interest, while xy and y, are equal to zero. The 
scale between the two coordinate systems is usually taken as 
one. The values of the rotation matrix R describe the attitude of