Legenstein, Dietmar 
  
distinguish these indices from the tensor indices they are written in capital letters on the left-hand side of the symbols. 
The basic equation for the co-ordinate transformation is the spatial similarity transformation. 
The transformation of the projection Ky from the image co-ordinate system to the reference system is given by, where s' 
is the vector in the reference system, ps’ is the vector in the image system and 5A'; is the rotation tensor of the similarity 
transformation: 
i 
s'=,R'; ,s! (2.1-4) 
The normal vector is transformed from the model co-ordinate System to the reference system. Instead of the rotation 
tensor sR’; the tensor MR}, of the model system has to be used, where n' is the vector in the reference system, y is the 
vector in the model system and MR; is the rotation tensor of the similarity transformation 
i 
nz, R; ín! (2.1-5) 
The direction of the vector gs in the image system is given by the difference of the centre of projection gXy and the 
image co-ordinates: 
B Sj7pX,7 goi (2.1-6) 
p aX BX pX2 image co-ordinates of the point of contour 
sXel s. 2Xex] do B Xo1> B-Yo2 principle point coordinates 
0 ó C principal distance 
The direction of the surface normal vector yn in the model system, in the case of an implicitly given surface ® is given 
by, where M® = 6 (yX) - 
  
n° = (2.1-7) 
M 2 x 
Mh is a function of 4X and has to be evaluated at the approximation for «X. In the next step the two vectors are 
transformed to the reference system according to (2.1-4, 2.1-5). 
2.2 Linearisation of the Condition of Contour 
The overdetermined system can be solved using least squares approximation [Kraus, 1994 p382ff]. Here the necessary 
linearisation is presented because of its importance in chapter 2.3: 
Observations, e.g. image co-ordinates or conditions of contour, are functions of the unknowns: 
DT York) (2.2-1) 
where /; are the observations and x; the unknowns. Observation equations like (2.2-1), cannot be used for the 
approximation right away because they have to be linearised beforehand. In the process of linearisation at the point x’; 
the approximate functions /;, that in general are non-linear, are substituted by linear ones. In the case of n unknowns this 
leads to an n-dimensional hyper-plane. 
  
474 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 
TI