3. S-TRANSFORMATIONS IN PHOTOGRAMMETRY 
In the previous section it was demonstrated that matrix S can be calculated when 
the null-space of N is known and a constraints-matrix B is chosen. 
Teunissen (ref.3) showed that the null-space of N is obtained by linearizing the 
two/three dimensional similarity transformation with respect to the assumed S- 
base. 
Consider a free-network photogrammetric bundle adjustment. For some reasons 
the S-base was chosen so that 
- the attitude of the first photograph which means w,,, and "X, is kept 
fixed; 
© c c 
- the position of the perspective center of the photograph X t y, and Z, is kept 
fixed; 
- the position of the x-coordinate of the perspective center of photograph n (X) is 
kept fixed. 
The three-dimensional similarity transformation is: 
X X t 
X 
(17) |Y| * AGRO. Y {+ ty 
Z z t 
I I 
where R,,, R, and R4 are the well-known orthonormal rotation matrices around 
the x-axis, y-axis and z-axis respectively, where t, , t, and t, are the translation 
vector components and where A isa scale factor. 
The differential similarity transformation assuming A lus 0,00 = 0, X= 0, t 
=0, t, = 0 and t, = 0 (approximate values) then becomes: 
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