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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