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distances in object space 
coordinate differences 
object point coordinates 
projection centres 
linear dependencies (planes, straight lines) 
The mathematical model of the bundle adjustment is based on the equations for perspective projection (Gründig, 
1985a): 
(X; T X, = AR, (x, r X,) (1) 
with 
BR... rotation matrix 
Xi aa object point 
Xo; - projection centre 
Xi -- image point 
Xoj ... elements of interior orientation 
AM ee scale factor 
In the first two steps of the bundle adjustment the projection centres, rotation matrices, scale factors, and object 
points are the unknowns. The image coordinates do not serve as observations, but there are pseudo observations: 
0- Qo Ry (X= x) -(X- Xj (2) 
In addition the first step uses some simplifications: 
e the scale factor A is assumed to be the same for all points of one image 
e the elements r of the rotation matrix are used as independent unknowns; there are constraints which are formu- 
lated as observation equations to softly force the matrix to be almost a rotation matrix. 
Obviously, this is not a correct geometrical model but the resulting equation system is linear (except for the constraint 
equations). The result is good enough to supply approximate values for the second step, which uses the correct geo- 
metrical model . The third step is a conventional bundle adjustment which is not described in detail. 
3.2.2 Calibration 
As in most bundle programs there is a set of parameters for camera calibration. The third step of the bundle adjust- 
ment provides an appropriate parameter set which can principally be applied in an image orientation, including simul- 
taneously a camera calibration. However, it makes sense to calibrate the cameras in a dedicated procedure with a 
certain geometrical configuration and a specially designed point field with a large number of signalized points. There- 
fore, the calibration can also be highly automated and can be repeated from time to time to check the camera parame- 
ters. 
There are 7 parameters for each camera: 
e Xo, yo principal point 
a8 0 — 3 focal length 
e aj, a2  ... parameters for radial distortion 
e 44804  .-- parameters for scale and shear 
The parameters a;...a4 are applied as follows: 
X= x-x-{a fr - 2) tarn) (3) 
y=y- y (a. = rn?) * aj(r T r^) 
with 
2 2 
2 
r zx x) +(y- yo) 
r,> = const 
and 
X-X-X:8,-y.a, : (4) 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences’, Zurich, March 22-24 1995