which must be checked continually and corrected for 
if not achieved. The corrections can be done as a self 
calibration of the principle point in the bundle 
adjustment. 
The high speed cameras normally used have a 
rotating prism and a continuously moving film. The 
errors caused by this type of camera can be modelled 
as a scale factor between the axis and separate radial 
distortion coefficients for x and y. 
Since the film is digitized with a scanner the 
calibration must also be able to pick up the errors 
caused by that procedure. The main type of error is 
most probably a scaling error between the coordinate 
axis and will be modelled as such. 
In the single camera calibration the type of calibration 
parameters can be any combination of the following: 
  
- Principle distance 
- Principle point 
- Radial distortion 
separate in x and y with a 3 ° polynomial 
a uniform 5 ? polynomial 
- Scale difference between x and y 
  
  
  
The cameras can also be calibrated in the bundle 
adjustment as additional parameters. Since there are 
as many cameras as there are camera stations the 
requirements for the ground truth is high even for 
the self calibration, but since some of the calibration 
parameters can be regarded as fix from previous 
calculations the ability is still of use. It is also possible 
to use other type of additional parameters such as 
polynomial deformations. 
43.1 Mathematical Formulation The  calcu- 
lations are based on the central projection model, i.e. 
all image rays are assumed to pass through one and 
the same point, the projection centre. This leads to 
the use of the collinearity equations, which states that 
the object point, the projection centre and the image 
point lie on a straight line. If the rotation matrix, the 
projection centre coordinates and the calibration 
parameters are unknown the observation equations 
will look like : 
  
  
a11(X-X0) + a12(Y-YO) + a13(Z-Z0) 
31(X-X0) + a32(Y-Y0) + a33(Z-Z0) 
  
VX = (-X' + Xpp + corr) * lec 
ca Le à A21(X-XO) + a22(Y-YO) + a23(Z-Z0) 
VXS Cy + VPP + come Sce e GCXO) + a32(V-VO) +. 233(Z-Z0) 
  
  
  
x, y the observed image coordinates 
Xpp.Ypp the principal point 
corr correction terms for the radial 
distortion 
SC scale difference in x and y 
€ the principal distance 
X Y,Z the object point coordinates 
Xo,Yo,Zo the projection centre coordinates in 
the object point coordinate system 
811.233 the elements of the rotation matrix of 
the image 
The two equations are not linear in the unknowns. 
To solve the equation system they are expanded in a 
Taylor series where only the first degree terms are 
used. The solution is iterated until stability. 
43.2 Requirements for the test field The point 
configuration for the test field is based on the 
geometrical conditions for determining the 
unknown parameters with as few points as possible 
without loosing the quality and security in the 
estimated parameters. The unknowns are the six 
orientation parameters, principal distance, principal 
point, radial correction terms (2) and a scale factor 
between x and y giving a total of 12 unknowns. 
The principal distance needs targets distributed 
in the outer parts of the image in as large depth 
difference as possible. 
The principal point needs points in a similar 
manner as the principal distance, but is 
strengthen if 3D information is available in the 
centre of the image as well. 
The radial distortion is dependent on the 
distribution of points over the whole image. It is 
not dependent on any 3D information. 
The scale difference have similar requirements 
as the radial distortion. 
The construction of a test field with this properties, 
which at the same time is stable, foldable and easy to 
move, is not a trivial matter. A three-folded test field 
with locking devices to keep the stability is a model 
which is considered. 
43.3 Approximate values To be able to run the 
calibration approximate values must be entered by 
the operator for 
- position (X,Y,Z in object coordinate system)