EXTENSION OF THE 11-PARAMETER SOLUTION 
FOR ON-THE-JOB CALIBRATIONS 
OF NON-METRIC CAMERAS 
Hanspeter Bopp 
and 
Herbert Krauss 
Institut für Anwendungen der Geodäsie im Bauwesen 
Universität Stuttgart 
ABSTRACT 
After a short characterization of the 11- 
parameter solution the basic equations in 
the case of on-the-job calibrations are 
given. For the application of this method 
to non-metric cameras it is necessary to 
extend the mathematical model to lens 
distortion. It is described how a special 
model for symmetrical radial and decentering 
lens distortion can be incorporated into the 
non-linear least-squares adjustment. 
The method is tested with data of some 
restituted slides taken with a non-metric 
camera during a photogrammetric control 
survey. 
1. INTRODUCTION 
The general projective relationship between 
the coordinates of points in a 3-dimensional 
object space and the corresponding image 
coordinates on an image plane can be re- 
presented by two linear fractional functions 
containing 11 transformation parameters. In 
this general singular collineation no 
restrictions concerning the coordinate systems 
are made. Since in photogrammetry orthogonal 
Systems are normally used, we have to accom- 
modate the transformation to this fact. 
This restriction does not affect any 
translation of the coordinate systems. 
That is why the image coordinates can be 
given in any orthogonal systems, e.g. in the 
System of a comparator. 
In some mn-topographic applications of 
photogrammetry, as the photogrammetric 
survey of complex structures, the general 
case with multiple arbitrarily chosen camera 
Stations, object space control and high 
oblique convergent photographs has to be 
used /3/, /4/. 
If the solution is based on the conventional 
collinearity equations the determination 
of initial values for the unknowns requires 
some efforts. The initial values of the un- 
knowns in the 11-parameter solution, 
however, can be determined out of a linear 
system of 11 equations. 
In /2/ the 11-parameter solution was in- 
troduced as an orientation and calibration 
method for non-topographic applications. 
There the relation between the 11 trans- 
formation parameters and the 9 parameters of 
the interior and exterior orientation in the 
conventional collinearity equations was 
shown. 
In this study we consider only the case of 
an on-the-job calibration and extend the 
11-parameter solution with regard to lens 
distortion. 
2. BASIC EQUATIONS OF THE 
11-PARAMETER SOLUTION 
The collinearity equations of the 11- 
parameter solution are represented by the 
linear fractional functions 
a4X + a„Y + az 
X = 
agX + anv + ay’ + 1 
Z + a) 
(1) 
es agX + acy + a7% + ag 
agX * anf + a2 + 1 
where X,Y,Z are the object space coordinates, 
Xy are the corresponding measured 
comparator coordinates of a point, 
a. are the 11 transformation 
i 
parameters. 
In the case of an on-the-job calibration, 
which is considered here, the trans- 
formation parameters have to fulfil the 
two constraints /2/ 
  
  
2 n4 
2 2 2 2 2 2 C"-B' . 
(a, + a, + az) (ag * ac a7)+ D = O 
(2) 
Boe Ç = 
A D = 0 
pz 8489 + 85849 + 83844 
C = agag + 86846 * 82844 
zn 2 
D = 89 * #40 * 944 
3. EXTENSION OF THE MATHEMATICAL 
MODEL 
In order to apply the 11-parameter solution 
for on-the-job calibrations of mn-metric 
cameras, an extension with regard to lens 
distortion is necessary. 
A possible model for the correction of 
symmetrical radial and decentering lens 
distortion is used in /5/. For this study we 
choose this model which corrects the 
symmetrical radial influence by the even- 
ordered polynomial 
d, = kr“ + kr + k,r (5) 
and the asymmetrical influence by