The distortion varies with the focussing distance. A 
modification in the focus distance is achieved by a 
change in the principal distance, resulting in a new 
lens distortion curve (Magill, 1955). But in this work 
we assume that the variation of the distortion curve is 
negligible. We expect moderate accuracy with non 
metric cameras, and our goal is the efficiency better 
than the full correctness. 
2. DETERMINATION OF THE RADIAL 
DISTORTION FOR NON METRIC CAMERAS 
The distortion is normally assessed during the camera 
calibration process. The calibrations procedures can be 
subdivided in single-frame methods and multi-frame 
methods. Cameras are normally calibrated in 
laboratory by means of specially designed goniometers 
(Petterson, 1978) or very special equipment for stellar 
survey, (Fritz, 1978). 
One of the main problems for non-metric cameras is 
the unflatness of the film. The modélisation of the 
particular deformation has been tried (Fraser, 1982). 
Another problem for non-metric cameras is the 
uncertainty of the interior reference system. 
The here proposed procedure is suited mainly for long 
focal lenses cameras in order to overcome numerical 
instability. The determination of the distortion apart 
from the calibration has other advantages also: a 
reduction of the amount of needed control points, and 
finally an improvement of the plotting accuracy. 
The distortion can be regarded as the part of the 
transformation that cannot be included in a linear 
transformation from the object plane to the image 
plane. A straight line in the object space should remain 
a straight line in the image space. 
3. THE RADIAL DISTORTION MODEL 
The radial distortion is function of the radial distance r 
from the principal point (Ziemann 1982) expressed 
with odd polynomials. 
3 5 7 
dr = k-^r + k^r + k^r + .... (3) 
This function is called characteristic curve, with an 
associated characteristic principal distance c (fig. 1). 
The polynome dr can be transformed getting a 
variation of the principal distance from c to c+dc. 
3 5 7 
dr c = dr + k 0 r = k 0 r + k l r + k 2 r + k 3 r + ~(4) 
The radial distortion can be set to zero dr c = 0 at a 
chosen distance from principal point. 
2 4 6 
*0 = "(Vo +k 2 r 0 + Vo + "> (5) 
c c 
dc = —(dr- dr c ) = — (~Vo) = " c • k 0 ( 6 > 
r 0 r 0 
obtaining the so-called calibrated principal distance c* 
* 
c = c + dc = c(l - ) (7) 
2 2 
F)g. 7 The radial distortion model 
The radial distortion for the tested calibrated camera, a 
Rollei 6008, equipped with a Distagon 40-mm lens, is 
expressed in the form (8) in the calibration certificate. 
4. THE EXPERIMENT 
A polyester sheet of regular grid, with 10cm interval, 
has been plotted in a TA10 Wild flat-bed plotter (1.x 
1. m 2 ) giving 121 points with 1/100 mm accuracy. 
The sheet, hanged on a flat vertical wall, has been 
photographed with a semi-metric camera Rollei 6008 
at 0.8 m of distance (fig.2). The photo-scale is 1/20. 
The measure of image co-ordinates has been carried 
out in a TA3 Omi comparator. Three images have 
been taken and measured, and they are called in this 
paper testi, test2 and test3. 
Fig. 2 - The layout of the test