A New Method for the Determination of the Distortion 
and the Inner Orientation of Cameras and Projectors 
by Bertil Hallert 
R. Institute of Technology, Stockholm, 70, Sweden. 
We assume a projector of a photogrammetric instrument to be adjusted with the aid 
of grid measurements. An accurate grid is put into the projector, with the projector in 
a vertical position 
(p = w = x = 0), 
Pig. 1. 
The coordinates of 
the projected grid are 
measured in the ma- 
chine coordinate sys- 
tem as usual at a pre- 
liminarily determined 
  
  
Fig. 1. The projec- 
  
e- "o, fea srsem end the enlargement. The dis- 
% orientation. crepancies between 
gd Da the measured machine 
coordinates and the 
corresponding  enlar- 
A yg ged grid coordinates 
are assumed ta depend 
upon the errors in the 
inner and outer orien- 
tation of the projector 
and the accidental er- 
rors of the measure- 
ments. 
Small errors in the 
inner and outer orien- 
tation of the projector 
can be substituted for 
  
  
  
  
  
each other in accor- 
dance with the formulas (5-9) below. We can therefore adjust the discrepancies by 
changes in either the outer or inner orientation. If superfluous observations have been 
made, the residual errors are assumed to be observational errors of accidental character. 
We perform the adjustment with the aid of the available elements of the outer orien- 
tation; namely, the three translations, dz,, dy,, dz,, and the three rotations, do, dw, dx 
of the projector. Translations dx, and dy, are identical with translations of the projected 
points in the system in which the coordinates are measured. See Fig. 1. 
The measured coordinates are compared with the grid coordinates enlarged by the 
ratio h/f into the projection plane. Errors of the measured coordinates are defined by 
+ x'h 
de = % — f = measured T7 given? (1) 
yh 
dy =y— ~~ = Ymeasured 7 Ygiven* (2) 
The coefficients of the differential formulas below can be transformed from the grid 
System to the projection system and vice versa by the simple relations 
  
  
  
     
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