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x 4 Ya , €q are distortion-corrected picture coordinates in the same plane. The 
parametres from the previous example are also used here. 
Using the exterior rotation elements to define the optical axis (and not the camera axis 
as in the previous example) and calling TRANS 1, we get the coordinates to the object point 
located in relation to the perspective center and optical axis. By reading these coordinates and 
cq into TRANS2 we get the coordinates xi . x , €g as output. 
Making a new routine TRANS4 (X, OR) with the dimension X (3) and OR (3) and reading 
X ; Ya >» Cq into X and ag, ag ; ap into OR the transformation will correct the picture 
coordinates for the radial symmetric distortion by performing the following expression : 
  
x R3 ls: (n e i ass n er Pas LO a er ya | 
xxv. hs: (x) e a eas. (e en, (a epu 
ei^ e! 
By calling TRANS4, a correction for radial symmetric distortion around the optical 
axis will be performed. 
In the next step we use TRANS 1 again, but this time we read x' 5 y; » Cz into X and 
; a a d 
0, O, 0,a.p. 0 into OR. 
After this transformation the picture coordinates still have a coordinate system located 
in the perspective center, but the tertiary axis (z) have changed from coinciding with the optical 
axis to going through the principal point. e 9 
By calling TRANS2 with the physical focal length c read into OR and the output from UN 
TRANSI read into X, the picture points are projected upon the real picture plane. Finally 
TRANS 3 is called to make the last displacement of the reduction point. 
This reduction point can be defined either by the fiducial marks, or, in cameras without 
fiducial marks, by any point suitable for reference. 
The evaluation of the fundamental photogrammetric equation by small simple geometrical 
transformations simplifies the programwork considerably and makes it very easy to add or 
remove geometrical properties without any great change in the program. It is, for example, 
quite easy to extend the program to include the problems of two-medium photogrammetry [4]. 
Linearization of the fundamental equation 
  
By using the above-mentioned small transformation modules we renounce the usual way 
of bringing the fundamental equations on a linear form. With no compound analytical expression 
we are unable to make the usual partial differentiation. But introducing numerical partial 
differentiation, the problem can easily be overcome. 
The usual fundamental equation describes the relation between object coordinates and 
picture coordinates with the general function 
PC; = Py (OR; 5 Xi) 
where PC, are the picture coordinates, 
OR; are the unknown orientation parametres, 
Xi are the object coordinates. 
Introducing approximated values OR ‚and corrections dOR; for the unknowns, they are 
evaluated as J ; ! \ ( 
OR, = OR, + dOR ; NS 
The picture coordinates PC, are divided into the measured picture coordinates pc; and 
the residuals Vi: 
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