EE 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   
AFFINE RECTIFICATION OF VERTICAL AERIAL PHOTOGRAPHS 
OF NON-FLAT GROUND 
by 
Vladimir Kraikv 
ABSTRACT 
The rectification method representing the optical solution of theoretical relations 
between two planes is in practice unfavourably influenced by altitude conditions of 
the ground. The ideal case of a flat ground whether level or inclined is very rare. 
Most often we meet a ground that shows small differences in height. If we rectify 
the photograph of such a ground to a true vertical photograph, there arise radial dis- 
placements - especially at the edge points of the rectified photographs - that make 
the following compilation of the rectified photographs into controlled aerial mosaic 
inaccurate or impossible. If we reach the affine rectification on a ground-plan pro- 
jection of fixed points similarly as in the case of an inclined ground the elements of 
rectification will be a function not only of the exterior orientation of the photograph 
but also of the general altitude conditions of the fixed points applied. Here the gene- 
ral additional displacement of the negative in its ground is necessarily of use. 
The displacement of the negative in the general direction makes affine changes 
in the projected image. The magnitude of these changes is independent of the direc- 
tion of the negative displacement and of the direction of the inclination of the easel 
plane or of the swing of the negative. The direction of the necessary displacement of 
the negative that can remove the affine deformation of the image depends both on 
the direction of the observed affine deviations and on the direction of the inclination 
of the easel plane or on the swing of the negative, 
The altitude differences between the given fixed points effects changes of recti- 
fying elements, viz. the change of the easel plane tilt in a general direction (or the 
change of the easel plane tilt and the swing of the negative) which is proportional 
to the general tilt of the adjusting plane interposed among four fixed points of the 
ground and further an additional displacement of the negative proportional to the 
altitude difference from one of the fixed points above the plane of the three remai- 
ning points. As the easel plane tilt depends also on the value of nadir distance of the 
photograph the resulting easel plane tilt is determined by the angle betwenn the 
photograph plane and the plane passing between the four fixed points. If the photo- 
graph is approximately parallel with this plane there occurs an insolubility of the 
affine rectification because the necessary displacement acquires hight values. 
Causes which make it necessary to introduce the displacement of the negatives at 
the rectification are - from the geometrical point of view - of three kinds: 
1. the variable inner orientation of the rectifier, 
4. the inclined flat ground, 
3. the ground with the altitude differences. 
The importance of individual factors depends on the practical conditions of the 
rectification. It is possible to neglect the displacement of the negative at practical 
rectification for a small nadir distance of the photograph, at a small inclination of 
the photographed ground and at small altitude differences between fixed points. 
From the analysis of these factors it follows that it is the necessity of additional 
displacement of affine rectification of ground with altitude differences that is most 
unfavourable at the vertical photographs. On the contrary it is possible to deduce 
that the displacement which appears necessary during rectification is at the verti- 
cal photographs nearly solely effected by the altitude differences between the fixed 
points. It is possible to draw the conclusion that the mechanical or numerical me- 
thods of introduction of the displacement of negatives cannot reach full success in 
practice if the altitude conditions among the fixed points are not taken into account. 
Meanwhile the best results in this direction are achieved by the rectification with 
empirical introduction of the displacement of negatives. 
The last section of the paper describes a practical method of setting for the cur- 
rent types of rectifiers that starts from the principles of affine rectification and ad- 
vantageously utilizes at the paper determined relations between the displacement of 
the negative and the affine deformation of the projected image 
ir 
li