Guerra, Francesco 
  
6.1 The warping algorithms 
6.1.1 Points of force 
The algorithm for points of force is simpler. Once vector movement for each pair of homologous points is calculated, 
each other point (pixel or node) will be displaced in ratio to the vector obtained as a weighted average of the reference 
vectors. 
The fundamental problems to resolve are the identification of a criteria for the selection of the weighted values of the 
reference vectors and the elevated number of these vectors. 
In the software implemented, the weight wi to give to each vector movement A; B; was calculated as 
where d is the distance between the point in discussion and the control point. 
A linear factor of reduction of the weight, s, has been introduced within the range of influence of each control point. 
The range defines a circular region beyond which the control point has no effect. The factor of reduction has the value 
s=0 per d>r 
Ad perOsdsr 
r 
The functions of weight and reduction can be changed in ratio to the characteristics of the elements presented on the 
images. Even being able to calibrate the system in this way, the fact remains that the method based on points of force is 
very sensitive to the distribution of the points themselves. It is difficult to utilise with raster images since it is 
practically inevitable that the straight lines in the initial image result curved after the transformation. At times this, in 
the case of tract cartography or architectural designs, leads to the production of unacceptable formulation: it is not 
possible, in fact, that corners of buildings, roads, technological lines, etc. assume curved courses. In the case of 
photographic images which contain landscapes or people the phenomenon is much less noticeable. With vector images 
the application turns out to be facilitated insofar as working only on the nodes and leaving the arches unaltered it is 
possible to avoid the problem. The reconstruction of the survey of the parchment of Siena presented in point 4 was 
realised in this way. 
In some specific cases, like for example in the antique maps where it is possible to recognise only the position of some 
objects, this algorithm is the only one that applies. In general, however, feature-based warping tends to be used. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
figure 6. Distortions induced by a point of force. 
6.1.2 Feature-based warping 
This technique requires that the relationships between the 
source image and the resultant image are no longer defined by 
pairs of points but rather by pairs of applied vectors. The 
advantage is that not only the position between the parts of the 
image is indicated but also their direction. As far as of 
definition of the relationships goes, the technique for lines of 
force generally requires a lesser definition (that is, fewer 
control points). The points outside of the lines undergo a 
displacement which is calculated according to the distance and 
the projection of the point on the line. This leads to, in the 
case of cartography, the possibility of maintaining the linearity 
some elements during the transformation. figure 7. Transformation of a point by using a line of 
force 
  
   
A 
  
  
The algorithm acts in the following way. Consider initially a 
  
  
344 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000.