Mw 
b ded 
Ma WM: v^ "- m» ww 
  
Guerra, Francesco 
single pair of vectors: AB in the first image and A'B' in the final image. A point P (pixel) will be tied to the initial 
vector and will be displaced by the movement of this vector. The relative position of P is determined with respect to 
  
AB and the co-ordinates are calculated for P' which, with respect to 4' B', is in a position homologous with P. The 
projection of the point onto itself is considered the position with respect to a vector. Two values are therefore 
calculated, the distance v from the point to AB and the length of the projection (normalised) u of AP onto AB. The 
  
difference between the positions and the directions of AB and A' B' defines a rototranslation with variations in scale. 
Expressing the vector AB in these terms: 
Xpy1 point of application 
O4 corner with the X axis 
li module; 
The vector A' B' in the terms: 
X»5y, point of application 
O3 corner with the X axis 
L module 
And the points P and P' respectively x,y and x ,y' the following is obtained: 
= (x-x,)*cos(@,) + (y — yı) * sin(&,) 
l, 
y z(x—- x,)*sin(0,) — (y — y1) *cos(a,) 
u 
  
x'—x,tu*l,*cos(0,)-- v*sin(o,) 
y'=y, +uxl, *sin(a,)—v *cos(a,) 
It can be assumed that if 4B and A'B'’ maintain the same direction but have different position, then even the final 
image will be translated with respect to the original. Changing the direction will produce a rotation and varying the 
length of one of the two vectors will produce a stepped image. 
The rototranslations necessary for the transformation of the image are difficult operations from a computing point of 
view: for each point 8 products and 6 sums are necessary. Since the rototranslations however are linear operations and 
use finite differences, it is possible to reduce the calculations to two sums. 
To effect the complex transformations it is necessary to introduce more vectors. In the transformations with various 
pairs of vectors, an elaboration is carried out which considers all the contributions, opportunely weighing the 
transformation induced by each pair. For each pair of vectors AB, and A'B' ; there will be rototranslation of the point 
PD. 
6.2 Transformation of a point utilising two lines of force 
The final co-ordinates of the point P will be given by the pondered average of the co-ordinates of all the points P’i. The 
weight is calculated in the following way: 
where 
p 
l 
a * d, 
wi is the weighted value to assign to the co-ordinates xi^ and yi", 
li is the model of the i vector 
di is the distance from the point to the vector 
a, b, p are constant parameters used for the control of the transformation, varying more or less incisively the effect of 
the vectors on the pixel. 
i 
  
If a tends toward 0 and if the distance P from the vector is 0, the weighted value tends toward infinity; this means that 
the pixels that are on the original vector will go exactly onto the destination vector. Values greater than a give a 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 345