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The transformation algorithm 
For the realization of this software the most generic form of the 
plane transformation was used extracting the transformations 
usually utilized as a particular case of the utilized 
transformation. 
The general equation which regulates the plane transformation 
between two systems of coordinates (in the specific case, whole 
pixel coordinates and coordinates of the external reference 
system) is represented by a n order polynomial such as: 
x =il 3 
/=0 J-0 
r-t, ivy 
/=0 y=o 
The procedures which utilize this polynomial are noted with the 
name rubber-sheeting; at times in the application packages (GIS 
o CAD) they are referred to with the more generic term of 
warp-trasformation. 
If the polynomial is developed, the following is obtained: 
X = «oo + «oiJF + «10* + + «02 y 2 + «20* + «i2^ 2 + «2i* 2 JF+- 
Y = boo + b 0l y + V + b^pcy + b^y 2 + + b [2 xy 2 + b 2l x 2 y+... 
4 
The figure 2 demonstrates the window of a simple program 
which indicates the development of the terms up to a certain 
degree. 
fig 2: the development of the terms 
As can be easily seen, the transformations commonly used in 
the survey discipline are obtainable considering the first terms 
of the developed polynomial. 
The similar transformation ( rototranslation with variations of 
scale) is obtained by the first 6 terms: 
X = aw+a ox y + a xo x s 
Y =boo+b ol y + b lo x 
where 
a oi ~ bio a io ~ - boi 
so the indipendent paramétrés are only 4. 
The affine transformatin (fig3) is obtained by the first 6 
without any conditions: 
x = a^+a Ql y + a xo x 6 
y = t> w +b 0i y+b lo x 
the bilinear transformation (fig.4) by the first 8: 
X = a 00 + a 0l y + a l0 x + a n yx i 
Y = \o+b ol y + b lo x + b n yx 
It would seem evident that acting on the number of the 
coefficients it is possible to utilize the program for typical 
applications in the survey and in the representation as an 
internal orientation of the photograms, the rectification, texture 
mapping, etc. In this sense, the software offers the same 
operative possibilities of other software for the geometric 
treatment of the images. 
In the case in which more than 8 coefficients are used, it is 
important to make some observations. The equations are 
satisfied exactly if the number of points known n = l A p, with 
the p number of the parameters taken into consideration. It 
seems evident, therefore, that if there is a large number of points 
known and if an exact solution is wanted then higher grade 
polynomials must be used. In general, it is better to use a larger 
number of points than what is strictly necessary in order to: 
• have an estimate in least squares of the parameters of the 
transformation to evaluate the results of the transformation; 
• avoid the problem of the overparameterization. 
This second aspect is particularly important insofar as often an 
excessive number of parameters, while allowing for a better 
solution from the numerical point of view (with equal control 
points, lesser residuals are obtained) it brings about a 
description of the phenomenon of transformation far from the 
physical reality. As an illustration of this phenomenon the 
figures 5 can be studied, which demonstrate what happens in the 
case of a one-dimensional function..