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International cooperation and technology transfer
Mussio, Luigi

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
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
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+...
The figure 2 demonstrates the window of a simple program
which indicates the development of the terms up to a certain
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
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..