In 
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e method of linear interpolation 
e method of parabolic diagram approximation 
using minimalization of deviations of diagram 
points along : 
— lines 
— columns. 
3.1 Linear interpolation along the luminosity 
profiles. 
On the fig.1a there is a diagram of the the luminosity 
profile perpendicular to the cross arm axis with the 
pixels marked on it with circles. The profile is 
determined point by point only by pixels, their posi- 
tion along the line (or column) and by their bright- 
ness expressed in the 0-255 levels scale. The 
definition of the axis of symmetry for that profile 
would provide one point of the cross arm axis on 
the xz-image-plane. The simplest way to determine 
the luminosity profile symmetry axis it is to inter- 
polate for each luminosity level of each existing 
pixel - the corresponding point on the opposite side 
of the diagram. For example, for the pixel marked 
by P3 would be interpolated linearly point P3Bis 
among pixels P13 and P14, and for pixel P13 would 
by linearly interpolated the corresponding P13Bis 
point among pixels P3 and P4. The center of sym- 
metry S13 of the vector P13 - P13Bis, center of 
symmetry S3 of the vector P3 - P3pis, and other 
centers Si calculated that way can be used to cal- 
culate the final mean value of their position along 
the line (or column). This mean center position 
mean-Xs would identify the point on the axis of 
the cross arm. 
To ensure better accuracy this method is modified 
as follows. The interpolation is performed not 
only for pixels, but additionally for each full value 
of brightness (eg. for 150, 149, 148 etc.). Symmetry 
center position Si is calculated as a mean value 
of all x - value of the points which were found 
on certain luminosity level by the interpolation. 
The levels containing less or more than 2 inter- 
polated points are neglected. After all symmetry 
centers were determinate, the mean value mean-Xs 
is calculated, and Si canters which differ more 
than 1 pixel are selected and rejected. The next 
iteration of mean-Xs calculation is performed with- 
out points rejected during first iteration. The itera- 
tion process is repeated with the successive 
selection of center points differing from the mean 
value more than 0.1 pixel and 0.01 pixel. The 
mean-Xs value calculated for the Si points which 
were not rejected in the third iteration is final. 
Final mean-Xs can be treated as the point of ana- 
lyzed line on the digital image surface which can 
be considered for determination the axis of the 
analyzed arm of a cross. 
393 
After the iterations for certain line were accom- 
plished we can analyse which pixels (luminosity 
levels) were not excluded during calculations. The 
pixels representative for certain luminosity diagram 
are those pixels which were accepted to determine 
vectors along which the interpolation for symmetry 
axis have been permitted. 
3.2 Approximation of the luminosity profile using 
the parabola equation. 
The parabola which approximates pixels along the 
luminosity profile can be determined by calculation 
coefficients a, b, c of the parabola equation for at 
least 3 points pixel of a profile. In case that the 
greater number of points determines the profile the 
correction equations of the type 
V,=ax"*+bx+c-Z 
can be solved applying the least squares method 
(fig.1.b). To get acceptable results the calculations 
should be based only on the pixels representative 
for the profile (see 3.1). 
The calculation of the parabola coefficients with the 
use of analysis of Vz deviations is very easy for the 
schematization, because normal equations for it are: 
x" a + 1b + PIC - [xz] =0 
ple * [2] b + [x Je — [xz] =0 
KPa + [x jb+ bc-{z]=0 
Considering that calculations are always performed 
for the predetermined image segment (eg. predeter- 
mined with the use of simple filtering; tresholding), 
one can assume a local coordinate system for each 
cross. In that local coordinate system the pixel 
numbers which identify representative pixels will be 
these same for many profiles (especially, when the 
size and shape of various cross images are these 
same). This gives possibility to use precalculated 
coefficients of the normal equation what speeds up 
very much the calculations. 
The position of the axis of the symmetry profile 
can be determined from the first derivative of the 
parabola equation : 
Oz b 
c mort and X = 
The method of determination of a parabola by the 
minimalization of the Vz deviations is very simple, 
but seemed to be not optimal for the parabola with 
the symmetry axis parallel to the z-axis of the coor- 
dinate system. Therefore additionally the parabolic 
approximation with the use of Vx deviations mini-