points to show an
ve,
is activated and
out.
nage of a rope. The
| their grey level
inal function of the
35 37 39 41
first derivative, c)
ed along the whole
ie upper catch, the
wer catch. The line
it and stops on the
ould be as long as
errors in the results
hg, the first step is
tioned points. This
course, inflection
ze positions of left
|. These are to be
? procedure. In the
tres of rope (as the
calculated at every
s carried out.
rt BS. Istanbul 2004
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXX V, Part B5. Istanbul 2004
In a more detailed description, in the first course, the procedure
starts from the starting point and forwards every single pixel
until the end point. The position on the rope (according to the
catenary equation) is calculated for every pixel. In this position,
raster image analysis is carried out in a strip of a pixel. The
length of the analysed strip constitutes the parameter of an
algorithm and can be changed from 21 to 49 pixels (only an odd
number of pixels in strips is allowed ).
k
Figure 5: The result of the line following algorithm execution.
Rope points identified by the algorithm are marked white. In
some areas points could not be identified properly and they
were filtered out.
The analysed strip of a pixel is placed vertically (along a
column — figure 6) or horizontally (along a row). The placement
of the strip depends on the angle between the lines made by the
first and the third point of the approximate rope placement and
the row direction (inclination angle). When the angle is bigger
than 45 degrees, the strip is placed horizontally, when it is
smaller than or equal to 45 degrees, the strip is placed
vertically. The middle of the strip is put exactly in the position
calculated by the catenary equation (for three approximate
points). Strip placements are calculated for every pixel along
abscissa (row or column depending on inclination angle). The
described situation is shown in figure 6. During the analysis,
inflection points of the function of the rope's image are found
by means of the first derivative (see figure 4 b — extremes of
function). In an actual implementation of an algorithm, two sets
of points are stored, one for the left edge of the rope and the
second for the right edge of the rope. The average positions (left
and right edges) are calculated according to the beginning of the
analysed strip (equation 3).
2
> di
=
n
where: D, — average position,
n — number of pixels,
d; — i" distance between beginning of strip and
inflection position, in pixels.
D (3)
av
a)
E
rows
columns
Figure 6: The scheme of the rope image analysis for an angle
between the 2-3 line and row direction of less than 45° (analysis
along the column). a) — À single analysed strip magnification,
the position of the left and right inflection points marked by
black points.
In the second course, after all approximate inflection
points positions for the left and right edge of the rope have been
recorded and the averages have been calculated, exact sub pixel
positions of the rope (highest points of the hump in figure 2 or
4a) are determined with a second derivative.
Practical experience shows that in some areas
miscalculations may occur. Such points are eliminated by the
filtering execution. Eliminated points are not taken into
consideration for further processing. Such filtering consists of
the eliminating of blunders by checking whether the value of
the deviation exceeds the threshold. The value of the threshold
is the parameter of the procedure and its value is set
empirically.
The filtering process takes place during the second course
of the algorithm. At every analysed strip placement (the same
as in the first course), sub pixel positions of the right and left
inflection points are determined by a linear interpolation of the
second derivative — figure 4c). Absolute differences between
such positions and the averages calculated in the first course are
compared to its threshold value. When at least one of them
exceeds it, both are eliminated. The exact sub pixel position of
the rope (ridge of the hump) for elementary strip placement is
the average of sub pixel positions of both inflection points. The
resulting set of rope points’ coordinates is stored for further
processing.
A more detailed description of the algorithm can be found
in my PhD thesis, which is available in the AGH library —
(University of Science and Technology in Krakow.)
Miscalculated points (which have to be filtered out) occur
in areas with a weak contrast between the rope and its
background. Very good results are achieved from the areas with
a homogeneous background and sufficient differences in grey
values of the pixels between the rope and the background. The
final result of the line following algorithm is shown in figure 5
(homogeneous and non-homogeneous areas can be identified).
As a significant magnification of image noise constitutes one of
the features of derivatives, a noise eliminating filter (low pass)
is advised to be run/employed beforehand [5].
Another detail worth mentioning here is that in the line
following algorithm, an assumption has been made that the line
in the image can be modelled with catenoid equation (equation
1). In other words, the catenoid after the projective
transformation still remains a catenoid. Although this has never
been proved in theory, practical results show that this may be
the case.
4. PRACTICAL RESULTS
Experiments for a practical verification of the elaborated
system have been carried out. Pictures were taken from the
terrain test field. The stay-out ropes of the laundry chimney of
the Military Hospital in Wroclawska Street in Kraków were
chosen as a test field (figure 7). The test field was quite small:
the vertical length component of the rope was 21.2 m long and
the horizontal length component of the rope was 17.6 m long.
Eight control points were set out into rope's plane, as marked in
figure 7. The rope was measured by three methods:
e The geodetic survey method (with theodolite),
