Russian 

Fig.7 
These reasons
also have underlain a suggested method of the description of contour
boundaries in terms of segments with fuzzy ends. In the carried
out researches as axial lines of approximating segments curves of
the simplest analytical type, namely  the central pieces of parabolas
were used. And each branch could have own curvature. For fine length
tuning of branches the mechanism similar to fine tuning of width of
segments was used. The conformity measure builted according to these
reasons, has the following mathematical expression:
Where
b_i, d_i  
the parameters defining a position of
a contour point in own frame of an approximating segment (fig.8);
these parameters are defined on coordinates of the point x_i,
y_i with the help of transformations 
b_i = (x_i
 X)*cos(A) + (y_i  Y)*sin(A), 

d_i =  (x_i  X)*sin(A) + (y_i  Y)*cos(A); 

Fig.8 
X,Y  
coordinates of the center of the segment; 
A  
the angle of declination of the segment;

P = 
P1, if b_i >= 0 (curvature
of the right branch), P2, if b_i < 0 (curvature of the left branch); 
B = 
B1, if b_i >= 0 (length
of the right branch), B2, if b_i < 0 (length of the left branch); 
D  
a halfwidth of the membership function.

A separate approximating
segment defined by such measure of conformity is symbolically
shown on fig.8. Here decrease of thickness of the segment reflects
decrease of its clearness, i.e. degrees of correspondence to an approximated
contour.
The contour image on which basic serviceability
of this procedure was checked is shown on fig.10a.

Fig.10a 
The size of an initial halftone picture
is equal  290x196 points. For detection of contour boundaries
the known Hueckel's operator [5,6] was used. In total, 800 contour
points were chosen approximately. Initial positions of contiguous
segments were defined by nodes of a regular grid with 12 raster points
step (reasons on the basis of which this step was set, are stated in
Appendix_2.2.A). Values of remaining
parameters were accepted as the following: B1 = B2 = 5, D = 2.5, A =
0, P1 = P2 = 0.
Initial approximations had the most
various positions concerning approximated pieces. Nevertheless,
in supressing majority, from 384 starts having here a place, procedure
of gradient "fine tuning" founded the maximum of the bellconformity
measure defining an actual contiguous segment (fig.10b).

Fig.10b 
The experiments carried out in this
way have shown that the maximums corresponding to the segments
in search have pure slopes. Really, on these slopes
there are no ejections (they are swallowed by the
basic extremums) in which gradient procedure could "lose the way".
These experiments have proved a basic possibility of deriving of
the description of contour figures in terms of contiguous (tangents)
of the segments corresponding to our intuitive representations.
At the same time, the analysis of behaviour
of the gradient procedure originally used for search of local
maximums of a bellconformity measure [2], has shown, that in neighbourhoods
of the maximums (i.e. at the final stages of search) the conformity
function looks like the long, narrow ridges prolated along axial
lines of contour boundaries. Thus, approximating segments can long
oscillate in the cross direction to these lines, with very slow movement
in the longitudinal direction. For fast transition of these ridges in
[3] it was offered to displace during search the central point of
an approximating segment not on axes of the image coordinate system
but along and cross of flowing direction of its axial line. As the axial
line of an approximating segment is gradually arranged under the axial
line of the represantation piece of contour boundary, longitudinal
moving of the segment is carried out directly along the axial line of
the mentioned ridges. Such organization of the search procedure of
local maximums of the conformity measure enabled rather fast and legiblly
to pass the above mentioned ridges of this measure. More in detail about
the choice of value of steps it is said in Appendix_2.2.B.
Further. As initial approximations in
the subsequent experiments, the most legible elements of contour
boundaries selected by the Hueckel's operator were accepted. Besides,
sparsed representation of contour boundaries was used. As
contour points the central points of contourboundary (edge)
patterns of the Hueckel's operator were assumed, instead of all
raster points placed under them as it was in the first series of experiments
were accepted.
According to sparsed representation,
the criterion of filling ratio of contiguous segments [Appendix 2.2.C] was modified also.
A contour image on which serviceability
of the advanced procedure of representation of contour boundaries
was checked by contiguous segments is shown on fig.13. A size
of the initial halftone picture  216x216. In this case, diameter
of the entering piece of the Hueckel's operator was equal to seven
(7) points of the raster. Overlapping at tracking of the contours
was equal to a half of diameter of this piece. Hence, a step between
"sparsed" contour points was 3.5. Contours of the exibited image contain
215 points.

Fig.13 
On the same picture the outcomes of
work of new procedure (resulting contiguous segments) are shown.
The proposed organization of the procedure
of gradient searching, sparsing of contour points and choice
of the initial approximations defined on the best pieces of contour
boundaries enabled to increase essentially stability of the process
of the description of these boundaries in terms of contiguous segments.
Necessary accounting time was sharply reduced also. Representation
of a contour picture (fig.13) with contiguous segments has demanded
50 sec of computation time, and a picture of the same order of the
complexity shown on fig.10 (the first series of experiments [2]), with the help of initial variant
of this procedure  over 20 minutes.
The carried out experiments confirmed
a possibility of practical realization of the concept of contiguous
segments  segments with fuzzy ends.
In Appendix_2.2.D
the some tactical problems of the organization of the search
process of local maximums of the bellconformity measure are
considered.
Further researches
on size optimization of steps of gradient procedure seems to be
expedient. In the considered variant the choice of their value was
carried out on the base of the most common reasons.
Now for detection
of false situations (in particular, at capture of several
segments),the final, rather bulky property criterion of filling
ratio of contiguous segments is used only. For more operative estimation
of the situations arising during the bellapproximation process, a
use of the information on behaviour of the gradient procedure  character
and number of steps  is possible. For the same purposes it is possible
to use a mutual coherence (confirmation) of the approximating segments
related to the same piece of contour boundaries. There is that natural
reason, that if one contiguous segment confirms by its branch the cross
position and orientation of the central piece of other segment (and
vice versa), then, most likely, there is the set of contour points under
these approximating segments corresponding to our concept about an uniform
line segment.
The applied mode
of preliminary choice of contour points is deintended for work in the
conditions of full absence of information on topological connections between
individual elementary contours. However, as an operational experience
shows for actual images, contour boundaries are traced through the help
of local operators on rather extended pieces. In this connection treatment
of the connected pieces of contour boundaries instead of acnodes from some
area is prospective. By the way, it can facilitate approximation of close
placed contours too.
Use of the Hueckel's
operator for search of contour boundaries of analysed images is
caused by a necessity of simplification of experiments. Most likely,
the suggested approximation method will be efficient at use of more
simple operators too, for example at the use of the simplified Hueckel's
operator [18], the Sobel's operator
and possibly of the Roberts simplest operator [7]. Besides, a possibility is supposed
to buildup of an operator of similar outcomes quality (and of operation
principle) to the Hueckel's operator, but more fastacting.
The basic share
of computation time in the procedure of detection of contiguous
segments is expended on a sequential evaluation of contributions
of contour points in a flowing value of the coincidence measure gradient.
The operating time of procedure can be reduced essentially if to calculate
these contributions in the parallel way using a specialized computer.
All evaluations
were carried out in format of numbers with floating point. There
are no principal difficulties in their transformation into integer
form. Thus byte (or slightly greater) representations would be enough
in most cases. Obvious replacement of direct analytical evaluations
by the table representation of belllike (onedimensional) functions
and their derivatives here will allow to reduce sharply calculation
time. Necessary for its products can also be determined employing tables.
Though the volume of such tables is great, but it does not exceed engineering
possibilities of used computing tools (for storage of these tables
memory about several tens kilobyte is required).
As a whole, the increase of operational
speed of the representation procedure of contour boundaries by contiguous
segments on several orders is quite real.
In this connection
an experimental examination of serviceability of the representation
procedure of contour boundaries by contiguous segments has revealed
the following problem. As appeared, parabolic contiguous segments
tend to run into smooth inflections of contour boundaries. The
example of representation of contour boundaries when movement of approximating
segments in a longitudinal direction is carried out in a direction
of corresponding component of conformity measure gradient is shown on
fig.14.

Fig.14 
And such arrival
is caused not by width of memberships functions as it was in the case
of rootmeansquare approximation of contour boundaries by rectilinear
segments. In order to prevent a possible disclosure of structure
delicacies of the conformity function caused by sparsing
of contour boundaries, the minimum width of memberships functions
was limited to a size approximately equal to a step between sparsed
contour points (D = 1.5). Experiments have shown noncriticality of
the obtained outcomes to a choice of this size (trial experiments
were carried out at D_min = 0.5  2.5).
The analysis
of this situation has allowed to draw a conclusion that arrival
here is defined by the form of approximating curves. Curvature of
parabolas is maximum in the center and decreases to the periphery,
therefore parabolas are easily entered in such pieces of contour
boundaries.
In work [3] for restriction of the running tendency
of parabolas it was offered to move aside branches of smaller
curvature. However, as appeared, thus counteraction is required
to slipping of contiguous segments to the ends of representation
pieces at fuzzy expressed linearity maximums (fig.15).
To solve this problem correctly was not possible,  it was not possible
to select univalently any strategy of switching of longitudinal
moving force which would enable to avoid simultaneously slipping
of approximating segments and arrivals on smooth inflections.

Fig.15 
In last version
of the procedure of approximation with contiguous segments a
soft restriction of width of rectilinear corridors, which these
segments can occupy, was suggested. Mathematically, such restriction
expresses that at an evaluation of individual components of a gradient
of the conformity measure the contour points are taken into account
with weight computed under the formula
Where
d_i  
deviation of a contour point from a
straight line tangenting the contiguous segment in its central
point, i.e. from the longitudinal axis of its own frame; 
U   a halfwidth of a rectilinear corridor occupied by the segment (fig.16). 

Fig.16 
The second series
of the experiments described in the previous section (their outcomes
are shown on fig.13) carried out with the similar count of contour
points, i.e. contributions of these points to values of components
of the conformity measure gradient were multiplied on their weights
with the count of width of an admissible corridor. Thus for an evaluation
of the component corresponding to longitudinal movement, the corridor
with a halfwidth equal 2.5, and for all remaining components  a corridor
with a halfwidth equal 5 was used.
Smooth (gradual)
restriction of width of the rectilinear corridors occupied by contiguous
segments, has allowed to avoid their explicit arrivals on
smooth inflections and slipping to the ends of represantation
pieces.
At the same time,
here again a legible solution of the problem of longitudinal fixing of
approximating segments to obtain it was not possible. At use of boundaring
rectilinear corridors the separate approximating segment can lose sensitivity
ends of a represented contour line. It can happen, if this line has a sufficient
expansion and curvature and consequently can leave for longitudinal boundaries
of the boundaring corridor. To increase width of the corridor it is impossible,
 the tendency of a parabola to run into smooth inflections begins to affect.
An example of such contour lines are contour boundaries of insoles
corresponding to its heel (fig.13), and also to lateral faces of
an iron when contours of this object are submitted full enough (fig.17).

Fig.17 
In similar cases
the approximating segments should be fixed arbitrarilycompulsorily.
The mechanism of such fixing consisted in that longitudinal movement
of the approximating segment in one direction was authorized only
on half of the distance between two points in which there was a change
of movement direction (in the beginning this distance was set equal
to a diagonal of the analysed image).
It is necessary
to pay attention that width restriction of the rectilinear corridors
occupied by approximating segments leads also to respective breakage
of these segments (on cross corridorboundaries). In outcome the
contour line which without cross restrictions could be submitted
by one segment, will be submitted by several ones (see, for example,
segments 4, 15 on fig.13).
Probably, to
resolve the problems of longitudinal fixing of approximating segments
is possible through the use of such curves (as the standards of the
simplest segments) which curvature will increase to their ends and
is minimum in the center (for example, curves of elliptic type). Apparently,
approximation of such type is capable to give objective (in our intuitive
understanding) subdivision of contour boundaries into pieces of minimum
curvature, but this problem demands the further researches.
The expediency
of further investigation in this direction as a whole is obvious.
In particular, legible longitudinal fixing of approximating segments
essentially would facilitate a solution of the subsequent problem
of integral recognitionmatching. Obviously, the more precisely such
fixing, the less final verifications of standards of search
objects with analysis pictures is required.
Despite of the some successes,
a final solution of the problem of longitudinal fixing of approximating
segments to obtain it was not possible. It was not possible to fix,
for example, these segments on lateral faces of an iron, and also
on a heel of an insole (see fig.13). Certainly it is possible to consider,
that these examples concern to some degenerated case  too smooth
curves. Nevertheless, in the exhibited examples basic unsteadiness
of the edge permitting to distinguish legibly a case when the contour
should be represented with an integral arc and when  composite one
(namely as the approximating segments should place in a true
case on the contour) is shown. Depending on initial approximations,
random interferences, and also not perfecteness of the form
of objects themselves can obtain various representations  subdivisions
 of the contour lines corresponding to identical elements of observation
objects.
This situation
is aggravated also with that in practice a full detection of the
contour boundaries meets rather seldom. Only scraps
of lines are selected. The blacked out pieces (or superlighted)
are badly reflected. Some pieces can be absent basically,  owing
to the possible partial overlapping by other objects. The last circumstance
represents one of the most characteristic peculiarities of a solution
of the problem of object recognition in actual conditions. Generally,
the approximating segments are not fixed on the defined places basically.
All these circumstances
induce to do a conclusion about necessity to buildup the procedures
of integral pose matching in the supposition of a possibility of
sliding of approximating segments on the contour boundaries of the
search objects. Speculations in this occasion as the whole lead to
a conclusion that the separate segment of the simplest form
(with feebly marked changes of curvature), inherently, legibly defines
the passing contour only in the cross direction. Often to understand
finally the longitudinal fixing it is possible only proceeding
from integral models of search contour figures.
Fig.18 illustrates
this position for the simplest cases, particularly,  an incomplete
detection of rectilinear contours (fig.18a).

Fig.18a,b,c,d 
Here the chosen
segments correspond to an angle of some figure. This angle can
slide concerning each of the chosen segments (fig.18b,c), and only
the combination of such segments catches this angle legibly
(fig.18d). The similar example for a curvilinear angle (iron),
which for example, is fixed at least by two approximating (contiguous)
segments (and by that defines these segments as the defined
parts of a contour of the search object), is represented on fig.18e.

Fig.18e 
Naturally, at
the so poorly fixing  fuzzy  primaryelementary matching of contours the
solution of the integral positional matching problem represents a nontrivial
problem. In this case, the separate segment generates as possible
all those positions of the search object, at which its standard contour
is only juxtaposed with this segment in the cross direction. To choose
a search position among so extensive set generated by each segment
of an observed picture is rather uneasy.
The following
stage of the given work also has been devoted to a solution of
this problem. At this stage the method [19] interpreting the positional
detection problem as accumulation of hypotheses [3], which in the case of objects defined
 fixed  forms allows to solve costructively the positional detection
problem in conditions of sliding primaryelementary matching of
contours has been developed.