Russian 
Where k  number
of points in each segment.
Extrema of such
measure are determined by zero values of its derivative:
The denominator
of resulting expression is positive at any values of entering variables.
The numerator can be transformed to the view
This equation
has five solutions, one of which is x_1 = 0, and four others :
Two solutions
which are determined by a difference of the expressions placed under
sign of exterior square root, have imaginary values at any values of
entering variables (as x_0> 0, under the initial agreement  see fig.5a).
Others two have real values, when
Simplifying this
expression, we shall obtain that at D < sqrt(3)*x_0 investigated conformity
function has three extremums: one central and two symmetric side ones. It
is easy to show, that:
It is obvious,
if D < sqrt(3)*x_0, then S"(x)_0 > 0, i.e. the central extremum
is a local minimum of considered conformity function of and, hence,
adjacent side exremums are local maximums of this function (see fig.5b).
At D > sqrt(3)*x_0, these maximums merge in one (see fig.5c). Positions
of maximums depending on a value of parameter D are shown on fig.5d
(see the formula for an evaluation of x_2... x_5).
1.7.A. Adding to
the Problem of the Automatic Correction of the Width of the Membership Function.
As it is affirmed,
the maximum of the function of conformity that is determined by expression
is reached at
the finite value of parameter D, when the exponent in which it is raised
in numerator  g (governing the amplitude growth rate) lays in the interval
1 < g < 2.
Let's prove this
statement.
Let's consider
an approximated segment (see fig.6b) as a continuous point set. In this
case the conformity measure is determined by an integral
Here the variable
v = d is entered to avoid confusion with differential label. The behaviour
of this measure depending on width of membership functions is defined
by a derivative on D:
Equating this
expression to zero, and introducing the variable gamma = V/D, we obtain:
In a range of
interest for us (0 < gamma < oo) this function has the maximum value
2 at gamma = 0 and asymptotically comes nearer to 1 at gamma > oo
(see fig.6d).

Fig.6d 
Its decrease
monotonicity is confirmed by that its derivative everywhere is negative
in the interval 0 < gamma < oo:

Fig.27 
Parameters of
X, Y initial sets were defined by nodes of some grid on the image plane.
Density of the grid was chosen so that in any circle in radius R_min/2 one
node of this grid hitted even. In the case of a square grid its step should
be less, than R_min/sqrt(2). Remaining parameters were prescribed identical
to all sets. A value of parameter B corresponding to length of initial
contiguous segments was chosen smaller than (R_min/2)*sqrt(3). This choice
is caused by the obtained theoretical estimation for a resolving power
of the suggested approximation method. Parameter D was chosen equal 0.5*B,
A  arbitrary, P1 = P2 = 0.
Such specificity
of variables has allowed to use effectively at the second stage of
experiments the gradient procedure in which moving on all variables
of eightparametrical conformity measure is carried out simultaneously
and basically is independent. During searching delays are possible
(misses of steps on some variables) if parameters with higher priority
were not arranged yet. In final variant the following system of priorities
was used: (+, A), (P1, P2), (D), (B1, B2, =) where  "+" and "=" designate
moving in cross and longitudinal directions accordingly. The step on
any variable increases half (with the count of restriction on maximum
value  see further) if the it is executed in the same direction, as previous,
and decreases otherwise. Thus it is considered, that those variables
were arranged, the step on which became less, than 1/4 from the maximum
admissible value on the given variable.
The last values
were chosen from the most common reasons so that as a result of each
step the new approximating segment not too overstepped the bounds of
old onethe. Steps in a longitudinal direction (and on length of the branches)
were limited to value B/2, and in cross  D/2. Steps on the angle and
curvature should not oscillate ends of the segment on value bigger than
D/2, i.e.
A*MAX(B1, B2)/0.7 and
P * (B/0.7)/2 should not exceed D/2. In order to prevent discretization
disclosures of analysed contours, the value of parameter D was limited
from below to the value ranging 0.5  2.5 distances between adjacent points
of the raster. From above parameter D was limited by value B/2.
Particularly
such restrictions are organized so, that at the approach to one of them
it is authorized to make the following step only on half of the distance
before this restriction. It simply enough ensures a possibility of increase
of a step if a necessity of movement in the opposite direction exists.
Initial steps on separate components were chosen equal to the half of
their greatest possible values. Thus initial values of parameters B and
D were accepted equal 10 and 4 accordingly, and as much as the possible
admissible step value on D equal 0.5. Search procedure of the next conformity
measure maximum stopped when steps on all variables became smaller, than
0.2 of their maximum possible admissible values.
The property
of filling ratio of a resulting segment is defined as follows:
Where
t_1, t_2... t_m
 coordinates of points of the raster;
E (t_i, h)  
membership function which is determined
by parameters of the found maximum (h = {X, Y, A, D, P1, P2, B1, B2}); 
e (t_i)  
the indicator function marking contour
points: = 1 if the given raster point is included into the number of contour points; = 0, otherwise. 
In the first
series of experiments those segments were only considered acceptable
which have filling ratio over 0.5.

Fig.9 
To avoid superfluous losses
of time, the operation of rough choice of the contour points concerning
to the flowing contiguous segment was applied. In the beginning for an
evaluation of coincidence measure gradient of chosen contour points (the
list of references to their initial set is formed) which lay inside a
wide rectangle enclosing the contiguous segment in its flowing position
(such neighbourhood is shown by fat primes on fig.12).

Fig.12 
Operations
of deletion and preliminary choice have not only engineering sense.
A solution of basic problems depends on their organization too. In particular,
ability of the procedure to understand complicated situations depends
on their organization,  at processing of false concentrations of
contour points, in the case of capture of several segments. In similar
cases the number of steps noticeably increased (sometimes so, that it was
necessary to discontinue compulsorily process of searching). Detection
of such situations was carried out by criterion of filling ratio of resulting
segments by contour points. An example of such situation is shown on fig.11.
On this picture it is shown the initial (dotted line) and finite (solid
line) positions of the contiguous segment, corresponding to process of search
of one local maximum. Apparently, the right branch of the initial contiguous
segment has not had time to be rounded along the corresponding piece
of the contour line and intersected an adjacent contour.

Fig.11 
At execution
of smoothing procedure numbers of neighboring zones are not calculated anew
but d are calculated by addition to components translating the number
of the central zone into number of one of 27 adjacent zones (3x3x3). These
components are calculated beforehand. The logic of their evaluation is
defined by logic of an evaluation of common linear number of an element
(zone) in a multivariate (threedimensional) array.

Fig.26 
In [3]
the operation of multiple smoothing has been suggested. In this case smoothing
is done{made} iterativelly until the condition will be satisfied that
at two sequential stages the same zone (besides the unique one) has the
maximum sum (among all zones at the given stage). The center of this zone
is accepted as a search cluster point. Though such the procedure has not
shown crucial advantages in comparison with once made smoothing, it is
expedient to have its in a collection of tools of the analysis of hypogram
intersection. Multiple smoothing will coordinate a cluster point (zone) with
its environment. If this point is on a cluster boundary it is gradually shifted
in its center. The stability of this point (zone) at several sequential
stages of smoothing is some performance of its common reliability. Besides,
if the cluster top is not acute, i.e. consists of several zones that (due
to the count of neighboring zones) one zone of maximum weight will be allocated
which position will depend on a context (i.e. weightinesses of enclosing
zones).