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 Determining the contours of images using the segmentation method



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4 Determining the contours of images using the segmentation method. 
Dividing the image in question into 

segments is performed based on the 
formation of subsets of related pixels. The basic idea of forming subsets of strongly 
coupled elements is that the elements of each segment will be closer to its “center” 
than to the “centers” of other segments. The problem of the formation of subsets of 
strongly coupled elements is considered solved if 

it was possible to determine the 
“centers” of segments and the boundaries of the corresponding subsets of elements. 


124 
Proximity to the central element of segments is determined on the basis of the 
concept of a fuzzy set [4, 6]. Consider the set of valid elements (i.e., valid pixels) of 
an image 

. Assume that each valid element 
Ƥ

Ƥ- ∈ R
)
 
matches feature vectors 
ᾱ = (a
1,…,
a
i,…,
a
n
),
 
calculated in the second stage.The algorithm for fuzzy 
segmentation of image elements can be described as follows. Let be given 
parameters 
Ɩ , C

(j = 2, Ɩ)
, which characterize the number of segments and center 
segments, respectively. Then, to determine the initial partition 
Ɍ
0
(ℜ) =
{S
j,
|S
j
⊂ ℜ},
it is possible to calculate the centers of the segments by the formula 
(1) and the value of the objective function by the formula (2). If for some 
J (J ∈
{2, … , l})
and some 
𝔯
xy
(𝔯
xy
∈ ℜ )
value 
∑ (𝔯
𝑥𝑦
𝑖
𝑣
𝑖𝑗
(𝑥, 𝑦)) = 0,
𝑁
𝑖=1
 
then for the corresponding fuzzy segment 
𝑆
𝑗
It is believed that 
𝜇
𝑗
(𝔯
xy
) = 1,
and for 
segments 
𝑆
𝑞 
(𝑞 = 1, 𝑙)¯
, j ≠q). It is believed that 
𝜇
𝑗
(𝔯
xy
) = 0
.
 
Further, for the obtained fuzzy segments, the refinement of the “centers” of the 
segments and the value of the objective function, respectively, by the formula (1) 
and (2). The calculation of the "centers" of the segments is carried out on the basis 
of an iterative method, which is based on the sequential refinement of the value 
𝑍
𝑖𝑗
(see formula 1) in each iteration. In this case, the value 
𝑀
𝑆
𝑗
(𝐽
𝑢
)
calculated as follows: 
𝑀
𝑆
𝑗
(𝐽
𝑢
) = 𝛿
𝑢
𝑗
/(∑
𝛿𝔯
𝑢
𝑣
𝑙
𝑣=
1
),
 
𝛿
𝑆
𝑗
(𝐽
𝑢
) = 1/(∑
𝔯
𝑖
𝑢
𝑛
𝑖=
1
),
 
𝐽 ∈ {1,2, … , 𝑙}𝑈 ∈ {1,2, … , 𝑚}.
 
𝐽 ∈ {1,2, … , 𝑙},
During the implementation of the fuzzy segmentation algorithm, the process 
of refining the “centers” of the segments is carried out. This process is terminated if: 
1) the condition is satisfied 
|𝔇(ℛ
𝑞−1
) − 𝔇(ℛ
𝑞
)| ≤ ∈;
2) the number of iterations performed 
exceeds the specified number 
q
0
.

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