KR101851695B1 - System and Method for Controlling Interval Type-2 Fuzzy Applied to the Active Contour Model - Google Patents

Abstract

An interval type-2 fuzzy control system and method applying an active contour model can distinguish an auroral oval image by easily expressing the uncertainty of the auroral oval image using an uncertainty membership value of an interval type-2 fuzzy set by combining the active contour model with an interval type-2 fuzzy logic.

Description

FIELD OF THE INVENTION The present invention relates to an interval type 2 fuzzy control system and an active contour model,

In particular, the present invention relates to an interval type 2 fuzzy control system, and more particularly, to a method and apparatus for controlling an interval type 2 fuzzy control system by combining an active contour model and an interval type-2 fuzzy logic to use an uncertainty membership value of an interval type 2 fuzzy set To an interval type 2 fuzzy control system using an active contour model that distinguishes aurora oval images by easily expressing the uncertainty of the aurora oval image.

Aurora is a phenomenon in which a part of the charged particles emitted from the sun is attracted by the Earth's magnetic field and reacts with air molecules to emit light as it enters the atmosphere.

Aurora is a natural phenomenon that appears in the sky in high latitude regions and is caused by the collision of charged particles initiated by solar wind and magnetism.

Aurora can radiate radio waves and have a strong impact on wireless communications, weather and biological processes.

The auroral research can analyze important physics processes in the magnetosphere by interpreting the behavior of high latitude, high power, and magnetic power.

Auroral burns have proven necessary to investigate several important physical processes in the Earth's proximity space.

Aurora images in particular convey vital information for the study of grid communication systems, meteorology, and complex biological systems.

The conventional image detection method has a problem in that it can not effectively distinguish between the background in the pixel intensity and the aurora ellipsoid.

In order to solve such a problem, the present invention combines an Active Contour Model and an Interval Type-2 Fuzzy Logic to generate an aurora oval shape using an uncertainty membership value of an interval type 2 fuzzy set. An object of the present invention is to provide an interval type 2 fuzzy control system and method in which an active contour model that distinguishes an aurora oval image is easily expressed by expressing an uncertainty of an image.

According to an aspect of the present invention, there is provided an interval type 2 fuzzy control system using an active contour model,

An image acquiring unit acquiring an aurora oval image using an image device;

A fuzzy logic unit for calculating a first membership value and a second membership value indicating whether or not each pixel of the Aurora oval image belongs to the Aurora oval region and the background region using the membership function of Equation 1;

A type reduction processing unit for performing Type Reduction processing for processing the calculated first membership value and the second membership value according to Equation (2) below; And

And an active contour modeling unit for performing hard partitioning in which pixels are divided and allocated to the Aurora oval region and the background region according to the first membership value and the second membership value, .

[Equation 1]

와

은 오로라 타원형 지역에 속하는 픽셀(x, y)의 하한 멤버쉽 값과 상한 멤버쉽 값이고, I(x, y)는 해당 픽셀(x, y)에 위치한 그레이 값(Gray Value)이고, m1과 m2는 최종 클러스터링의 퍼지화 정도를 결정하는 퍼지화 계수를 나타냄.Where V1 is the Prototype of the Aurora Oval region, V2 is the Prototype of the background region,

Wow

(X, y) is a gray value (gray value) of a pixel (x, y) belonging to the aurora oval region and I (x, y) is a gray value Represents the fuzzification coefficient that determines the degree of fuzzification of the final clustering.

&Quot; (2) "

&Quot; (3) "

In the interval type 2 fuzzy control method applying the active contour model according to the feature of the present invention,

Acquiring an aurora oval image using an imaging device;

Calculating a first membership value and a second membership value indicating whether or not each pixel of the obtained auroral oval image belongs to the Aurora oval region and the background region using the membership function of Equation 1;

Performing Type Reduction processing for processing the calculated first membership value and the second membership value using Equation (2): < EMI ID = 2.0 > And

And performing hard partitioning in which pixels are divided and allocated to the Aurora oval region and the background region according to the first membership value and the second membership value, .

[Equation 1]

와

은 오로라 타원형 지역에 속하는 픽셀(x, y)의 하한 멤버쉽 값과 상한 멤버쉽 값이고, I(x, y)는 해당 픽셀(x, y)에 위치한 그레이 값(Gray Value)이고, m1과 m2는 최종 클러스터링의 퍼지화 정도를 결정하는 퍼지화 계수를 나타냄.Where V1 is the Prototype of the Aurora Oval region, V2 is the Prototype of the background region,

Wow

(X, y) is a gray value (gray value) of a pixel (x, y) belonging to the aurora oval region and I (x, y) is a gray value Represents the fuzzification coefficient that determines the degree of fuzzification of the final clustering.

&Quot; (2) "

&Quot; (3) "

With the above-described configuration, the present invention combines the active contour model and the interval type 2 fuzzy logic, is less sensitive to the initial contour condition dividing the aurora oval image, and uses the uncertainty membership value of the interval type 2 purge set, There is an effect that the uncertainty of the image can be easily explained.

FIG. 1 is a block diagram briefly showing the construction of an interval type 2 fuzzy control system to which an active outline model is applied to distinguish auroral oval images according to an embodiment of the present invention.
FIG. 2 is a diagram illustrating a section of a membership function of an interval type 2 fuzzy logic set according to an embodiment of the present invention.
FIG. 3 is a view illustrating formation of a purge region according to a fuzzy coefficient in two clusters having different volumes according to an embodiment of the present invention.
4 is a diagram illustrating a Type Reduction processing method for estimating a left value and a right value of a prototype C ₁ according to an embodiment of the present invention.
5 is a diagram illustrating a process of distinguishing contours of an aurora oval image according to an embodiment of the present invention.
6 is a view showing an interval type 2 fuzzy control method in which an active outline model is applied to distinguish aurora oval images according to an embodiment of the present invention.

Throughout the specification, when an element is referred to as " comprising ", it means that it can include other elements as well, without excluding other elements unless specifically stated otherwise.

In most pattern recognition or clustering problems, there is always uncertainty about the parameters because we can not know all the information about the pattern we want to distinguish.

Type 1 Fuzzy Set is an algorithm used to express the uncertainty of a pattern. However, Type 1 fuzzy sets have limitations in expressing uncertainties such as external noise or auroral image images.

The present invention applies a Type 2 fuzzy set which expresses uncertainty information as a membership function and treats uncertainty efficiently.

This Type 2 Fuzzy Set assigns a number of Secondary Memberships to define the Primary Membership and controls the uncertainty of the data more effectively than the Type 1 Fuzzy Sets.

 The Type 2 Fuzzy Set proposes an Interval Type-2 Fuzzy Set that extends the fuzzy set to the Interval Set to reduce the computational complexity of the computation.

Information about Aurora oval images is difficult to obtain completely.

As a solution to this problem, the present invention exemplifies a method of applying a Type 2 fuzzy set to an active contour model applied to the segmentation of the aurora oval image. In this way, an interval that can express the uncertainty of the auroral oval image Type 2 fuzzy ACM (Interval Type-2 Fuzzy Active Contour Model).

FIG. 1 is a block diagram briefly showing the construction of an interval type 2 fuzzy control system in which an active outline model is applied to distinguish an aurora oval image according to an embodiment of the present invention. FIG. 2 is a block diagram of an interval type FIG. 3 is a view showing a formation of a purged region according to a fuzzy coefficient in two clusters having different volumes according to an embodiment of the present invention, and FIG. 4 FIG. 5 is a diagram illustrating a Type Reduction processing method for estimating a left value and a right value of a prototype C ₁ according to an embodiment of the present invention. FIG.

The interval type 2 fuzzy control system 100 applying the active contour model according to the embodiment of the present invention includes an image acquisition unit 110, a fuzzy logic unit 120, a type reduction processing unit 130, An active contour modeling unit 140, a control unit 150, and a purge energy function calculating unit 160.

The image acquiring unit 110 acquires the aurora oval image using the image equipment.

The fuzzy logic unit 120 uses an interval type 2 fuzzy set such as the following Equation 1 with a second membership value of all 1 in order to apply the interval type 2 purging.

Where U is the second membership function and Jx is the first membership of x, the second membership function region.

In Equation (1), since the second membership values are all the same, it can be used as a Type 1 Fuzzy Set.

(X'),

(x')]와 같이 구간으로 표현될 수 있다. 도 2에서 음영으로 된 부분이 불확실성을 표현한 부분이다.The membership function of the interval type 2 fuzzy logic set has the same membership value as the second membership value,

(X '),

(x ')]. In Figure 2, the shaded portion represents the uncertainty.

The fuzzy logic unit 120 obtains an upper membership function and a lower membership function according to the fuzzing coefficients m1 and m2 in two clusters having different volumes, .

와

은 오로라 타원형 지역에 속하는 픽셀(x, y)의 하한 멤버쉽 값과 상한 멤버쉽 값이고, I(x, y)는 해당 픽셀(x, y)에 위치한 그레이 값(Gray Value)이고, m1과 m2는 최종 클러스터링의 퍼지화 정도를 결정하는 퍼지화 계수를 나타낸다.Here, V1 and V2 are the prototype of the Aurora oval region and the prototype of the background region,

Wow

(X, y) is a gray value (gray value) of a pixel (x, y) belonging to the aurora oval region and I (x, y) is a gray value And a fuzzy coefficient that determines the degree of fuzzification of the final clustering.

The above-described V1 and V2 are obtained from the following equation (6).

m1 and m2 are calculated by an approximation of the membership value by Equation (2).

The membership value can represent how many pixels (x, y) belong to the Aurora oval region.

In other words, the fuzzy logic unit 120 generates a first membership value and a second membership value indicating whether or not each pixel of the aurora oval image belongs to the aurora oval region and the background region, using the membership function of Equation (1) .

As shown in FIG. 3, the fuzzy logic unit 120 gives an uncertainty using m1 and m2 to form a proper purging region according to the cluster volume according to the fuzzification coefficient.

As shown in FIG. 3, in the patterns of the Aurora oval region and the background region, two clusters have different volumes and luminance, and the values of m1 and m2 are adjusted rather than C1 and C2 equally based on the center line of the boundary line As shown in FIG. 3 (b), it is an ideal structure to form the left region wide and the right region narrow based on the center line of the boundary line in the maximum purge region.

In other words, the fuzzy logic conversion unit 120 calculates by the prototype (Prototype), C ₁ and the prototype (Prototype), C ₂ of the background area of Aurora oval area in the following formula 3].

는 주어진 초기 영상을 나타내는 것으로 최소화(Minimization) 과정을 거치면서 커브 C가 정해진다.Here, u (x, y) represents a membership value of a pixel (x, y) belonging to a specific region (∈ [0, 1]), m is a fuzzy coefficient,

The curve C is determined by minimizing the initial image.

The fuzzy logic unit 120 controls the fuzzification coefficient to improve the segmentation of the region superimposed on the aurora oval image.

The membership values of each pixel have upper and lower bounds.

The Type Reduction processor calculates the Type Reduction process for applying the membership value according to the following equation (4).

As a result of the final segmentation, the pixels of the Aurora elliptical image are assigned to a destination area or other background area corresponding to the Aurora oval image, depending on the membership value of each pixel.

Hard Partitioning is considered defuzzification of the fuzzy set.

The active contour model 140 is calculated as Equation (5) to perform hard partitioning, which is the final partition of each pixel.

The present invention applies an interval type 2 fuzzy set to represent uncertainty, and the membership value of each pixel has an upper limit approximation and a lower limit approximation.

Since the active contour modeling unit 140 can not directly apply the upper limit approximation value and the lower limit approximation value to the above-described expression (5), the type reduction processing (expression 4) by the type reduction processing unit 130 is first performed Hard partitioning is performed.

The active contour modeling unit 140 may generate a pixel in the auroral oval region and the background region according to a first membership value and a second membership value indicating whether each pixel of the auroral oval image belongs to the aurora oval region and the background region And performs hard partitioning for partitioning and allocation.

In applying the Interval Type 2 Fuzzy Set to the active contour model, the active contour modeling unit 140 expresses hard partitioning as shown in Equation (5).

The control unit 150 when performing a Type Reduction processing method for estimating the value of the left and right values of the prototype (Prototype), C ₁ as follows.

The controller 150 calculates the membership value of each pixel according to the above-described [Equation 2] and [Equation 4] (S100).

Controller 150 calculates a prototype C ₁ using the membership values and the formula 3 obtained in the above step S100 (S102).

Controller 150 and then sorting the pixels in ascending order, find the index k-th pixel having the minimum distance from the prototype C ₁ calculation of all the pixels (S104).

The membership value of a pixel having an index smaller than k (i <k) is set equal to the lower limit membership value. If the membership value of a pixel having an index smaller than k is not set equal to the lower membership, the membership value is set equal to the upper membership value.

The control unit 150 calculates the right value C _1r of the prototype using the membership value obtained in step S104 and the above-described formula (3) (S106).

The control unit 150 determines whether C _1r = C ₁ If it is not the same, C ₁ = C _1r is set. Then, the flow goes to the above-described step S104 to repeat the type reduction processing procedure (S108, S110, S112).

)을 전술한 S104 단계에 적용한다.The left value C _1l of the prototype is subjected to Type Reduction processing by repeating steps S100 to S112 described above. However, the difference is that i is greater than or equal to k (

) To the above-described step S104.

The control unit 150 calculates using the following Equation (6), which is a method of demarcating the Crisp prototype.

Where i is 1 or 2, C _il is the left value of the prototype, and C _ir is the right value of the prototype.

The Interval Type-2 Fuzzy Set affects the fuzzy energy function.

The fuzzy energy function calculator 160 forms a purge energy function as shown in the following equation (7) to distinguish the auroral oval image.

는 오로라 타원형 지역에 속하는 픽셀(x,y)의 타입 2 퍼지 멤버쉽 값이고, I(x,y)는 (x,y)에 위치한 픽셀의 그레이 값이고, m은 퍼지화 계수를 나타낸다.Where V1 and V2 are prototypes for the Aurora oval and background regions,

Is the type 2 fuzzy membership value of the pixel (x, y) belonging to the aurora oval region, I (x, y) is the gray value of the pixel located at (x, y), and m is the fuzzification coefficient.

The fuzzy energy function is repeatedly moved in the direction in which the energy function is minimized, and the outline of the auroral oval image is traced by converging the boundary between the object and the background (FIG. 5).

5, the interval type 2 fuzzy control system 100 includes an image acquisition unit 110, a fuzzy logic unit 120, a type reduction processing unit 130, an active contour model unit 130, The control unit 150, and the purge energy function calculator 160. The contour of the auroral elliptical image is distinguished from the contour of the auroral elliptical image.

6 is a diagram illustrating an interval type 2 fuzzy control method in which an active outline model is applied to distinguish auroral oval images according to an embodiment of the present invention.

The image acquiring unit 110 acquires the aurora oval image using the image equipment.

> 0으로 설정하고, 배경 지역을

< 0으로 설정하여 분할 초기화를 수행한다(S200).The controller 150 controls the Aurora oval area

> 0, and set the background area to

&Lt; 0, and initialization of division is performed (S200).

The control unit 150 calculates the prototype V1 of the aurora oval region and the prototype V2 of the background region using the above-described steps S100 to S112 and (Equation 6) (S202).

In other words, using steps S100 to S112, the controller 150 calculates the right value C _1r of the prototype of the aurora oval region, the left value C _1l of the prototype, the right value C _2r of the prototype of the background region, After calculating the value C _2l , the prototype V1 of the aurora oval region and the prototype V2 of the background region are calculated using the above-described Equation (6).

과 그레이 값 Io을 전술한 [수학식 2]와 [수학식 4]를 이용하여 계산하며, 픽셀의 새로운 멤버쉽 값

을 전술한 [수학식 2]와 [수학식 4]를 이용하여 계산한다(S204).The controller 150 determines whether the current pixel membership value

And the gray value Io are calculated using the above-described Equations (2) and (4), and the new membership value

Is calculated using the above-described Equations (2) and (4) (S204).

The controller 150 calculates an energy change amount, which is a difference value between the existing energy and the new energy of each pixel, using the following equation (8) (S206).

와

를 나타낸다.here,

Wow

.

S1 is the part corresponding to the Type-2 fuzzy membership in the entire image, and S2 is the outside part which does not correspond to the Type-2 fuzzy membership in the entire image.

라고 판단하는 경우,

를

으로 대체하며,

이 아니라고 판단하는 경우, 다음 S210 단계로 진행한다(S208).The control unit 150

If so,

To

And,

The process proceeds to the next step S210 (S208).

The control unit 150 calculates the total energy function F according to the above-described Equation (7) in conjunction with the purge energy function calculation unit 160 (S210).

라고 판단하는 경우, 인터벌 타입 2 퍼지 제어 절차를 종료하며(S212), 그렇지 않은 경우, 전술한 S202 단계로 이동하여 인터벌 타입 2 퍼지 제어 절차를 반복한다.The control unit 150

The interval type 2 fuzzy control procedure is terminated (S212). Otherwise, the flow goes to the above-described S202 and the interval type 2 fuzzy control procedure is repeated.

Equation (8) is an iterative method for obtaining a solution by repeating a predetermined process.

는 Old Energy F_old와 New Energy F_new의 차분을 보는 것으로 반복법의 중단 조건을 찾기 위한 것이다.

Is to find the interruption condition of the iteration by looking at the difference between Old Energy F _old and New Energy F _new .

Ε is more than when performing the iterative approach, still looking for _new F value, and stops when the difference between the _new and the F F _old is less than ε. The threshold value ε = 10 ^-2 is used as a fuzzy stop condition.

Interval Type 2 Fuzzy Active Contour Model (ACM) is a combination of active contour model and interval type 2 fuzzy logic. It has the advantage of being less sensitive to the initial contour conditions that divide the auroral oval image. The interval type 2 fuzzy set It is possible to easily explain the uncertainty of the auroral oval image using the uncertainty membership value of the auroral oval image.

The embodiments of the present invention described above are not implemented only by the apparatus and / or method, but may be implemented through a program for realizing functions corresponding to the configuration of the embodiment of the present invention, a recording medium on which the program is recorded And such an embodiment can be easily implemented by those skilled in the art from the description of the embodiments described above.

While the present invention has been particularly shown and described with reference to exemplary embodiments thereof, it is to be understood that the invention is not limited to the disclosed exemplary embodiments, It belongs to the scope of right.

100: Interval type 2 fuzzy control system
110:
120: fuzzy logic unit
130: Type reduction processor
140: active contour model part
150:
160: Fuzzy energy function calculation unit

Claims (11)

An image acquiring unit acquiring an aurora oval image using an image device;
A fuzzy logic unit for calculating a first membership value and a second membership value indicating whether or not each pixel of the Aurora oval image belongs to the Aurora oval region and the background region using the membership function of Equation 1;
A type reduction processing unit for performing Type Reduction processing for processing the calculated first membership value and the second membership value according to Equation (2) below; And
And an active contour modeling unit for performing hard partitioning in which pixels are divided and allocated to the Aurora oval region and the background region according to the first membership value and the second membership value, And an interval type 2 fuzzy control system to which an active contour model is applied.
[Equation 1]

Where V1 is the Prototype of the Aurora Oval region, V2 is the Prototype of the background region,

Wow

(X, y) is a gray value (gray value) of a pixel (x, y) belonging to the aurora oval region and I (x, y) is a gray value Represents the fuzzification coefficient that determines the degree of fuzzification of the final clustering.
&Quot; (2) &quot;

&Quot; (3) &quot;

The method according to claim 1,
The fuzzy logic unit calculates the first prototype (C1) of the aurora oval region and the second prototype (C2) of the background region by using the following Equation (4) 1 &lt; / RTI &gt; prototype and the volume of the second prototype. &Lt; RTI ID = 0.0 &gt; 1 &lt; / RTI &gt;
&Quot; (4) &quot;

Here, u (x, y) represents the membership value of a pixel (x, y) belonging to a specific region (∈ [0,1]), and m is a fuzzy coefficient. 3. The method of claim 2,
Further comprising a controller for performing Type Reduction processing for estimating a right value and a left value of the first prototype (C1) and the second prototype (C2), respectively,
The controller calculates the membership value of each pixel using Equations (1) and (2), substitutes the calculated membership value into Equation (4), and outputs the first prototype (C1) Calculating a type (C2), arranging the pixels in ascending order, and obtaining an index kth pixel having a minimum distance from the first prototype (C1) and the second prototype (C2) active contour model, characterized in that the values are substituted in equation (4) calculating the first prototype (C1) and wherein the right value of the second prototype (C2) (C _ir) and left value (C _il) Type 2 fuzzy control system. The method of claim 3,
The controller _divides the right value C _ir and the left value C _{il of} the first prototype C1 and the second prototype C2 into the following equation 5 to define the aurora oval region And a prototype (V1) of the aurora oval region and a background region prototype (V2), each of which is a Crisp prototype of the background region, are calculated.
&Quot; (5) &quot;

Here, i = 1 or 2, C _il is the left value of the prototype, C _ir is the right value of the prototype, V 1 is the prototype of the aurora oval region and the background region. 5. The method of claim 4,
And a fuzzy energy function calculation unit for assigning the prototype (V1) of the aurora oval region and the prototype (V2) of the background region to the fuzzy energy function of Equation (6) below to distinguish the aurora oval image An interval type 2 fuzzy control system with an active contour model.
&Quot; (6) &quot;

Where V1 and V2 are prototypes for the Aurora oval and background regions,

Is the type 2 fuzzy membership value of the pixel (x, y) belonging to the aurora oval region, I (x, y) is the gray value of the pixel located at (x, y), and m is the fuzzification coefficient. Acquiring an aurora oval image using an imaging device;
Calculating a first membership value and a second membership value indicating whether or not each pixel of the obtained auroral oval image belongs to the Aurora oval region and the background region using the membership function of Equation 1;
Performing Type Reduction processing for processing the calculated first membership value and the second membership value using Equation (2): &lt; EMI ID = 2.0 &gt; And
And performing hard partitioning in which pixels are divided and allocated to the Aurora oval region and the background region according to the first membership value and the second membership value, An interval type 2 fuzzy control method using an active contour model.
[Equation 1]

Where V1 is the Prototype of the Aurora Oval region, V2 is the Prototype of the background region,

Wow

(X, y) is a gray value (gray value) of a pixel (x, y) belonging to the aurora oval region and I (x, y) is a gray value Represents the fuzzification coefficient that determines the degree of fuzzification of the final clustering.
&Quot; (2) &quot;

&Quot; (3) &quot;

The method according to claim 6,
Wherein the step of calculating using the membership function comprises:
Each pixel computing a first prototype (C1) of the aurora oval region and a second prototype (C2) of the background region by: &quot; (4) &quot; And
And forming a purging region according to the volume of the first prototype and the second prototype according to the fuzzy coefficient.
&Quot; (4) &quot;

Here, u (x, y) represents the membership value of a pixel (x, y) belonging to a specific region (∈ [0,1]), and m is a fuzzy coefficient. 8. The method of claim 7,
The step of performing by the above-mentioned formula (2)
A first step of calculating a membership value of each pixel using Equation 1 and Equation 2;
A second step of calculating the first prototype C1 and the second prototype C2 by substituting the calculated membership value into Equation 4;
A third step of aligning the pixels in ascending order and obtaining an index k-th pixel having a minimum distance from the first prototype (C1) and the second prototype (C2); And
The right value C _ir and the left value C _{il of} the first prototype C1 and the second prototype C2 are calculated by substituting the membership value of the kth pixel into the equation 4, And a fourth step of performing a reduction process on the input image data. 9. The method of claim 8,
The right value (C _ir ) of the first prototype (C1) and the second prototype (C2) sets a condition when i < k when the index kth pixel is calculated in the third step, Performing the fourth step sequentially in a first step,
The left value (C _il ) of the first prototype (C1) and the second prototype (C2) is obtained when the index k-th pixel is obtained in the third step

And the fourth step is sequentially performed in the first step. The method according to claim 1, further comprising: 9. The method of claim 8,
After the fourth step,
A right value C _ir and a left value C _{il of} the first prototype C1 and the second prototype C2 are defined using Equation 5 below and the aurora oval region and the background And calculating a prototype (V1) of the Aurora oval region and a prototype (V2) of the background region, each of which is a Crisp prototype of the region.
&Quot; (5) &quot;

Here, i = 1 or 2, C _il is the left value of the prototype, C _ir is the right value of the prototype, V 1 is the prototype of the aurora oval region and the background region. 11. The method of claim 10,
After each step of calculating the Crisp prototype,
And a step of assigning the prototype (V1) of the Aurora oval region and the prototype (V2) of the background region to the fuzzy energy function of Equation (6) to distinguish the Aurora oval image Model - based Interval Type 2 Fuzzy Control Method.
&Quot; (6) &quot;

Where V1 and V2 are prototypes for the Aurora oval and background regions,

Is the type 2 fuzzy membership value of the pixel (x, y) belonging to the aurora oval region, I (x, y) is the gray value of the pixel located at (x, y), and m is the fuzzification coefficient.

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