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Images Segmentation based on Interval Type-2 Fuzzy C-Means

ASSAS Ouarda* Department of Computer Science,

Laboratory Analysis of Signals and Systems (LASS) University of M’sila

M’sila, Algeria e-mail: assas_warda@yahoo.fr

Abstract—Segmentation process helps to find region of interest in a particular image. The main goal is to make image more simple and meaningful. This work is an improvement of an existing method which is Fuzzy C-Means (FCM) to partitioning an image into several constituent components – type 2 Fuzzy C- Means-. First, membership function defined by Hamid R Tizhoosh is used to measure the image fuzziness. Second, new membership functions are proposed. The evaluation of adopted approaches was compared using the validity functions: Partition Coefficient Vpc, Partition Entropy Vpe and Peak Signal and Noise Ratio PSNR. The experimental results on real images prove that the proposed approaches are more accurate and robust than the standard FCM approach.

Keywords—component; Segmentation; Fuzzy Logic; Type-2 Fuzzy Sets; Fuzzy C-Means

I. INTRODUCTION Image segmentation plays an important role in vision and

image processing applications. It is a widely employed technique in many fields like: document image analysis, scene or map processing, Signature Identification, Biomedical Imaging, and Target Identification [1]. The images segmentation has remained a challenge. Many approaches have been studied, including Methods based edge, methods based region, methods based on thresholding, methods based artificial neural networks, data fusion methods, Markov random field methods and hybrid Methods.

In fuzzy segmentation, the image pixel values can belong to more than one segment, and associated with each of the points are membership grades that indicate the degree to which the data points belong to different segments.

Segmentation process also helps to find region of interest in a particular image. The main goal is to make image more simple and meaningful. Fuzzy C-Means (FCM) is a unsupervised fuzzy classification algorithm. Resulting from the C-means algorithm (C-means), it introduces the concept of fuzzy set in the class definition: each point in the data set for each cluster with a certain degree, and all clusters are characterized by their center of gravity.

In this paper, the use of fuzzy logic is extended to a higher order, which called type-2 fuzzy logic. The fuzzy C-Means using type-2 fuzzy logic with new membership functions is proposed to segment human MR Brain images.

The organisation of the paper is as follows. In section 2 the fuzzy C-Means technique of segmentation is reviewed and in section 3 describes briefly the type-2 fuzzy sets. Section 4 present a complete description of proposed segmenting approach using tye-2fuzzy logic, where each step of the algorithm is developed in detail. Ssection 5 illustrates the obtained experimental results and discussions and section 6 concludes this paper.

II. FUZZY C-MEANS TECHNIQUE Modeling inaccuracy is done by considering gradual

boundaries instead of clear borders between classes. The uncertainty is expressed by the fact that a pixel has attributes that assign a class than another. So, Fuzzy clustering assigns not a pixel a label on a single class, but its degree of membership in each class. These values indicate the uncertainty of a pixel belonging to a region and are called membership degrees. The membership degree s in the interval [0, 1] and the obtained classes are not necessarily disjoint. In this case, the data Xj are not assigned to a single class, but many through degrees of membership Uij of the vector Xj to class i. The purpose of classification algorithms is not only calculating cluster centers bi but all degrees of membership vectors to classes. If Uij is the membership degree of Xj to class i, the matrix U(CxN, C number of cluster and N is the data size) is called fuzzy C-partitions matrix if and only if it satisfies the conditions (1) and (2):

[ ] [ ] [ ]∈

∈∀∈∀ =

N

j ij

ij

Nu

u NjCi

1

0

1,0 ,1,,1 (1)

[ ] =

=∈∀ C

i ijuCi

1

1 ,1 (2)

The objective function to minimize J and the solutions bi, Uij, of the problem of the FCM are described by the following formulas:

= =

= C

i

N

j ij

m ij bxdUXUBJ

1 1

2 ),()(),,( (3)

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774 | P a g e 978-1-4673-7606-8/15/$31.00 ©2015 IEEE

=

== N

j

m ij

N

j j

m ij

u

Xu bi

1

1

)(

.)( (4)

1

)1( 2

1 2

2

),( ),(

−

−

=

= mC

k kj

ij ij bXd

bXd u (5)

With the variable m is the fuzzification coefficient which takes values in the interval [0, + [. The FCM algorithm stops when the partition becomes stable

Like other unsupervised classification algorithms, it uses a criterion minimization of intra-class distances and maximizing inter-class distances, but gives a degree of membership of each class for each pixel. This algorithm requires prior knowledge of the number of clusters and generates classes through an iterative process by minimizing an objective function. Thus, it allows to obtaining a fuzzy partition of the image by providing each pixel with a membership degree (between 0 and 1) to a given class. The cluster which is associated with a pixel is one whose degree of membership is the highest.

The main steps of the Fuzzy C-means algorithm are:

1) Input the image Xj: j=1..N, N: size of image. 2) Set the parameters of the algorithm: C: number of

cluster, m: fuzzy coefficient, : convergence error. 3) Initialize the membership matrix U with random values

in the range [0,1]. 4) Update the centers bi using the equation (4) and

evaluation of the objective function Jold using the formula (3). 5) Update the membership matrix U using the equation

(5) and evaluation of the objective function Jnew using the formula (3).

6) Repeat steps 4 and 5 until satisfaction of the stopping criterion which is written: || Jold-Jnew:||

7) The outputs are the membership matrix U and the centers bi.

III. TYPE-2 FUZZY SETS [2] A new area in fuzzy logic is introduced in this section,

which called type-2 fuzzy logic. Type-2 fuzzy set theory was introduced by Zadeh in [3] to solve the problem in defining the complex uncertainty which the problem is unable to define the existing type-1 fuzzy set theory. A type-2 fuzzy set is a set in which we also have uncertainty about the membership function. Type-2 fuzzy logic is a generalisation of conventional fuzzy logic (type-1) in the sense that uncertainty is not only limited to the linguistic variables but also is present in the definition of the membership functions.[4]

Speaking of uncertainty, there are two main types of uncertainty, linguistic and random. The first is associated with the word and the fact that it can mean different things to different people while the second is associated with unpredictability. Probability theory is used to treat the random uncertainty and fuzzy set is used to treat the linguistic

uncertainty. As the variance provides a measure of dispersion around the average of probabilistic uncertainty, a fuzzy set needs a dispersion measurement of linguistic uncertainty. A type-2 fuzzy set provides precisely this measure of dispersion.

A Type-2 fuzzy set is an extension of the type-1 fuzzy set. It has the degrees of membership which are themselves fuzzy. For each value of the primary value (pressure, temperature …etc.), membership is a function and not a value (secondary membership function), whose domain, is the primary membership in the interval [0, 1] and whose row (second degrees) must also be in the interval [0, 1]. The membership function of a type-2 fuzzy set is three-dimensional, and it is this new third dimension that provides new degrees of freedom in the design to treat uncertainty. These sets are very useful in situations where it is difficult to determine the exact membership function for a fuzzy set.

The term footprint of uncertainty (FOU) is used in the literature to verbalize the shape of type-2 fuzzy sets(shaded area in Fig.1)[5][6]. The FOU implies that there is a distribution that sits on top of that shaded area. When they all equal one, the resulting type-2 fuzzy sets are called interval type-2 fuzzy sets. For which, the membership function provides an interval [7].

(a) (b)

Fig. 1. (a)Type-1 membership function and (b) FOU for an interval type-2 fuzzy set [6]

Definition: A type-2 fuzzy set à is defined by a type-2 membership function μÃ(x, u), where x X and u Jx [0,1] [6].

Ã={((x, u), μÃ(x, u)) ∀ x X, ∀ u Jx [0,1]} (6)

in which 0 μÃ(x, u) 1. à can also be expressed in the usual notation of fuzzy sets as:

(7) where the double integral denotes the union over all x and

u. In order to define a type-2 fuzzy set, one can define a type-1 fuzzy set and assign upper and lower membership degrees to each element to (re)construct the footprint of uncertainty (Fig.1). A more practical definition for a type-2 fuzzy set can be given as follows:

Ã={(x, μU(x), μL(x)) ∀ x X, μL(x)) μ(x) μU(x), μ [0,1]} (8)

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Tizhoosh [7] has defined the upper and lower membership degrees μU(x) and μL(x) of initial (skeleton) membership function μ by means of linguistic hedges:

μU(x) = [μ(x)] 1/ , (9)

μL(x) = [μ(x)] (10)

Where ]1,+ [. In the conducted experiments, ]1, 2] has been used because >> is usually not meaningful for image data.

For =2, the upper and lower membership degrees represent dilatation and concentration:

μU(x) = [μ(x)] 0.5, (11)

μL(x) = [μ(x)] 2 (12)

Of course, other linguistic hedges such as de-accentuation and accentuation can also be employed:

μU(x) = [μ(x)] 0.75, (13)

μL(x) = [μ(x)] 1.25. (14)

IV. FUZZY C-MEANS USING TYPE-2 FUZZY SETS

In the proposed approaches, first, μL=μ and μU= μ 1/ are

taken. The membership functions can be calculated by three ways as follows:

μ(x) = (μL+ μU)/2 (15)

Or μ(x) = (μL * μU) 1/2 (16)

Or μ(x) = (μL + μL*μU) (17)

Second, new upper and lower membership degrees μU(x) and μL(x) of initial (skeleton) membership function μ are proposed by the flowing expressions:

μU(x) = μ(x)*( μ(x)+1), (18) μL(x) = μ(x)/( μ(x)+1). (19)

And

μU(x) = μ(x) 2*( μ(x)2+1)/2, (20)

μL(x) = μ(x) 1/2/2( μ(x)1/2+1). (21)

The general algorithm for the type-2 fuzzy C-means approach can be formulated as follows:

1) Input the image Xj: j=1..N, which N is the image size. 2) Set the parameters of the algorithm: C: number of

cluster, m: fuzzy coefficient, : convergence error. 3) Initialize the membership matrix U with random values

in the range [0, 1]. 4) Update the centers bi using the equation (4) and

evaluation of the objective function Jold using the formula (3). 5) Update the membership matrix U using the equation

(5) and Compute the upper and lower membership using the equations couple (9) and (10) or (18) and (19) or (20) and (21).

6) Calculate the membership functions using one of the formula (15), (16) or (17) then Evaluate the objective function Jnew using the formula (3).

7) Repeat steps 4 and 5 until satisfaction of the stopping criterion which is written: || Jold-Jnew:||

8) The outputs are the membership matrix U and the centers bi.

9) Assign all pixels to clusters by using the maximum membership value of every pixel.

V. EXPERIMENTAL RESULTS The proposed algorithm for a fuzzy 2-partition thresholding

has been tested on many images with various histogram distributions to ensuring its efficiency. Each image is presented by eight bits, that is, grey levels are ranging from 0 (the darkest) to 255 (the brightest). The experimental results of the proposed method are presented and discussed through the dataset of standard 512×512 grayscale test images (Fig 2).

Validation functions of resulting classes of fuzzy partitioning are often used to evaluate the performance of different methods of classification. These functions are: partition coefficient Vpc and partition entropy Vpe. They are defined as follows:

(22)

(23)

The idea of the validity of these functions is that the partition with less fuzzy means better performance. Therefore, the best partition is reached when the Vpc value is maximum and Vpe is minimum.

Also, Peak signal to noise ratio (PSNR) is used to determine the quality of the segmented image. The PSNR give the similarity of an image against a reference image based on the mean square error (MSE) of each pixel:

(24) ) 255

(log20 10 RMSE PSNR =

Where, RMSE is the root mean-squared error, defines as:

( ) ( )[ ] (25) ,,1 2

−= M N

jiÎjiI MN

RMSE

Here I and Î are the original and segmented images of size MxN, respectively.

Partition coefficient Vpc, partition entropy Vpe and PSNR are used to compare the performance of the adopted techniques for segmentation. Performance of the proposed methods is compared with fuzzy c-means method using type-1 fuzzy sets. The numerical results obtained using type-1 Fuzzy c-Means and type-2 Fuzzy c-Means with c=4 are presented in Table 1.

Using the three proposed membership functions and upper and lower membership degrees, performance value tends to increase. Tables II and III show the obtained values of

97

partition coefficient Vpc, partition entropy Vp signal to noise ratio (PSNR) using the propose the test images. A mean (μ) and standard d calculated on efficiency in order to show the the proposed and other method as in table I , I table I, as is apparent, for Vpc a biggest m standard deviation 0,1327 and for Vpe a low and standard deviation 0,1253 are obtained proposed membership function which confirm improvement. Figure 3 shows segmented imag

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Fig. 2. Dataset of standard test images

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pe and the Peak ed approaches for deviation (σ) are e effectiveness of II, and III. From

mean 3,0320 and west mean 0,0538 d from the third ms the qualitative ge for test images

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TABLE I. EXPERIMENTAL RESULTS USING TYPE-1FCM AND TYPE-2 FCM (TIZHOOSH)

Image Type-1 FCM

Type-2 FCM (Tizhoosh)

μ(x)=(μL+μU)/2 μ(x)=(μL * μU) ½ μ(x)=μL+μL*μU

pvc Pve PSNR Pvc pve PSNR pvc pve PSNR pvc pve PSNR

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

0,7666 0,7501 0,7659 0,7229 0,7604 0,7446 0,7784 0,8406 0,7427 0,4899 0,8233 0,7995 0,7939 0,7466 0,4898 0,7720 0,7599 0,8053 0,7819 0,7528 0,8425 0,7321 0,7723 0,7984 0,7630 0,7365 0,8028 0,7590 0,7453 0,8098 0,7704 0,8128 0,7643 0,7656 0,7663 0,7703 0,7372 0,7782 0,8380 0,7434 0,7455 0,7588 0,8046 0,7458 0,7720 0,7420 0,7601 0,7787 0,8335

0,8931 0,9174 0,8876 0,9812 0,8913 0,9186 0,8685 0,6957 0,8942 1,4696 0,7628 0,8256 0,8174 0,9238 1,4696 0,8442 0,9028 0,7860 0,8554 0,9073 0,6527 0,9475 0,8296 0,8003 0,8507 0,9581 0,7752 0,9043 0,9465 0,8008 0,8757 0,7870 0,8886 0,8984 0,9014 0,8885 0,9323 0,8703 0,7386 0,9347 0,9222 0,8907 0,7855 0,8939 0,8372 0,9362 0,9052 0,8203 0,7424

59,9219 59,8711 54,6581 55,6092 53,4672 54,3089 57,0702 62,8419 56,3880 54,1540 56,2574 55,6175 58,7398 56,6427 52,4359 52,3838 51,7095 55,6994 55,2753 56,7824 57,2201 54,6739 57,4169 55,9750 55,4341 56,6102 53,7848 53,7657 52,9606 56,6207 55,6399 51,8087 54,2598 54,7323 53,4955 56,5148 55,3828 55,0082 56,0596 57,3124 55,9885 56,8288 56,7750 57,9561 53,5282 55,5564 59,2656 54,0085 56,2768

0,7666 0,7501 0,7659 0,7230 0,7604 0,7446 0,7784 0,8406 0,7427 0,7852 0,8233 0,7995 0,7939 0,7466 0,7062 0,7720 0,7599 0,8053 0,7819 0,7528 0,8425 0,7321 0,7723 0,7984 0,7630 0,7365 0,8028 0,7590 0,7453 0,8098 0,7704 0,8128 0,7643 0,7656 0,7663 0,7703 0,7372 0,7782 0,8380 0,7434 0,7455 0,7588 0,8046 0,7458 0,7720 0,7420 0,7602 0,4901 0,8335

0,8931 0,9174 0,8876 0,9800 0,8913 0,9186 0,8685 0,6958 0,8943 0,7835 0,7628 0,8256 0,8174 0,9238 0,9921 0,8442 0,9028 0,7860 0,8554 0,9073 0,6527 0,9475 0,8296 0,8003 0,8507 0,9581 0,7752 0,9042 0,9465 0,8008 0,8757 0,7870 0,8886 0,8984 0,9014 0,8885 0,9323 0,8703 0,7386 0,9347 0,9222 0,8907 0,7855 0,8939 0,8372 0,9362 0,9051 1,4686 0,7424

56,1735 53,9295 53,4129 54,5694 54,3653 53,8159 50,8998 60,6100 58,9479 56,2691 57,0273 54,1700 58,8416 58,0339 59,6437 59,2650 59,0343 65,8172 59,1305 55,9023 54,6723 56,6540 54,3464 54,3215 53,8422 54,8598 58,1743 58,3142 54,6892 57,8771 52,7551 56,8707 54,7562 58,3036 59,2990 57,5287 59,0841 54,3958 61,8339 57,8382 61,1035 58,9056 57,6529 59,4435 56,1949 54,2426 53,7222 56,2187 58,4327

0,6945 0,6486 0,6745 0,6499 0,6524 0,6714 0,6680 0,7818 0,7345 0,6899 0,7426 0,7084 0,7206 0,6482 0,6681 0,6699 0,6638 0,7286 0,6843 0,6441 0,7597 0,6320 0,7112 0,7368 0,6826 0,6152 0,7100 0,6876 0,6462 0,7462 0,6875 0,7301 0,7002 0,6728 0,6725 0,7116 0,6596 0,6950 0,7777 0,6398 0,6335 0,7182 0,7328 0,6525 0,7309 0,6365 0,6458 0,6768 0,7727

0,3303 0,3792 0,3521 0,3795 0,3764 0,3554 0,3559 0,2356 0,2880 0,3277 0,2735 0,3138 0,3012 0,3802 0,3586 0,3536 0,3614 0,2908 0,3404 0,3821 0,2586 0,3951 0,3102 0,2825 0,3393 0,4175 0,3106 0,3386 0,3823 0,2732 0,3365 0,2914 0,3236 0,3531 0,3537 0,3099 0,3681 0,3293 0,2398 0,3899 0,3969 0,3025 0,2851 0,3752 0,2883 0,3931 0,3834 0,3442 0,2469

53,6640 54,6443 54,0172 54,0910 59,1329 53,8223 64,1287 52,4717 56,2632 62,2692 50,7993 58,9390 58,7882 54,2536 52,5250 56,7600 54,1686 62,2923 55,3751 53,8923 59,6786 54,5118 55,1036 53,4889 55,4598 54,3100 55,6546 59,4443 56,5714 52,4261 54,3140 57,9523 52,2208 54,8520 57,0213 54,3461 56,2751 53,9393 62,2455 58,4859 59,6311 59,9732 60,1866 62,0558 54,2636 57,7441 57,7412 56,8627 51,9249

3,0043 2,9353 3,0002 2,8275 2,9816 2,9128 3,0564 3,3228 2,9282 3,0853 3,2432 3,1459 3,1216 2,9201 2,7614 3,0293 2,9789 3,1696 3,0711 2,9488 3,3301 2,8648 3,0312 3,1465 2,9912 2,8801 3,1596 2,9766 2,9176 3,1884 3,0188 3,2042 2,9933 3,0021 3,0036 3,0261 2,8832 3,0548 3,3100 2,9092 2,9176 2,9744 3,1673 2,9175 3,0263 2,9025 2,9794 3,0575 3,2915

0,1019 0,1480 0,0968 0,2541 0,1046 0,1492 0,0598 -0,2270 0,1032 -0,0587 -0,1205 -0,0170 -0,0267 0,1569 0,2830 0,0307 0,1212 -0,0789 0,0365 0,1344 -0,3084 0,1960 0,0077 -0,0598 0,0503 0,2095 -0,0871 0,1179 0,1862 -0,0595 0,0755 -0,0822 0,0993 0,1091 0,1132 0,0903 0,1789 0,0608 -0,1672 0,1747 0,1589 0,1027 -0,0758 0,1276 0,0281 0,1832 0,1214 -0,0067 -0,1606

57,0920 54,2034 53,7383 56,0729 53,4672 53,8986 51,8257 58,5402 56,6325 54,3553 54,9643 52,5391 53,7258 55,6121 58,4502 63,7951 56,6779 55,0349 52,0729 60,0354 57,2684 53,9845 53,6453 55,9060 54,7053 54,4536 55,7208 58,8689 57,2568 54,0072 54,8326 54,6798 54,0912 54,5045 55,4136 57,9124 58,6629 59,0623 59,3004 57,8382 59,4168 54,2835 52,0865 54,9039 57,1192 53,4033 58,2266 52,4531 52,4697

Mean(μ) 0,7620 0,8863 55,7285 0,7665 0,8757 56,8611 0,6902 0,3338 56,3466 3,0320 0,0538 55,6982

STD(σ) 0,0642 0,1403 2,1728 0,0512 0,1132 2,7741 0,0418 0,0456 3,1775 0,1327 0,1253 2,5130

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TABLE II. EXPERIMENTAL RESULTS USING TYPE-2 FCM (FIRST SET OF LOWER AND UPPER MEMBERSHIP FUNCTION)

Image

Type-2 FCM (first set of lower and upper membership function)

μ(x)=(μL+μU)/2 μ(x)=(μL * μU) ½ μ(x)=μL+μL*μU

pvc pve PSNR pvc pve PSNR pvc Pve PSNR

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

0,7866 0,7580 0,7732 0,7480 0,7575 0,7635 0,7737 0,8463 0,8092 0,7879 0,8274 0,7989 0,8041 0,7551 0,7601 0,7736 0,7648 0,8128 0,7798 0,7501 0,8334 0,7392 0,7935 0,7629 0,7794 0,7297 0,8000 0,7890 0,7551 0,8237 0,7831 0,8113 0,7880 0,7699 0,7730 0,7980 0,7587 0,7875 0,8452 0,7426 0,7432 0,7926 0,8134 0,7574 0,8108 0,7454 0,7550 0,7788 0,8403

0,4435 0,4908 0,4639 0,5126 0,4943 0,4837 0,4528 0,3174 0,4053 0,4131 0,3580 0,4159 0,4099 0,4978 0,4841 0,4579 0,4779 0,3891 0,4551 0,4949 0,3480 0,5164 0,4157 0,4677 0,4428 0,5396 0,4032 0,4342 0,5043 0,3724 0,4418 0,3895 0,4370 0,4708 0,4669 0,4182 0,4932 0,4416 0,3304 0,5200 0,5182 0,4222 0,3826 0,4922 0,3905 0,5142 0,4983 0,4380 0,3355

58,0114 54,1158 54,1943 56,2675 55,8964 52,9515 54,4158 60,4551 54,0658 57,6955 56,9141 56,3290 52,9021 55,6785 56,7441 53,1870 55,4249 55,2785 53,4788 54,5379 54,4867 53,7188 57,1086 56,5554 56,7251 55,7708 52,0194 56,5601 52,0970 54,0422 52,4676 56,6268 57,1359 52,9191 56,7614 52,9470 58,6128 56,3935 60,4722 54,5518 56,3790 55,4692 56,0689 56,6179 60,7960 55,0983 56,9498 54,1289 51,6725

0,7829 0,7532 0,7686 0,7444 0,7530 0,7601 0,7733 0,8435 0,8066 0,7829 0,8240 0,7948 0,8004 0,7502 0,7563 0,7690 0,7604 0,8093 0,7755 0,7449 0,8296 0,7351 0,7906 0,7575 0,7752 0,7241 0,7958 0,7853 0,7505 0,8205 0,7787 0,8076 0,7843 0,7655 0,7685 0,7946 0,7546 0,7835 0,8424 0,7379 0,7380 0,7895 0,8098 0,7526 0,8078 0,7404 0,7499 0,7741 0,8375

0,3984 0,4458 0,4206 0,4644 0,4480 0,4366 0,4078 0,2838 0,3599 0,3776 0,3210 0,3736 0,3677 0,4525 0,4394 0,4152 0,4326 0,3485 0,4104 0,4523 0,3119 0,4701 0,3750 0,4295 0,4012 0,4953 0,3650 0,3899 0,4561 0,3317 0,4000 0,3496 0,3938 0,4258 0,4222 0,3751 0,4467 0,3968 0,2920 0,4735 0,4729 0,3796 0,3435 0,4473 0,3490 0,4685 0,4535 0,3989 0,2990

56,1009 54,1158 52,8344 54,8775 55,0472 54,7517 51,1537 60,7640 60,7482 55,7561 56,9141 54,1326 58,8055 57,6285 62,4823 58,2009 59,0757 67,4584 59,1707 55,0355 54,5979 57,0306 55,0569 53,4375 53,3964 54,1858 57,8649 56,9489 54,9167 57,4828 52,3924 56,5985 54,9776 59,5421 59,2929 58,8171 60,3189 53,9600 62,1836 56,8584 63,0499 58,0777 58,1505 59,2349 55,9368 54,1975 54,2658 55,4020 58,0624

2,7408 2,5738 2,6509 2,5579 2,5695 2,6438 2,6234 3,0973 2,9051 2,7190 2,9326 2,7899 2,8463 2,5384 2,6378 2,6299 2,6095 2,8770 2,6934 2,5283 3,0071 2,4739 2,8095 2,8612 2,6862 2,4029 2,7966 2,7362 2,5189 2,9508 2,7045 2,8818 2,7627 2,6468 2,6417 2,8072 2,5949 2,7372 3,0798 2,5174 2,4834 2,8474 2,8972 2,5628 2,8882 2,4942 2,5281 2,6570 3,0584

-0,5971 -0,4917 -0,5400 -0,4803 -0,4884 -0,5354 -0,5222 -0,8202 -0,6994 -0,5922 -0,7262 -0,6293 -0,6663 -0,4691 -0,5307 -0,5339 -0,5166 -0,6882 -0,5697 -0,4637 -0,7662 -0,4299 -0,6419 -0,6782 -0,5676 -0,3798 -0,6339 -0,6023 -0,4580 -0,7329 -0,5754 -0,6842 -0,6122 -0,5375 -0,5349 -0,6417 -0,5064 -0,5951 -0,8108 -0,4546 -0,4328 -0,6677 -0,7011 -0,4840 -0,6941 -0,4396 -0,4611 -0,5503 -0,7927

64,3974 63,6257 54,2071 53,6361 54,3348 55,8766 55,2204 64,7776 57,6872 57,9310 56,2191 55,0321 58,4994 57,4087 55,5968 52,1124 52,1464 55,2456 54,3341 56,2381 57,0828 53,7559 54,6017 56,4853 56,4605 55,7733 53,8747 56,0625 52,7059 54,0088 59,4106 51,8460 54,5465 56,1485 53,8518 56,6778 55,5568 55,5481 56,0457 56,5197 55,5099 58,5497 58,1479 56,6943 53,3449 54,4854 59,3397 53,9827 55,7968

Mean(μ) 0,7823 0,4442 55,5040 0,7783 0,4014 56,9651 2,7183 -0,5842 56,0682

STD(σ) 0,0295 0,0554 2,1449 0,0300 0,0525 3,1282 0,1723 0,1095 2,7697

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TABLE III. EXPERIMENTAL RESULTS USING TYPE-2 FCM (SECOND SET OF LOWER AND UPPER MEMBERSHIP FUNCTION)

Image

Type-2 FCM (Second set of lower and upper membership function)

μ(x)=(μL+μU)/2 μ(x)=(μL * μU) ½ μ(x)=μL+μL*μU

Pvc pve PSNR pvc pve PSNR pvc Pve PSNR

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

0,7990 0,7726 0,7867 0,7632 0,7722 0,7776 0,7868 0,8555 0,8189 0,8017 0,8373 0,8112 0,8155 0,7699 0,7746 0,7876 0,7790 0,8241 0,7826 0,7656 0,8431 0,7542 0,8048 0,7785 0,7926 0,7463 0,8123 0,8013 0,7696 0,8339 0,7963 0,8237 0,8001 0,7837 0,7866 0,8095 0,7724 0,8001 0,8542 0,7580 0,7595 0,8031 0,8244 0,7718 0,8208 0,7609 0,7699 0,7933 0,8498

0,4395 0,4874 0,4601 0,5104 0,4897 0,4809 0,4475 0,3117 0,4028 0,4065 0,3533 0,4097 0,4051 0,4946 0,4815 0,4517 0,4737 0,3823 0,4695 0,4912 0,3422 0,5154 0,4139 0,4626 0,4382 0,5387 0,3978 0,4289 0,5008 0,3664 0,4372 0,3816 0,4337 0,4672 0,4630 0,4142 0,4914 0,4365 0,3233 0,5179 0,5146 0,4205 0,3770 0,4895 0,3865 0,5117 0,4950 0,4312 0,3301

56,3236 56,0583 54,7588 57,7972 55,5695 55,7860 52,7795 64,5674 52,2592 55,7561 61,7700 54,4593 52,8105 56,0758 59,7442 56,7740 60,2183 54,7455 60,2128 54,2830 53,0893 56,9982 54,6687 55,2864 56,2108 55,4325 53,8458 55,2230 54,7296 56,5287 51,7657 52,5377 52,4483 55,5154 55,2493 55,2819 55,4197 58,1013 53,1381 56,7719 62,8947 56,6196 56,5632 54,1339 57,0435 54,0836 53,7265 60,2106 52,8832

0,7885 0,7599 0,7748 0,7508 0,7595 0,7659 0,7807 0,8477 0,8114 0,7897 0,8290 0,8004 0,8058 0,7570 0,7633 0,7755 0,7669 0,8145 0,7818 0,7521 0,8343 0,7417 0,7956 0,7635 0,7812 0,7320 0,8014 0,7911 0,7576 0,8253 0,7849 0,8133 0,7900 0,7718 0,7748 0,8001 0,7613 0,7892 0,8466 0,7449 0,7452 0,7953 0,8152 0,7592 0,8123 0,7475 0,7567 0,7807 0,8416

0,4095 0,4566 0,4312 0,4765 0,4593 0,4488 0,4171 0,2914 0,3710 0,3848 0,3297 0,3835 0,3776 0,4636 0,4493 0,4252 0,4434 0,3578 0,4207 0,4620 0,3202 0,4819 0,3853 0,4388 0,4110 0,5059 0,3738 0,4002 0,4673 0,3411 0,4097 0,3577 0,4042 0,4367 0,4331 0,3853 0,4579 0,4075 0,3006 0,4850 0,4840 0,3891 0,3522 0,4584 0,3595 0,4797 0,4644 0,4075 0,3077

52,8140 55,8919 52,6392 56,9631 61,6082 56,2946 52,5695 60,4551 55,2960 55,6238 56,2191 52,7332 55,6251 54,5281 62,4823 61,0080 56,4124 54,3773 57,5570 53,8276 52,7048 54,8569 56,6002 54,4250 54,6339 54,1858 55,6894 53,7690 52,0970 56,9665 52,7867 53,1172 53,2713 54,3618 54,9815 63,0127 57,6230 54,6672 55,1217 56,3772 56,7796 55,4070 56,0440 56,6179 59,8215 56,0687 61,5534 54,3100 51,8908

3,0222 2,8917 2,9589 2,8504 2,8900 2,9229 2,9610 3,2970 3,1312 3,0125 3,1938 3,0733 3,0977 2,8742 2,9080 2,9548 2,9203 3,1354 2,9865 2,8526 3,2322 2,8118 3,0566 3,0880 2,9848 2,7619 3,0777 2,9941 2,8582 3,1881 2,9995 3,1358 3,0298 2,9439 2,9555 3,0729 2,8957 3,0242 3,2881 2,8207 2,8209 3,0594 3,1374 2,8873 3,1308 2,8328 2,8730 2,9787 3,2693

-0,5123 -0,4110 -0,4646 -0,3689 -0,4038 -0,4318 -0,4841 -0,7639 -0,5996 -0,5521 -0,6777 -0,5645 -0,5779 -0,3932 -0,4275 -0,4717 -0,4376 -0,6191 -0,4853 -0,3945 -0,7028 -0,3587 -0,5642 -0,5534 -0,5049 -0,3029 -0,5848 -0,4843 -0,3754 -0,6567 -0,5063 -0,6203 -0,5238 -0,4525 -0,4592 -0,5632 -0,4078 -0,5153 -0,7436 -0,3481 -0,3492 -0,5598 -0,6290 -0,4071 -0,6190 -0,3600 -0,3911 -0,5090 -0,7311

53,2956 59,5215 54,5085 59,5984 56,9342 53,3775 60,9337 51,2168 56,2447 60,1496 56,6787 55,1438 52,3275 56,6076 58,0196 58,2009 60,1757 52,0311 56,8419 56,3215 57,8601 53,1958 53,6429 58,8791 54,2159 58,9863 56,7557 57,5123 52,1138 60,1435 56,2712 53,1566 56,9685 53,2451 55,2860 52,9466 58,6236 56,4354 54,5059 56,5060 55,0264 54,9645 55,4124 59,1778 56,0283 56,6617 55,9110 54,2513 53,9538

Mean(μ) 0,7950 0,4403 55,9010 0,7843 0,4115 55,8095 3,0029 -0,5066 56,0565

STD(σ) 0,0276 0,0566 2,7716 0,0292 0,0534 2,7640 0,1341 0,1144 2,4731

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VI. CONCLUSION In this work, an approach is proposed to segment images

based on the concept of type-2 fuzzy c-means. The image can carry on as much information as possible when the image is transformed to the fuzzy domain. The main idea of this work was to use of type-2 fuzzy sets into fuzzy c-means. For this purpose, new membership functions, upper and lower membership degrees are proposed. The experimental results have shown the effectiveness and usefulness of the proposed methods for image segmentation. The type-2 fuzzy c-means approach can deliver satisfactory performance to segmenting images.

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Algorithm (ICCS’12 International Conférence en Complex Systems

Agadir, Moroco. 978-1-4673-4766-2/12/$31.00 ©2012 IEEE. http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=6422825

[2] O. Assas, Threshold Selection Based On Type-2 Fuzzy 2-Partition Entropy Approach (WCCS’14 2nd World Conférence en Complex Systems Agadir, Moroco. 978-1-4799-4647-1/14/$31.00©2014 IEEE.

[3] L. A. Zadeh, The concept of linguistic variable and its application to approximate reasoning-Part I-II-III Information Science , 8,8,9 (1979), 199-249,301-357,143-180.

[4] O. Castillo, Type-2 Fuzzy Logic: Theory and Applications, 2008 Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-76283-6]

[5] J. M. Mendel , R.I Bob John, “Type-2 fuzzy sets made simple”, IEEE Trans. Fuzzy Syst. 10(2) (2002) 117-127.

[6] J.M. Mendel, Uncertain Rule based Fuzzy Logic Systems, Prentice- Hall, Englwood Cliffs, NJ,2001.

[7] H. R. Tizhoosh, “Image thresholding using type II fuzzy sets”,Pattern recognition, 38(2005), 2363-2373.