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

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 u N u i C j N 1 0 0,1 1, , 1, E E (1) [ ] ¦= ∀ ∈ = C i C uij i 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 j i m ij J B U X U d x b 1 1 2 ( , , ) ( ) ( , ) (3) SAI Intelligent Systems Conference 2015 November 10-11, 2015 | London, UK 774 | Page 978-1-4673-7606-8/15/$31.00 ©2015 IEEE ¦ ¦ = = = N j m ij N j j m ij u u X bi 1 1 ( ) ( ) . (4) 1 ( 1) 2 1 2 2 ( , ) ( , ) − − = » » » ¼ º « « « ¬ ª ¸ ¸ ¹ · ¨ ¨ © § = ¦ C m k j k j i ij d X b d X b u (5) With the variable m is the fuzzification coefficient which takes values in the interval [0, + Ğ [. 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, ߝ :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:||İߝǤ 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 ę X and u ę Jx ك [0,1] [6]. Ã={((x, u), μÃ(x, u)) ̮ ∀ x ę X, ∀ u ę Jx [0,1]} (6) in which 0İ μÃ(x, u) İ 1. à can also be expressed in the usual notation of fuzzy sets as: ! ൌ ׬ ׬ ஜ!ሺ୶ǡ୳ሻ ୶אଡ଼ ୳א୎౮ ሺ୶ǡ୳ሻ ǡ ୶ ك ሾͲǡͳሿ (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 ę X, Ɋ μL(x)) İ μ(x) İ μU(x), μ ę [0,1]} (8) SAI Intelligent Systems Conference 2015 November 10-11, 2015 | London, UK 775 | Page 978-1-4673-7606-8/15/$31.00 ©2015 IEEE Tizhoosh [7] has defined the upper and lower membership degreesɊμU(x) and ɊμL(x) of initial (skeleton) membership function μ by means of linguistic hedges: μU(x) = [μ(x)]1/Į , (9) ɊμL(x) = [μ(x)] Į (10) Where ĮЫ]1,+ Ğ[. In the conducted experiments, Įę]1, 2] has been used because Į>> is usually not meaningful for image data. For Į=2, the upper and lower membership degrees represent dilatation and concentration: ɊμU(x) = [μ(x)]0.5, (11) ɊμL(x) = [μ(x)]2 (12) Of course, other linguistic hedges such as de-accentuation and accentuation can also be employed: ɊμU(x) = [μ(x)]0.75, (13) ɊμL(x) = [μ(x)]1.25. (14) IV. FUZZY C-MEANS USING TYPE-2 FUZZY SETS In the proposed approaches, first, μL=μĮ and μU= μ1/Į 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, ߝ :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:||ืߝǤ 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: ܸ݌ ܿൌ ଵ ே σ σ ݑ ௝௜ே ଶ ௝ୀଵ ௖ ௜ୀଵ (22) ܸ݌ ݁ൌ ଵ ே σ σ ݑ ௝௜ே ௝ୀଵ ௖ ௜ୀଵ Ž‘‰ ݑ) ௝௜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 20log ( 10 RMSE PSNR = Where, RMSE is the root mean-squared error, defines as: [ ] () () , , (25) 1 2 = ¦¦ − M N I i j Î i j 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 1 2 3 9 10 11 17 18 19 25 26 27 33 34 35 41 42 43 Fig. 2. Dataset of standard test images SAI Intellig Novem 8-1-4673-7606-8/15/$31.00 ©2015 IEEE 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 based on type-2 fuzzy c-means (Į=1 type-2 fuzzy sets algorithm is very segmentation (Fig. 3). For exampl main features such as sea, sky and b it can be seen, the type-2 fuzzy c equally well in terms of the quality leads to a good visual result.

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