Intuitionistic fuzzy entropy clustering algorithm for infrared image segmentation

Abstract

In order to discover the detailed information contained in the infrared image, this paper proposes an intuitionistic fuzzy entropy clustering algorithm for image segmentation. Because of the blurred characteristic of the infrared image, the intuitionistic fuzzy sets are selected for infrared image segmentation. The object function of the intuitionistic fuzzy entropy clustering algorithm is constructed by the intuitionistic fuzzy distance and the intuitionistic fuzzy entropy based on the regularization technique. The condition of the ntuitionistic fuzzy entropy clustering algorithm is researched. The Lagrange multiplier method is employed to calculate membership functions and the centroids. An iterative algorithm is deduced to calculate Lagrange multiplier coefficient and membership. Finally, experimental results demonstrate the ability of the intuitionistic fuzzy entropy clustering algorithm for infrared image segmentation.

Keywords

Intuitionistic fuzzy C-means clustering infrared image segmentation membership degree

1 Introduction

Infrared image analysis has a wide range of application in military operation, such as target examination, recognition and tracking. Infrared image segmentation is a basic and critical technology in the fields of infrared image processing. It aims at providing evidence for the subsequent identification, classification, tracking and retrieval. Infrared image contains rich information, but the infrared image has the advantage of more blurred and strong noise, so how to segment infrared image and pick up useful information is an important research field.

Fuzzy clustering algorithm is an important segment algorithm for infrared image, which has been researched by many researchers [1, 2]. The intuitionistic fuzzy sets as the extension of the fuzzy sets, have been successfully applied to image processing. The method of applying the intuitionistic fuzzy sets in the filed of image processing is discussed [3], and an algorithm for constructing the intuitionistic fuzzy sets histograms of a gray-scale digital image is present in [4]. Some researchers have investigated intuitionistic fuzzy-based thresholding techniques for image segmentation. Pedro Couto [5] researched the optimal threshold value for gray-level image segmentation based on intuitionistic fuzzy entropy. Ioannis K. Vlachos [6] researched the intuitionistic fuzzy cross-entropy. Applications of the proposed measures of cross-entropy to image segmentation are also presented. Tamalika Chaira [7] researched using intuitionistic fuzzy set theory to segment blood vessels and also the blood cells in pathological image.

Some researchers also have investigated intuitionistic fuzzy C-means clustering techniques for image segmentation. Z.S. Xu [8] introduced some methods for calculating the association coefficients of IFSs, and proposed a clustering algorithm for IFS. Z.S. Xu [9] introduced an intuitionistic fuzzy hierarchical algorithm for clustering IFSs. Dimitris K. Iakovidis [10] investigated the issue of intuitionistic clustering fuzzy image representation and proposed an intuitionistic fuzzy clustering with applications in computer vision. Tamalika Chaira [11] studied the intuitionistic fuzzy clustering method for edges detection. In this algorithm, at every iteration, the cluster center are updated by modified membership degree. Prabhjot Kaur [12] present a robust intuitionistic fuzzy C-means and a robust kernel intuitionistic fuzzy C-means algorithm.

This paper proposes an intuitionistic fuzzy entropy clustering algorithm for infrared image segmentation. Intuitionistic fuzzy entropy clustering proposed in this paper is an alternative generalization of intuitionistic C-means clustering. Same as the fuzzy entropy clustering algorithm, the objective function of the intuitionsitic fuzzy entropy algorithm is defined as the sum of the intuitionistic C-means clustering criterion and the intuitionistic fuzzy entropy terms. Minimization of the objective function can minimize the intuitionistic fuzzy distance between points and centroids, and maximize the intuitionistic entropy, which leads to reduce the hesitation. The Lagrange multiplier method is employed to calculate membership functions and the centroids. Because the number of the variables need to be solved is big, but the conditions of the optimization problem is limited, so an iterative algorithm is deduced to calculate Lagrange multiplier coefficient. Some tests are carried out to show that a fast convergence with a fixed number of iterations about 10 is sufficient. Finally some experiments and performance evaluation are carried on several military infrared images. The results are shown that the intuitionistic fuzzy entropy clustering algorithm present in this paper can discover more information then the intuitionistic fuzzy C-means clustering algorithm and fuzzy C-means clustering algorithms.

The organization of the paper is as follows: Section 2, briefly reviews the basic notions, concepts and definitions of intuitionistic fuzzy sets. Section 3 describles the proposed algorithms, the intuitionistic fuzzy entropy clustering algorithm. Section 4 describes the intuitionistic fuzzy image representation considered. The experimental results are apposed in Section 5. The last section, Section 6, summarizes the conclusions of this study.

2 Preliminaries

Atanassov’s Intuitionistic fuzzy sets are extension of fuzzy sets, which have some own feathers, and provide a flexible and intuitive framework to deal with imperfect information. Intuitionistic fuzzy sets are characterized by degrees of membership, degrees of non-membership and degrees of hesitance to elements belonging the a set. In this section, the basic notions, concepts, and definitions of intuitionistic fuzzy sets are described.

2.1 Intuitionistic fuzzy set

Definition 1. An intuitionistic fuzzy set A in a universal set X is given by [13, 14]: $A = {x, μ_{A} (x), ν_{A} (x) | x \in X}$ (1) Where μ_A (x) → [0, 1] and ν_A (x) → [0, 1]. With the condition 0 ≤ μ_A (x) + ν_A (x) ≤ 1, for all x ∈ X. The values of μ_A (x) and ν_A (x) denote the membership and non-membership degrees of an element x to the set A. For each element x in A, the hesitation degree is given by π_A (x) = 1 - μ_A (x) - ν_A (x). It expresses a lack of knowledge whether x belongs to A or not.

2.2 Intuitionistic fuzzy generators

In order to construct Attanassov’s intuitionistic fuzzy set (IFS) from fuzzy set, the intuitionistic fuzzy generators are used. From the definition of intuitionistic fuzzy generators by Bustince et al. [15]:

Definition 2. (Bustince et al. [15]) A function φ : [0, 1] → [0, 1] will be called intuitionistic fuzzy generator (IFG) if $φ \leq 1 - x$ (2)

For all x ∈ X.

Theorem 1. (Butince et al. [13]) Let φ : [0, 1] → [0, 1]. Then, φ is a continuous IFG if and only if there exists a continuous f : [0, 1] → [0, 1] such that

— f (x) ≤ x for all x ∈ [0, 1].

— φ (x) = (f ∘ N) for all x ∈ X.

Where N denotes the standard negation, N : [0, 1] → [0, 1] given by N (x) = 1 - x for all x ∈ X. Furthermore, an IFS can be constructed form an FS and an IFG by according to the following theorem.

Theorem 2. (Butince et al. [15]) Let $\bar{A}$ be an FS on the universe X ≠ Θ, and let φ be an IFG. Then, the set: $A = {〈 x, μ_{\bar{A}} (x), φ (μ_{\bar{A}} (x)) 〉 | x_{i} \in X}$ (3) is an IFS on X.

2.3 Intuitionistic fuzzy distance measures

Szmidt and Kacprzyk [16] introduced some popular distance measures between two intuitionistic fuzzy sets A and B that take into account the membership degree μ, the non-membership degree ν, and the hesitation degree π in universal set X ={ x₁, x₂, …, x_n }. Some of the intuitionistic fuzzy distance measures are defined as follows:

Intuitionistic Hamming distance: $H_{IFS} (A, B) = \sum_{i = 1}^{n} (\begin{matrix} | μ_{A} (x_{i}) - μ_{B} (x_{i}) | + \\ | v_{A} (x_{i}) - v_{B} (x_{i}) | + \\ | π_{A} (x_{i}) - π_{B} (x_{i}) | \end{matrix})$ (4)

Intuitionistic Euclidean distance: $E_{IFS} (A, B) = \sqrt{\sum_{i = 1}^{n} [\begin{matrix} {(μ_{A} (x_{i}) - μ_{B} (x_{i}))}^{2} + \\ {(v_{A} (x_{i}) - v_{B} (x_{i}))}^{2} + \\ {(π_{A} (x_{i}) - π_{B} (x_{i}))}^{2} \end{matrix}]}$ (5)

Weight Euclidean distance: ${WE}_{IFS} = {(\frac{1}{2} \sum_{i = 1}^{n} ω_{i} (\begin{matrix} {(μ_{A} (x_{i}) - μ_{B} (x_{i}))}^{2} + \\ {(v_{A} (x_{i}) - v_{B} (x_{i}))}^{2} + \\ {(π_{A} (x_{i}) - π_{B} (x_{i}))}^{2} \end{matrix}))}^{\frac{1}{2}}$ (6)

With ω_i ≥ 0, $\sum_{i = 1}^{n} ω_{i} = 1 (i = 1, 2, \dots, n)$ .

2.4 Intuitionistic fuzzy entropy

The intuitionistic fuzzy entropy is defined by Szmidt and Kacprzyk [17] based on the fuzzy entropy definition [18].

Definition 3. [17] An entropy on IFS(X) is a real-valued function E : IFS (X) → [0, 1], satisfying the following axiomatic requirements [19]:

1) E (A) = 0 iff A is a crisp set, i.e., μ_A (x_i) = 0 or μ_A (x_i) = 1 for all x_i ∈ X.

2) E (A) = 1 iff μ_A (x_i) = v_A (x_i) for all x_i ∈ X.

3) E (A) ≤ E (B) if A ≤ B, i.e., μ_A (x_i) ≤ μ_B (x_i) and μ_B (x_i) ≤ v_B (x_i), for μ_B (x_i) ≤ v_B (x_i) or μ_A (x_i) ≥ μ_B (x_i) and v_A (x_i) ≤ v_B (x_i), for μ_B (x_i) ≥ v_B (x_i) for any x_i ∈ X.

4) E (A) = E (A^C).

Theorem 3. [20] For a set A ∈ IFS (X): $E_{LT}^{IFS} (A) = - \frac{1}{n ln 2} \sum_{i = 1}^{n} [\begin{matrix} μ_{A} (x_{i}) ln μ_{A} (x_{i}) + \\ v_{A} (x_{i}) ln v_{A} (x_{i}) - \\ (1 - π_{A} (x_{i})) * \\ ln (1 - π_{A} (x_{i})) - \\ π_{A} (x_{i}) ln 2 \end{matrix}]$ (7) is an intuitionistic fuzzy entropy measure.

2.5 Intuitionistic Fuzzy Image representation

The definition of a gray-scale image is given by FSs of Pal [12]. Based on this definition, Vlachos and Sergiadis proposed the following representation of a digital image in terms of elements of IFS theory [13].

Definition 4. A gray-scale image A is described by the A-IFS $A = {〈 g_{ij}, μ_{A} (g_{ij}), v_{A} (g_{ij}) 〉 | g_{ij} \in {0, 1, \dots, l - 1}}$ (8) With i∈ { 1, 2, …, M } and j∈ { 1, 2, …, N }, where μ_A (g_ij) is membership of the (i, j)th pixel to the set A, and v_A (g_ij) is non-membership, and l is the number of gray levels.

The membership and non-membership function are described by: $μ_{\bar{g}} (x : λ) = 1 - {(1 - max {0, 1 - \frac{| x - g |}{p}})}^{γ - 1}$ (9) and $v_{\bar{g}} (x : λ) = \frac{1 - μ_{\bar{g}} (x : λ)}{1 + β μ_{\bar{g}} (x : λ)}$ (10) With γ ≥ 1, β ≥ 0, g∈ { 0, 1, …, L - 1 } is the gray level of the image, p is a positive real control parameter.

3 Intuitionistic fuzzy clustering algorithms

3.1 Fuzzy C-means clustering for IFSs (FCM-IFSs) algorithms

The fuzzy C-means clustering algorithms for IFSs is researched by Z.S. Xu [8]. The distance measure WE_IFS is used as the proximity function of the FCM-IFSs algorithm, then the objective function of the algorithm can be formulated as follows [8]: ${\begin{matrix} min J_{1} (U, V) = \sum_{j = 1}^{p} \sum_{i = 1}^{c} μ_{ij}^{m} {WE}_{IFS}^{2} (Z_{j}, V_{i}) \\ s . t . \sum_{i = 1}^{c} μ_{ij} = 1, 1 \leq j \leq p \\ μ_{ij} \geq 0, 1 \leq i \leq c; 1 \leq j \leq p \\ \sum_{j = 1}^{p} μ_{ij} > 0, 1 \leq i \leq c \end{matrix}$ (11) Where Z ={ Z₁, Z₂, …, Z_P } are pIFS s each with n elements, c is the number of clusters (1 ≤ c ≤ p), and V ={ V₁, V₂, …, V_c } are the prototypical IFSs, i.e., the centroids of the clusters. The parameter m is the fuzzy factor (m > 1), μ_ij is the membership degree of the jth sample Z_j to the ith cluster, U = (u_ij) _c×p is a matrix of c × p.

The membership degree of the jth sample to the ith cluster is calculated as following: $μ_{ij} = \frac{1}{\sum_{r = 1}^{c} {(\frac{{WE}_{IFS}^{2} (Z_{j}, V_{i})}{{WE}_{IFS}^{2} (Z_{j}, V_{r})})}^{\frac{1}{m - 1}}}$ (12)

The centroids are calculated as following: $μ_{V_{i}} (x_{l}) = \frac{\sum_{j = 1}^{p} μ_{ij}^{m} μ_{Z_{i}} (x_{l})}{\sum_{j = 1}^{p} u_{ij}^{m}}$ (13) $v_{V_{i}} (x_{l}) = \frac{\sum_{j = 1}^{p} μ_{ij}^{m} v_{Z_{i}} (x_{l})}{\sum_{j = 1}^{p} μ_{ij}^{m}}$ (14) $π_{V_{i}} (x_{l}) = \frac{\sum_{j = 1}^{p} μ_{ij}^{m} π_{Z_{i}} (x_{l})}{\sum_{j = 1}^{p} μ_{ij}^{m}}$ (15)

3.2 Intuitionistic fuzzy C-means (IFCM) clustering algorithms

Tamalika Chaira [11] presents a novel intuitionistic fuzzy C-means clustering using intuitionistic fuzzy set theory. In the algorithm, in order to take into account of the hesitation, a modified membership degree is defined as $μ_{ik}^{'} = μ_{ik} + π_{ik}$ , where μ_ik denotes conventional fuzzy membership and π_ik, hesitation degree of kth element in cluster i, π_ik is given: $π_{ik} = 1 - μ_{ik} - (1 - μ_{ik}) / (1 + β μ_{ik})$ (16) Where β ≥ 0.

Based on the definition of the modified membership, the modified cluster center $v_{i}^{'}$ is calculate as: $v_{i}^{'} = \frac{\sum_{j = 1}^{p} μ_{ij}^{'}^{m} x_{ik}}{\sum_{k = 1}^{n} μ_{ij}^{'}^{m}}$ (17)

The membership degree of the jth sample to the ith cluster is calculated as following: $μ_{ik} = \frac{1}{\sum_{j = 1}^{c} {[\frac{d_{ik}^{2}}{d_{jk}^{2}}]}^{\frac{1}{m}}}$ (18) Where d (x_k, v_i) is Euclidean distance measure between cluster center v_i of each region and data x_k, n is number datas, m is degree of fuzziness or fuzziness index.

4 Intuitionistic fuzzy entropy (IFE) clustering algorithms

4.1 The objective function and conditions

In the intuitionistic fuzzy entropy clustering algorithm, the data for clustering are IFSs. To incorporate the intuitionistic fuzzy entropy term, the objective function of the intuitionistic fuzzy entropy clulstering based the regularization technique is defined as $J_{m} (U, V) = J_{1} + λ J_{2}$ (19) Where, J₁ is intuitionistic fuzzy distance term and J₂ is the intuitionistic fuzzy entropy term, which is defined as: $J_{1} = \sum_{j = 1}^{p} \sum_{i = 1}^{c} μ_{ij}^{2} {WE}_{IFS}^{2} (Z_{j}, V_{i})$ (20) $J_{2} = - \sum_{i = 1}^{n} [\begin{matrix} μ_{ij} ln μ_{ij} + v_{ij} ln v_{ij} - \\ (1 - π_{ij}) ln (1 - π_{ij}) - π_{ij} ln 2 \end{matrix}]$ (21) Where μ_ij is the membership degree of the jth sample Z_j to the ith cluster, v_ij is the non-membership degree of the jth sample Z_j to the ith cluster, and π_ij is the hesitation degree of the jth sample Z_j to the ith cluster. λ is the degree of intuitionistic fuzzy entropy.

Assuming that μ_ij, v_ij, π_ij, λ satisfy the following conditions: $\sum_{i = 1}^{c} μ_{ij} = 1, 1 \leq j \leq p$ (22) $μ_{ij} \geq 0, 1 \leq i \leq c; 1 \leq j \leq p$ (23) $\sum_{j = 1}^{p} μ_{ij} > 0, 1 \leq i \leq c$ (24) $μ_{ij} + v_{ij} + π_{ij} = 1 \leq i \leq c; 1 \leq j \leq p$ (25) $0 \leq λ$ (26)

4.2 The iterative algorithm to solve parameters

To solve the optimization problem, the Larange multiplier method is employed [21, 22]. Let

$\begin{matrix} L & = & \sum_{j = 1}^{p} \sum_{i = 1}^{c} μ_{ij}^{m} {WE}_{IFS}^{2} (Z_{j}, V_{i}) \\ + λ \sum_{j = 1}^{p} \sum_{i = 1}^{c} [\begin{matrix} μ_{ij} ln μ_{ij} + v_{ij} ln v_{ij} \\ - (1 - π_{ij}) ln (1 - π_{ij}) - π_{ij} ln 2 \end{matrix}] \\ - \sum_{j = 1}^{p} λ_{j} (\sum_{i = 1}^{c} μ_{ij} - 1) \\ - \sum_{j = 1}^{p} \sum_{i = 1}^{c} λ_{ij} (μ_{ij} + v_{ij} + π_{ij} - 1) \end{matrix}$ (27)

Furthermore let ${\begin{matrix} \frac{\partial L}{\partial μ_{ij}} = 0, 1 \leq i \leq c; 1 \leq j \leq p \\ \frac{\partial L}{\partial v_{ij}} = 0, 1 \leq i \leq c; 1 \leq j \leq p \\ \frac{\partial L}{\partial λ_{j}} = 0, 1 \leq j \leq p \\ \frac{\partial L}{\partial λ_{ij}} = 0, 1 \leq i \leq c; 1 \leq j \leq p \end{matrix}$ (28)

To solve the Equation (28), we have

$\begin{matrix} λ_{j} \\ = - λ ln (\sum_{i = 1}^{c} \frac{1 - π_{ij}}{exp (\frac{{WE}_{IFS} (Z_{j}, V_{i})}{λ}) + exp (- \frac{λ_{j}}{λ})}) \end{matrix}$ (29) $μ_{ij} = \frac{1 - π_{ij}}{exp (\frac{{WE}_{IFS} (Z_{j}, V_{i}) - λ_{j}}{λ}) + 1}$ (30)π_ij is hesitation factor, which is given as $π_{ij} = 1 - μ_{ij} - \frac{(1 - μ_{ij})}{1 + β μ_{ij}}$ (31)

Substituting the Equation (31) into the Equations (29) and (30). $λ_{j} = - λ ln (\sum_{i = 1}^{c} \frac{μ_{ij} + \frac{(1 - μ_{ij})}{1 + β μ_{ij}}}{exp (\frac{{WE}_{IFS} (Z_{j}, V_{i})}{λ}) + exp (- \frac{λ_{j}}{λ})})$ (32) $\begin{matrix} μ_{ij} = \frac{- (1 + exp (\frac{{WE}_{IFS} (Z_{j}, V_{i}) - λ_{j}}{λ}))}{2 β} + \\ \frac{\sqrt{{(1 + exp (\frac{{WE}_{IFS} (Z_{j}, V_{i}) - λ_{j}}{λ}))}^{2} + 4 β}}{2 β} \end{matrix}$ (33)

The computation of λ_j and μ_ij are solved from Equations (32) and (33) with a special recursion scheme which yields a very quick convergence. The block diagram showing the λ_j and μ_ij equations is given in Fig. 1. This yields a fast convergence with a fixed number of iterations about 10 is sufficient.

Next we compute $V = {x, μ_{V} (x), ν_{V} (x) | x \in X}$

Let $\frac{\partial L}{\partial μ_{V_{i}} (x_{l})} = \frac{\partial L}{\partial v_{V_{i}} (x_{l})} = \frac{\partial L}{\partial π_{V_{i}} (x_{l})} = 0$ (34)

We have $μ_{V_{i}} (x_{l}) = \frac{\sum_{j = 1}^{p} μ_{ij} μ_{Z_{i}} (x_{l})}{\sum_{j = 1}^{p} u_{ij}}$ (35) $v_{V_{i}} (x_{l}) = \frac{\sum_{j = 1}^{p} μ_{ij} v_{Z_{i}} (x_{l})}{\sum_{j = 1}^{p} μ_{ij}}$ (36) $π_{V_{i}} (x_{l}) = \frac{\sum_{j = 1}^{p} μ_{ij} π_{Z_{i}} (x_{l})}{\sum_{j = 1}^{p} μ_{ij}}$ (37)

The membership functions μ_ij and the centroids V are updated iteratively.

4.3 The procedure of IFE clustering algorithm

The procedure of IFE algorithm is given as follows:

Step 1: Initialize seeds V (0), π, let k = 0, and set ɛ > 0.

Step 2: Calculate λ_j and μ_ij based on iterative algorithm described as Fig. 1.

Step 3: Calculate $V (k + 1) = {V_{1} (k + 1), V_{2} (k + 1), \dots, V_{c} (k + 1)}$

Where $\begin{matrix} V_{i} & = & {x_{l}, \frac{\sum_{j = 1}^{p} μ_{ij} μ_{Z_{i}} (x_{l})}{\sum_{j = 1}^{p} μ_{ij}}, \frac{\sum_{j = 1}^{p} μ_{ij} v_{Z_{i}} (x_{l})}{\sum_{j = 1}^{p} μ_{ij}}}, \\ 1 \leq i \leq c \end{matrix}$

Step 4: if $\sum_{i = 1}^{c} d (V_{i} (k), V_{i} (k + 1)) / c < ɛ$ , then go to Step 5; Otherwise, Let k : = k + 1, and return to step 2.

Step 5: end.

5 Example

5.1 Experiments

The experiments and performance evaluation were carried on several military infrared images. Results are compared with fuzzy C-means clustering algorithms for IFSs (FCM-IFSs) [6, 23], and intuitionistic fuzzy C-means clustering algorithms (IFCM) [9]. All algorithms were implemented with Matlab R2010a in a Windows XP system.

(a) ship infrared image. Figures 2–6 illustrate the case of ship infrared image segmentation by FCM-IFSs, IFCM and IFN. Figures 2(a)–6(a) is the original infrared image. Figure 2(b–d) are segmentation results by FCM-IFSs clustering algorithms. Figure 3(b–d) are segmentation results by IFCM clustering algorithms. Figure 4(b–d) are segmentation results by the proposed IFE clustering. The FCM-IFSs clustering algorithms for IFSs in Fig. 2 and the IFCM clustering algorithms method in Fig. 3 are similar and all the clustered images are obtained, but the result are not good as compared with the images for proposed IFE clustering with λ = 0.001, β = 1, c = 3 as shown in Fig. 4. It can be seen that some background information of ship can be observed in Fig. 4(b), more details of background are presented, but those information are missed in Figs. 2(b) and 3(b). The segmented images obtained are better than compared methods in Fig. 4(c–d). The clustered results in Fig. 4(c–d) are more clearly and more defined.

Increasing the parameter λ from 0.001 to 0.05, the clustered results using proposed intuitionistic fuzzy entropy clustering algorithm become bad as shown in Figs. 5–6. When the λ > 0.04, the clustered images Fig. 6(b) and (c) are same, it means that the number of clusters n (initial value is 3) is reduced to 2. It is the results of the intuitionistic fuzzy entropy term taking main affect. When λ < 0.0005, there are some NA values in the iterative computation, and the program stop. λ = 0.001 is the best value for ship infrared image segmentation by a large number of experiments. It can be observed that the clustering results are strongly affected by the parameter λ.

(b) helicopter infrared image. Figures 7–9 illustrate the case of helicopter infrared image segmentation by FCM-IFSs (Fig. 7), IFCN (Fig. 8) and proposed IFE clustering algorithms (Fig. 9). It can be observed that the segmented images obtained by proposed IFE clustering algorithms are better than compared methods, the image regions of body and engine exhausting gas of helicopter corresponding to the different objects are better discriminated. And the images of body and engine exhausting gas of helicopter are more clear and defined, and edges are more distinct in Fig. 9(c–d). The edge of helicopter are picked out more reasonable in Fig. 9(b), and the details of engine exhausting gas are obtained In the Fig. 9(d). More information of engine exhausting gas can be observed in the Fig. 9(d), and cannot obtained in Figs. 7(d) and 8(d).

(c) aircraft infrared image. In the third example, we are interesting in segmentation of aircraft infrared image. Figures 10–12(a) are original infrared image. Comparing with other two methods, the proposed IFE clustering method can describe more detail of background information of aircraft in Fig. 12(b), and pick up the aircraft body and engine exhausting gas accurately in Fig. 12(c) and (d). In the Fig. 12(b), the airflow around the aircraft is shown, but there is no information of the airflow around the aircraft in Figs. 10(b) and 11(b), so the proposed IFE clustering method can discover some information behind in the infrared image.

5.2 Performance

The the generalized Misclassification error(ME), Probability error(PE) and Relative ultimate measurement accuracy(RUMA) was used to evaluate the performance of the intuitionistic fuzzy entropy algorithm and compared method [11].

Figure 13 shows the ME, PE and RUMA for three methods for the image Figs. 10–12 are shown in the Fig. 13. From Fig. 13 one may observe that the intuitionistic fuzzy entropy algorithm produces segmented images that are closer to the ideally segmentedones.

6 Conclusion

In this paper we presented an intuitionistic fuzzy entropy clustering algorithm. Using the intuitionistic fuzzy entropy term, the objective function of the intuitionistic fuzzy entropy clulstering based the regularization technique is defined. A weight factor of the intuitionistic fuzzy entropy function is introduced as the degree of the intuitionistic fuzzy entropy. A special recursion scheme is provided to calculate the membership of the sample to the cluster. Finally, we demonstrated the efficiency of the proposed intuitionistic fuzzy entropy clustering algorithm on several infrared images and the results are observed to be far better as compared to fuzzy C-means clustering algorithms and intuitionistic fuzzy C-means clusteringalgorithms.

Further work is intended on the adaptation of the proposed intuitionistic fuzzy entropy clustering algorithm towards color image segmentation.

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