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Automatic Optimal Thresholding Using Generalized Fuzzy Entropies and 
Genetic Algorithm 

 
Gh. Reza Atazandi  

Department of Electrical Engineering, Mashhad Branch, Islamic Azad University 
Razavi Khorasan Province, Mashhad, Kuy-e-Honar, Ostad Yousefi Boulevard, Iran 

Phone: +98 51 3525 1317 
atazandi@gmail  

 
S. Ehsan Razavi 

Department of Electrical Engineering, Mashhad Branch, Islamic Azad University 
Razavi Khorasan Province, Mashhad, Kuy-e-Honar, Ostad Yousefi Boulevard, Iran 

Phone: +98 51 3525 1317 
ehsanrazavi@mshdiau.ac.ir 

 
Fariba Nobakht 

Faculty member of Asrar Institute of Higher Education, Mashhad, Iran 
Razavi Khorasan Province, Mashhad, District 11, 69, Iran 

Phone: +98 51 3866 1771 
 

Abstract 
The use of fuzzy entropy for image segmentation is one of the most popular methods, which 

is used today. In a classical fuzzy entropy, using a fuzzy complement with an equilibrium point of 
0.5 is a limitation, which reduces the chances of obtaining an optimal result. We use generalized 
fuzzy entropy phrases in this paper, which uses fuzzy complements of Sugeno and Yager, and 
corresponds the equilibrium point to the m parameter (0<m<1), and increases the chance of finding 
the optimal threshold. So, we will have many pictures depending on the points of balance, and by 
the genetic algorithm, we choose the best decision among them. The effect of this method have 
considered in medical images to find the brain tumors. Results have shown that the use of 
generalized fuzzy entropy and the genetic algorithm can greatly be used to find the optimal 
threshold. Presented method is very effective for reducing the number of intensity levels. Problems 
may cause images with height amount of unwanted information which is saved to the expanse of 
subjective more important information. 
 

Keywords: Image Segmentation; Fuzzy Entropy; Generalized Fuzzy Entropy; Fuzzy 
Complement Operator; Genetic Algorithm. 
 
1. Introduction  
The fuzzy entropy determines the fuzziness level and ambiguity of a fuzzy set, it is also a 

basic concept for the theory of fuzzy sets. The Fuzzy entropy has been widely used for image 
segmentation and threshold selection (Huang, & Wang, 1995; Li, Zhao, & Cheng, 1995; Cheng, 
Chen, & Sun, 1999). The first definitions of fuzzy entropy were presented by Deluca and Termini 
(De Luca, & Termini, 1972). The basic functions of union, intersections, and fuzzy 
complementation were used by them in their definitions which were introduced by Zadeh (Zadeh, 
1965). One of the important characteristics of using a fuzzy supplement in the definition of entropy 
is that there is the greatest ambiguity at the equilibrium point of x = 0.5. This issue in real 
applications always confronts us with a truth. If we use the concept of fuzzy uncertainty to calculate 
the darkness of an image, as Pal uses it (Pal, 1982). A single point of equilibrium for a basic fuzzy 
complement does not necessarily result in acceptable results. By generalizing the concept of fuzzy 
entropy, Zenzo (Zenzo, Cinque, & Levialdi, 1998) created a condition in which we have the highest 
amount of fuzziness and uncertainty for the equilibrium points of x=m (m∈(0,1)) and the 
application of this generalized fuzzy entropy for threshold selection in image processing, increases 
the chances of obtaining an appropriate image.  



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But the fuzzy entropy generalized by Zenzo did not refuse the fourth condition of a fuzzy 
entropy, which includes a complement operator for a fuzzy set. Therefore, the generalized fuzzy 
entropies were then presented (Fan, & Zhao, 2008). in which the comprehensiveness of the four 
conditions of the fuzzy entropy definition was taken into account.  To apply these generalized fuzzy 
entropy definitions was made to select the optimal threshold with m membership degree, ( 0<m<1 ). 

But using these methods for image segmentation, provides m image which ultimately to 
choose the best picture is done by a person. Depending on the individual view, the selected final 
images may also be different. In this study we present a method in which a variety of fuzzy 
complements are used for fuzzy entropies, to improve the previous methods. in addition, at the end, 
the genetic algorithm is used to select the best image from the m image provided by the extended 
fuzzy entropy, to have the optimal image selection by an intelligent method.  

The structure of this article is as follows: In the second section, we give an overview of the 
definition of fuzzy entropy, In Section 3 we examine the various definitions of generalized fuzzy 
entropy. The fourth part introduces the method of image segmentation using generalized fuzzy 
entropy and the use of various fuzzy supplements. In Section 5, an intelligent selection of optimal 
image based on genetic algorithm is proposed. The performance of the proposed method in images 
related to the identification of brain tumors has been done in section 6. Finally, the conclusions and 
recommendations are presented in Section 7. 

 
2. Fuzzy Entropy  
In this paper, we show the set of all fuzzy sets on universal set X by means of F(X), and the 

set of all crisp sets by means of P(X). 
- μ  (x) is the membership function of A∈F(X). 
- [a] is the fuzzy set of x, so that 𝜇[ ](𝑥) = 𝑎. 
Complement, union and intersection operators for these sets are defined as follows:  
𝐴  Is the indicator of completes𝐴, so that c(x) = 1− x (x∈ [0, 1]). The union of two fuzzy 

sets of A and B is shown as 𝐴 ∪ 𝐵 and defined as 𝜇 ∪ (x) = max 𝜇 (𝑥) , μB (𝑥)) and intersection 
of two fuzzy sets of A and B is shown as 𝐴 ∩ 𝐵 and defined as 𝜇 ∩ (x) =min 𝜇 (𝑥) , μB (𝑥)). A 
fuzzy set A∗ is called a sharpening of A. For the set of A* which is sharper than A, If 𝜇  (x) ≥0.5, 
then 𝜇 ∗ (x) ≥ 𝜇  (x) and if 𝜇 (x) ≤ 0.5 then 𝜇 ∗ (x) ≤ μ A (x). Deluca and Termini.4, defined the 
fuzzy entropy for the calculation of the uncertainty level of a fuzzy set, defined on the universal set 
X as follows:  

 
e(A) = 0 if A∈P(X); 
e(A) reaches to its maximum if A=[0.5]; 
If A∗ is a sharpening of A, then e(A*) ≤ e(A); 
e(𝐴 ) = e(A). 

 
3. Generalized Fuzzy Entropy 
As mentioned, the equilibrium point of the base complement operator is only at x =0.5, and 

this is a limitation to use to select the threshold. Other complement operators have been proposed by 
researchers, which eliminate the limit of the unique equilibrium point (Sugeno, 1977; Lowen, 1978; 
Yager, 1980; Malinowski, 1993).    
For example, Sugeno has presented a fuzzy complement function as follows: 
 

 
In which the equilibrium point of this function is equal to m = (-1±√(1+λ)) / λ. 
yager has provided the following fuzzy complement function: 

 



Gh. R. Atazandi, S. E. Razavi, F. Nobakht - Automatic Optimal Thresholding Using Generalized Fuzzy Entropies and 
Genetic Algorithm 

 

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Which has the equilibrium point of m = 1 / √(w&2). Based on the fuzzy complement with 
the non-unique equilibrium point, various functions for generalized fuzzy entropy (em: F(X) → 𝑅 ) 
are presented (Khotanloua et. al., 2009), which refuses the four conditions below: 

 
(1)  e m (A) =0 if A∈P(X); 
(2)  e m (A) attains its maximum if A=[m] ;  
(3)  If A* is shaper than A , then em (A*) ≤  em (A).  
Here A* satisfies that μA (x) ≥ μA (x) when μA (x) ≥ m ; μA (x) ≤ μA (x) 
when μA (x) ≤ m ; 
(4)  e(Acm ) = e(A). 
 
But in all of them, only one fuzzy complement has been used. In this paper we use 

generalized fuzzy entropy functions with multiple complements to improve performance. Some of 
the generalized fuzzy entropy relations for a fuzzy set A on a finite x set are given below: 

Where  

 
Where 

    
 

4. Fuzzy Entropy-Based Image Segmentation Method 
To choose the threshold for image processing based on generalized fuzzy entropy has been 

one of the most commonly used methods in recent research. In this method we consider the 
parameter m in the interval (0,1) with steps 0.05, and we will find for each image 19 outcomes and 
then, by intelligent genetic algorithm, the best decision will be chosen from these results. We first 
consider the image of size M × N with a Matrix Q, where q xy shows the gray level of the pixel (x, 
y). If we look at the image as a fuzzy set, μQ qxy represents the membership value, which indicates 
the brightness level of the pixels (x,y) in the Q matrix. To show the gray level brightness, we use 
the s function, which is defined as follows: (Li & Yang, 1989) 

 

 
 
(7) 

In which x is the gray level in Q and a, b, d are the parameters that identifies the shape of S 
function, and a > b > d.  

The image segmentation based on fuzzy entropy is based on the selection of maximum 
fuzzy entropy as threshold. The shape of the function s is determined by the parameters a, b, d, so 
the threshold selection problem is related to the combination of these parameters, So that maximum 
fuzzy entropy is obtained (Li, Zhao, & Cheng, 1995, Cheng, & Chen, 1997, Hrubes & Kozumplik, 
2007). This optimal combination of parameters is represented by (a*, b*, d*), Then, using the 
generalized fuzzy entropy for image segmentation, the optimal threshold value is selected as 
follows:  

 



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When we consider the fuzzy complement of the Sugeno class instead of the base fuzzy 

complement, the relation between m, and ƛ is obtained as follow: 

 

  (11) 
Which is a mapping of m∈(0,1) to ƛ∈(-1,∞). 
 
If we use the fuzzy complement of the yager class, the relationship between m and w is as 

follow:  

 

  (12) 
which is a mapping of m∈(0,1) to w∈(-1,∞). 
 
By respect to these relations, the relations introduced in (3-6) for generalized fuzzy entropy 

include parameters m (m∈(0,1)) . Then we divide m in the internal (0,1) with steps 0.05. 19 results 
are obtained for each image. the best visual image will be selected by a smart algorithm. 

 
5. Genetic Algorithm 
Genetic algorithm is an intelligent tool for optimization, In this method, the problem is 

transformed from decimal space to the binary space in which each row is a set of parameters as an 
answer to the problem. The quality of all the answers in each generation is evaluated by a proper 
function. The best answer is selected with the highest probability along with a random exchange of 
information in two parts for intersection. These solutions are then used with the genetic mutation 
used to maintain genetic diversity and avoid local extremes. The stimulus operator returns randomly 
selected bits to a genetic sequence. An example of a flowchart genetic algorithm is shown in Fig 1. 
In this case, an answer, containing the parameters a, b, c, will be the membership function of 
µ(a,b,c). 

All of these parameters are integers from 0 to 255, similarly, an answer of the problem will 
be encrypted with 24 bits. 

 
6. Experimental Results 
In this section, we compare our method performance with that of the classical fuzzy 

entropy-based method and efficient methods used today for image tumor description. The 
generalized fuzzy entropy formulas mentioned in (3) ~ (6). The complement functions are segeno 
and Yager’s complement. 
 



Gh. R. Atazandi, S. E. Razavi, F. Nobakht - Automatic Optimal Thresholding Using Generalized Fuzzy Entropies and 
Genetic Algorithm 

 

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Figure 1. Flow Diagram of Genetic Algorithm 

 
6.1. Compare with Traditional Fuzzy Entropy Based 
The experimental images are meningioma and glioblastoma. For each of this images, first, 

we have extracted 19 Retrieved images with generalized fuzzy entropy and then with Genetic 
Algorithm we find the optimal image. 

For meningioma (Figure 2), with  the generalized fuzzy entropy, the threshold of fuzzy 
entropy is 104, and the final threshold of the generalized fuzzy entropy with (GA) is located at m = 
0.75 (T=149). For glioblastoma (Figure 3), wuth the generalized fuzzy entropy, the threshold of 
fuzzy entropy is 117, and the final threshold of this method is located at m = 0.85 (T=157). 
 

 
Figure 2. Meningioma image: (a) Prime image; (b) Processed with fuzzy entropy; (c) Processed with 

generalized entropy and (GA) 



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6.2. Compare with a Fuzzy Neural Network  
In this part we compare the results obtained from this method with a fuzzy kohonen 

clustering network (N.I.Jabbar & M.Mehrotra, 2008). The result of the segmentation with our 
method obtained in m = 0.9 (T=200) (Figure 4). 
 

 
Figure 4. The image results : (a) prime image; (b) processed with a fuzzy kohonen clustering network; (c) 

processed with generalized entropy and (GA) 
 
6.3. Compare with a 3D Brain Tumor Segmentation in MRI Using Fuzzy 
Classification, Symmetry Analysis and Spatially Constrained Deformable Models 
Based on this method, first we will have a segmentation in the presence of tumors. Then a 

tumor detection stage will be implemented with selection asymmetric regions, considering the 
approximate brain symmetry and fuzzy classification (H. Khotanloua, O.Colliotb, J.Atifc & 
I.Blocha, 2009, Moumen T El-Melegy & Hashim M Mokhtar, 2014, Chalumuri Revathi & B. 
Jagadeesh, 2017). 

Compare this method with the method that we presented, gives the following result in m 
=0.7 (T=151) (Figure 5). 

 
 

 
Figure 5. The image results: (a) prime image; (b) processed with a fuzzy classification, symmetry analysis 

and spatially constrained deformable models; (c) processed with generalized entropy and (GA) 
 
 



Gh. R. Atazandi, S. E. Razavi, F. Nobakht - Automatic Optimal Thresholding Using Generalized Fuzzy Entropies and 
Genetic Algorithm 

 

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7. Conclusions 
With generalized fuzzy entropy and (GA)-based method we have segmentation for an image 

that the threshold can accept the membership value m. This method increases the chance of 
choosing optimal thresholds, so it provides better performance than the fuzzy entropy-based 
method.  

Presented method is very effective for reducing the number of intensity levels. Problems 
may cause images with height amount of unwanted information which is saved to the expanse of 
subjective more important information. Results have shown that the use of generalized fuzzy 
entropy and the genetic algorithm can greatly be used to find the optimal threshold. 

However, doing this method with other well-known algorithms such as ant colony 
optimization, particle swarm optimization and imperialist competitive algorithm, worth to research 
further. 

 
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Revathi, C., Jagadeesh, B. (2017). Segmentation of Brain Images for Tumor Detection Using 
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Seyed Ehsan Razavi (b. September 05, 1981) received his BSc in Electrical 
Engineering (2006), MSc in Control Engineering (2008), PhD in Control 
Engineering (2014). Now he is Assistant Professor of Electrical Engineering 
Department in Islamic Azad University, Mashhad Branch, Mashhad, Iran. His 
current research interests are Control Theory, Biomedical Engineering, 
Nanoparticles, Image Processing and Hyperthermia. He has authored several 
book chapters and published more than 30 papers in journals and conferences.