Performance Enhancement of Differential Evolution by Incorporating Lévy Flight and Chaotic Sequence for the Cases of Satellite Images

International Journal of Applied Metaheuristic Computing, 6(3), 69-81, July-September 2015 69

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ABSTRACTDifferential Evolution (DE) is a simple but powerful evolutionary algorithm. Crossover Rate (CR) and Mutation Factor (F) are the most important control parameters in DE. Mutation factor controls the diversification. In traditional DE algorithm CR and F are kept constant. In this paper, the values of CR and F are modified to enhance the capability of traditional DE algorithm. In the first modified algorithm chaotic sequence is used to perform this modification. In the next modified algorithm Lévy Flight with chaotic step size is used for such enhancement. In the second modified DE, population diversity has been used to build population in every generation. As a result the algorithm does not converge prematurely. Both modified algorithms have been ap-plied to optimize parameters of the parameterized contrast stretching function. The algorithms are tested for satellite image contrast enhancement and the results are compared, which show that DE via chaotic Lévy and population diversity information outperforms the traditional and chaotic DE in the image enhancement domain.

Performance Enhancement of Differential Evolution by

Incorporating Lévy Flight and Chaotic Sequence for the Cases of Satellite Images

Krishna Gopal Dhal, Department of Engineering and Technological Studies, University of Kalyani, Kalyani, India

Md. Iqbal Quraishi, Department of Information Technology, Kalyani Government Engineering College, Kalyani, India

Sanjoy Das, Department of Engineering and Technological Studies, University of Kalyani, Kalyani, India

Keywords: Chaotic Sequence, Contrast, Differential Evolution, Image Enhancement, Lévy Flight

DOI: 10.4018/ijamc.2015070104


70 International Journal of Applied Metaheuristic Computing, 6(3), 69-81, July-September 2015

1. INTRODUCTION

The goal of image enhancement is to process an image using some transformation function such that the resultant image is more suitable than the original one for some specific ap-plications (Gorai and Ghosh, 2009). Image enhancement is essential in image processing field for various image processing applications like contrast enhancement, noise reduction, edge enhancement, edge restoration etc. dur-ing pre-processing of various kind of images especially those which having poor contrast. In the case of color image enhancement, one of the simple techniques is to separate the image into the chromaticity and intensity component and then apply transformation function on intensity component (Garg et al., 2011; Yang et al., 2003). One of the basic processes to enhance the im-ages is histogram transformation (Leandro and Viviana, 2009). Histogram Equalization (HE) is a simple mechanism for image enhancement, but it has no control over the rate of enhance-ment. The enhanced image always follows the uniform distribution. The controlled enhance-ment can be done by putting limitations on the probability density function with the bin underflow (BU) and bin overflow (BO) (Yang et al., 2003). In literature different image en-hancement techniques are proposed based on the histogram information; but enhancement in a controlled way is still a challenging problem. As a solution soft computing oriented meth-ods have been applied recently. Evolutionary Algorithms (EAs) have been successfully ap-plied in image enhancement and segmentation field where both these two are considered as optimization problem (Paulinas and Ušinskas, 2007; Snyers and Petillot, 1995; Coelho et al., 2009). Genetic Algorithm (GA) has been suc-cessfully applied to enhance the images in a controlled way (Snyers and Petillot, 1995; Pal et al., 1994; Hashemi et al., 2010). GA is also effectively used in image segmentation domain to give an optimal segmented image (Chun and Yang, 1996). GA performs well in medical im-age segmentation. Tissue of ultrasound image is segmented in a prominent way by genetic

based incremental neural network (Dokur and Olmez, 2008). Differential evolution (DE) is a supreme version of GA. DE has the ability to grip non-differentiable, nonlinear, multi-modal cost functions and also has good convergence property (Liu et al., 2011). The efficiency of DE has also been proved in image enhancement domain (Yang, 2010). Application of DE is also found in the field of image fusion where DE is used for multi-focus image fusion to determine the suitable sizes of the block (Feng et al., 2011). An interactive DE algorithm has been applied for automatic image enhancement tool in smart phone (Lee and Cho, 2012). Mutation factor has been modified by chaotic sequence (Coelho et al., 2009) and the results show that modified DE is far better than traditional DE in image enhancement field maintaining faster convergence rate and better diversity property. Recently chaotic sequence has been used in metaheuristic algorithms to make it more pow-erful (Leandro and Viviana, 2009; Caponetto et al., 2003). Other metaheuristic algorithms have been successfully applied in image en-hancement and segmentation domain (Braik et al., 2007; Singh and Pandey, 2012; Gorai and Ghosh, 2011; Gupta and Gupta, 2012; A., 2012; Ma et al., 2011; Yun-Fei et al., 2012). Hybrid meta-heuristic algorithms which combine two metaheuristic algorithms give more promising results, viz. harmony search which mimics the process of a music player (Liu et al., 2011).

In this paper, a simple parameterized contrast stretching technique has been used to for contrast enhancement of gray level satellite images which is described in section 2. DE with Lévy Flight and DE with chaotic sequence are proposed in section 3 to enhance those images by maximizing the fitness value. Here mutation factor and crossover rate of DE algorithm have been taken as a gray numbers and are modified by chaotic Lévy Flight and chaotic sequence to improve the efficiency the traditional DE algorithm. Lévy Flight can be useful to modify any parameter to get a global best parameter as it is a random walk (Yang, 2012; Yang and Deb, 2010). Chaotic sequences are also good to get global best solution in stochastic optimization



techniques (Leandro and Viviana, 2009; Capo-netto et al., 2003). Proposed DE combined with chaotic Lévy Flight is compared with traditional DE and DE combined with chaotic sequence. Simulation result in section 4 proved that modi-fied DE via chaotic Lévy Flight is better than traditional DE, DE via chaotic sequence and HE to enhance the image.

2. PARAMETERIZED IMAGE ENHANCEMENT TECHNIQUE

Image enhancement tools improve the contrast and features of images, which in turn smoothens the progress of further image processing appli-cation like segmentation and finally improves the ability of human or machine recognition system to understand the useful information in the images (Coelho et al., 2009).

The contrast of an image can be modified by contrast stretching technique (Braik et al., 2007; Gonzalez and Woods, 2002) as given below:

g i j f i j

c m i j max m i j

, ) [{ ,

, } / , ]

= ( )( )− × ( ) − ( )( )

× ×( ) ( ) +( )M k i j b/ ,σ (1)

where, g i j,( ) and f i j,( ) are the gray level intensity of pixels in the output and input im-ages respectively and max is the maximum gray scale value. m i j,( ) and σ i j,( ) are the mean and standard deviations of the input im-age, computed using 3 3×

window. M is the

number of pixels in horizontal line. In Equation (1) c, k, b are the associated three parameters to obtain a large variation in resultant image. Range of three parameters are same as (Gorai and Ghosh, 2009). c ∈ [0, 1], b ∈ [0, 0.5], k ∈ [0.5, 1.5]. Initially k and c have been taken as 1 and b as 0. Parameter c always has been takes as a fractional value so that a fraction of the mean always subtracted from the pixel’s grey level intensity value.

2.1. Objective Function

Necessity of Objective Function of any optimi-zation algorithm, used for image enhancement is to select a criterion that is associated to a fitness function which will express all about the image features. In this paper three perfor-mance measurement parameters are taken into account viz. Entropy, Sum of the Edge Intensity and the Number of Edge Pixels or Edgels. It is obvious that good contrast enhanced image has more edgels and higher intensity of the edges (Gorai and Ghosh, 2009). If the allotment of the intensities is homogeneous, then histogram is equalized and the entropy of the image will be more. The Objective Function Fit(z), later expressed as Fit in this paper is the fitness value of enhanced image and is proposed as:

Fit z log

n M N H z

e

E I

edgels I

e

e

( ) =× ×( )( )× ( )

( )( )

( )� /

(2)

where, E Ie( ) is the sum of pixel intensities

of Sobel edge image Ie

, nedgels Ie( ) is the num-

ber of edge-pixels whose intensity value is above a threshold in the Sobel edge image. Based on the histogram, entropy value H z( ) is calculated on the enhanced image z . M, N are the numbers of pixels in horizontal line and vertical line of the image respectively.

2.2. Gray Theory

Gray theory, initiated in 1982 by Deng Julong is effectively used to solve mathematical prob-lems containing uncertainty (1989). “Gray” means poor, incomplete, uncertain etc. (Julong, 1989). The successful applicable of fields of gray theory are society, economics, finance, agriculture, industry, mechanics, meteorology, ecology, hydrology, geology, medicine etc. (Julong 1989).

In gray theory, gray number is a random variable and a random process is considered as



a gray process. Gray system is considered as a system containing information presented as gray numbers. In this paper, a simple use of this theory is used in image enhancement.

A gray number ⊗x is defined as an unknown value that has known upper and lower bounds.⊗ ∈[ ]x a b, , where, a b, are the lower and upper limits respectively (Ma et al., 2011). The whitening of a gray number means to specify a deterministic value for it in its defined inter-val. The whitened value �⊗x of gray number ⊗x is a deterministic value within the range a b, (Ma et al., 2011), i.e. mathematically ⊗ = ′x x , where, ′ ∈

x a b, .

In this paper, crossover rate (CR) and mutation factor (F) are considered as gray numbers and image enhancement method as an optimization problem is taken as a gray process. Here crossover rate (CR) and mutation factor (F) are both real and have constant values, where CR∈[ ]0 1, and F ∈

0 2, (Storn and

Price, 1997). So lower and upper limits of CR and F are known and have been taken as gray numbers.

3. THEORY OF DE ALGORITHM

3.1. Introduction

DE is a very powerful and simple evolutionary algorithm, proposed by Storn and Price which reveal regular and dependable performance in nonlinear optimization applications [16]. It has a few numbers of control parameters and faster convergence rate with more certainty than many other global optimization methods like genetic algorithm [29].

3.2. Lévy Flight

Lévy Flight has been used here as a random number generator or random walk. Certain species of birds and insects show this type of motion while gathering food (Yang, 2010). A random walk is a mathematical method of representing a series of consecutive random steps. It has huge applications in the area of computer science, physics, statistics, econom-ics and engineering (Yang, 2010). It can be expressed by the formula as follows:

S XNi

N

i==∑

1

where, Xi is a random step size drawn from a

random distribution and SN is the sum of each of these consecutive random steps. Here, step length of Lévy Flight is obtained from the Lévy distribution. It has the ability to explore bulky amount of search space. Mantegna’s algorithm, Rejection algorithm and McCulloch’s algo-rithms are admired to produce Lévy distribution. In this study Mantegna’s algorithm has been used. It produces random numbers according to a symmetric stable Lévy distribution (see Box 1) where, Γ is the gamma function (Yang, 2010; Yang and Deb, 2010), 0< α ≤ 2 (Yang, 2010), in this study it is taken as 1.5 which is same as (Yang and Deb, 2010). σ is the stan-dard deviation.

As per Mantegna’s algorithm the step length v can be calculated as:

v xy

= � �/1 α

(4)

Box 1.

σ α πα α α α α= +( ) ( ) +( ) )−( )Γ Γ1 2 1 2 2

1 2 1sin / / ( / ]/ / (3)



Here, x and y are taken from normal dis-tribution and σ σ σx y= =, 1 (Yang, 2010), where σ is the standard deviation.

The resulting distribution has the same behavior of Lévy distribution for large values of the random variables (Leccardim; Yang and Deb, 2010). Lévy Flight has been used in metaheuristic algorithms for the diversifi-cation and intensification (Yang, 2010; Yang and Deb, 2010; Yang, 2010). For the case of diversification the step length has been taken larger than in the case of intensification. It has good searching property because the repeti-tion of the same position in its space by Lévy Flight is less than in the Brownian motion (Yang, 2010).

3.3. Making of Initial Population

Initially n numbers of individuals are generated using the equation given below:

x low up lowi = + −( )×∂ (5)

xi is the ith individual. up and low are

the upper and lower bounds of the search space of objective function. ∂ is the random variable belongs to [0, 1]. If the initial solutions are generated in this way then the solutions become diversely distributed (Jamil and Zepernick, 2013).

3.4. DE Algorithm for Image Enhancement

Step 1: Initialize the population of enhanced image using Equation (5).

Step 2: Take the Objective function Fit as per Equation (2).

Step 3: Mutation step: For every enhanced image, take any three randomly chosen images and for every parameter, get a new parameter by:

V X F X Xi

t

r

t

r

t

r

t+( ) ( ) ( ) ( )= + × −( )1

3 1 2 (6)

where:

r r r1 2 3≠ ≠

F is the mutation factor, which controls the amplification of the difference between two individuals so as to avoid search stagnation.

Step 4: Crossover: Crossover increases the diversity of the population:

H Vi

t

i

t+( ) +( )=1 1

if:

rand CR0 1, ≤

= ( )Xit

if:

rand CR0 1, > (7)

Step 5: Selection: Using parameter Hi create new enhanced image:

X Hit

it+( ) +( )=1 1

if:

H Fit Xi

t

i

t+( ) ( )> ( )1

= ( )Xi

t

if:



H Fit Xit

it+( ) ( )≤ ( )1

It means if modified parameters create an enhanced image which fitness value is greater than previous one replace it otherwise not.

Step 6: Find the current best.Step 7: Repeat steps 3 to 6 until maximum

generation.

3.5. Chaotic DE Algorithm

This is another approach to design mutation factor and crossover rate in this paper. The complex behavior of non-linear deterministic system is defined by chaos (Boccaletti et al., 2000). Chaos has non-repetition property, hence ergodic (Leandro and Viviana, 2009). Several optimization algorithm use chaotic sequences to provide the standard algorithm good di-versification property (Leandro and Viviana, 2009; Caponetto et al., 2003). Particle swarm Optimization (PSO) used it for enhance the diversification property (Leandro and Viviana, 2009). Evolutionary optimization algorithms can enhance the capability of searching global best solution using chaotic sequences (Capo-netto et al., 2003).

There are several chaotic generators like logistic map, tent map, gauss map, sinusoidal iterator, lozi map, chua’s oscillator etc (Capo-netto et al., 2003). Among those simple logistic equation that based on logistic map is used in this paper to generate mutation factor. The equation of logistic map is given below:

L aL Lm m m+ = −( )1

1 (8)

a is a control parameter and 0< a ≤ 4, Lm is the chaotic value at mth iteration. The behav-ior of the system is mostly depends on the variation of a. Value of a is 4 and L

0 does not

belong to {0, 0.25, 0.5, 0.75, 1} otherwise the logistic equation does not show chaotic behav-ior (Coelho et al., 2009). The range of Lm is transformed to [0, 1] in this study.

The mutation factor (F) and crossover rate (CR) have been replaced by value generated by logistic equation. So the Equation (6) and (7) are replaced by Equation (9) and (10) respectively:

V X L

tt

X

i

t

r

t

m

maxr

t

+( ) ( )

(

= +

× −

×

1

3

1exp )) ( )−( )X

r

t

2

(9)

H Vi

t

i

t+( ) +( )=1 1

if:

rand Lm0 1,[ ] ≤

= ( )Xi

t

if:

rand Lm

0 1, > (10)

3.6. DE Algorithm with Chaotic Lévy Flight

The basic DE algorithm and the above modified DE algorithm do not look up on the popula-tion diversity which plays an important role in population based algorithms so that they do not converge prematurely. Greater population diversity means greater searching power in population based algorithms (Fister et al., 2013). Selection of individuals to make the population of search process plays an important role in any metaheuristic algorithm. If the population diversity has reached to saturation very quickly then the algorithm converges prematurely



(Fister et al., 2013). So, if the population di-versity is computed and depending upon that the individuals is selected to construct the set of population then algorithm neither converges too early nor become trapped in the local minima and also gives a balance between exploration and exploitation. This technique also depends upon the fitness value of each solution. The whole idea is depended on the Fitness Diversity Metric (FDM). This FDM is used to measure the population diversity (Fister et al., 2013). On the other hand, this metric helps to maintain balance between the diversification and intensification in metaheuristic algorithms (Fister et al., 2013). It is defined as follows:

ϕ =

− −( ) −( )

1

mod fit fit fit fitavg min max min/

(11)

where, fitavg

, fitmin , fitmax

are the average, minimum and maximum fitness values within the population (Fister et al., 2013).

So, from the equation it can be deduced that � ,�ϕ ∈

0 1 .

When the value of ϕ is close to zero the population diversity is low and when close to one population diversity is high. After popula-tion diversity has been calculated, the following steps are executed to select individuals to make population.

Step 1: At first 2Q numbers of individuals are created. Where, Q is the number of indi-viduals that have been taken for evaluation in algorithm.

Step 2: First Q individuals are taken as parent set (P

parent) and next Q individuals are

taken as offspring set (Poffspring ).Step 3: Sort the 2Q numbers of individuals

according to their fitness value. Make two sets, P

high and Plow . Where, P

high contains

Q individuals with higher fitness values

and Plow

contains Q individuals with lower fitness values.

Step 4: Select Q numbers of individuals from Phigh and P

low by the following rule: A

random number r is generated in the inter-val [0, 1]. If r < −( )0 5. ϕ then take in-

dividual from Plow . Otherwise take indi-vidual from P

high. By using the above rule

make the evaluation set Peval with Q in-dividual to evaluate in search process.

Step 5: In the next generation the evaluated Q individuals from the P

eval are replaced by

the off-spring set Poffspring .

There are three control parameters which can control the effectiveness of DE algorithm, namely Population number, Crossover rate (CR) and Mutation factor (F). F controls the diversification property of DE algorithm. If the choice of these two parameters is not so good the algorithm converges prematurely. In traditional DE these three parameters are taken as constant values in the whole search process and it is a difficult task there to choose the proper values.

Application of Lévy Flight to devise the mutation factor and crossover rate is a good strategy to provide the traditional DE algorithm a superior diversification and intensification capability as it has less repetition property than Brownian motion (Yang, 2010; Yang 2010). As a result, it does not converge prematurely.

Initially F and CR have been taken as 0.5, and then they are modified by Lévy Flight by the given equation:

Y Y

t ti

t

i

t+( ) ( )=

+ × −( ) ×1

α λexp Lévymax/ ) ( )

(12)

Here, Yit+( )1 is the value of the F and CR

parameter at (t+1)th iteration. �tmax

represents the maximum generation andα is a randomization



parameter derived from chaotic sequence within the range [0, 0.5]. rand is a random value be-tween 0 and 1 and Lévy( )λ is the value determined by Lévy fight (Yang and Deb, 2010).

Step size α is small for both F and CR. If the values of F and CR are changed in large volume then the algorithm will be trapped in the local minima. Here step size α derived using chaotic sequence. Drawback of Lévy Flight is the choice of appropriate step size. If the step size is fixed then it does not show the ergodicity property which signifies that it does not pass through every point of the solution space (Walton et al., 2013). This step length is chosen depending upon the scale of the problem. In this paper chaotic sequence is used as step size because chaotic sequence has ergodicity property (Leandro and Viviana, 2009; Capo-netto et al., 2003). Thus it amplifies the chance to search with every possible scale in solution space (Walton et al., 2013).

4. EXPERIMENTAL RESULTS

In Table 1, Img (a) is the original Image, Img (b) is the result of DE combined with chaotic Lévy Flight sequence, Img (c) is the result of DE com-bined with chaotic sequence, Img (d) is the result of traditional DE of the corresponding image, Img (e) is the result of histogram equalization (HE).

4.1. Enhancement Factor

Enhancement factor (EF) is calculated us-ing variance and mean of the image (Jha and Chouhan, 2014) as follows:

EF e e

o

=( )( )σ

σ

µ

µ

2

20

/

/ (13)

where, σe2 , .. represent the variance and the

mean of the enhanced image and σ o2 , µo are

the variance and the mean of the original image (see comparison in Table 2).

4.2. Comparison of the Number of Maximum Generations

Three algorithms have been applied over 100 images with initial population number being varied from 10 to 50 and maximum generations up to 100. In this study, num-bers of initial population has been set as 20. From the experiment, the generations have been optimally put 90 for traditional DE, 70 for DE via chaotic sequence, 55 for DE via chaotic lévy. The graphical interpretation has been given below in Figures 1, 2, and 3.

From the above graphs it is clear that DE combined with chaotic Lévy Flight gives more accurate global best solution in less number of iterations than DE combined with chaotic sequence.

5. CONCLUSION

This paper introduces two approaches to enhance the capability of DE algorithm to better the contrast of gray level images. One based on chaotic sequences and other based on chaotic Lévy Flight. Modifica-tion of mutation factor and crossover rate by chaotic Lévy Flight improves the effi-ciency of traditional DE algorithm. Devised mutation factor and crossover rate using logistic equation also better the quality of traditional DE algorithm. From visual analysis and quantitative results it is clear that DE combined with chaotic Lévy Flight is better than DE combined with chaotic sequence with respect to fitness values and number of maximum generations. But both two modified DE give far better result than traditional DE algorithm in image enhance-ment field.

ACKNOWLEDGMENT

This research work is funded by DST-PURSE.



Table 1. Comparisons of resultant image of different methods



Figure 1. Maximum generation (x axis) vs. fitness value (y axis) for traditional DE for Img1

Table 2. Comparison of number of edgels, entropy, sum of the edge intensity pixels and fitness value

No. of Edgels Entropy Sum of the Edge Intensity Pixels

Fitness Value EF

Img1

DE via chaotic Lévy Flight

2765 7.8497 1521.2 2.4267 6.34

DE via chaotic sequence

2696 7.6371 1478.3 2.2930 4.70

Traditional DE 2622 7.2133 1189.8 2.0437 3.44

HE 2645 7.8875 1412.1 2.3088 5.03

Img2


3402 7.6037 1704.5 2.9371 5.33


3352 7.5360 1526.6 2.8250 4.58

Traditional DE 3301 7.3340 1235.2 2.6298 2.84

HE 3226 7.7699 1637.1 2.8306 4.47

Img3


2413 7.2806 1554.9 1.9701 7.41


2389 7.4450 1206.4 1.9257 5.71

Traditional DE 2313 6.9049 1215.6 1.7310 3.64

HE 2302 7.7769 1408.9 1.9806 2.01



Figure 2. Maximum generation (x axis) vs. fitness value (y axis) for DE via chaotic sequence for Img1

Figure 3. Maximum generation (x axis) vs. fitness value (y axis) for DE via chaotic Lévy Flight for Img1



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Krishna Gopal Dhal completed his B.Tech and M. Tech from Kalyani Government Engineering College. Currently he is working as Research Scholar in University of Kalyani. His research interests are image Processing and Nature inspired Metaheuristics.

Iqbal Quraishi completed his M.Tech from West Bengal Univ. of Technology. He is currently working as Assistant Professor in Dept. of Information Technology, Kalyani Govt. Engineering College. His research interests are Image Processing and Soft-Computing.

Sanjoy Das completed his B.E. from Regional Engineering College, Durgapur, M.E. from Bengal Engineering College (Deemed Univ.), Howrah, Ph.D. from Bengal Engineering and Science University, Shibpur. Currently he is working as Associate Professor in Dept. of Engineering and Technological Studies, University of Kalyani. His research interests are Tribology and Optimization Techniques.

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http://dx.doi.org/10.1016/0167-8655(94)90058-2

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http://dx.doi.org/10.1109/CISP.2010.5647237

http://dx.doi.org/10.1002/9780470640425

http://dx.doi.org/10.1504/IJMMNO.2010.035430

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Performance Enhancement of Differential Evolution by Incorporating Lévy Flight and Chaotic Sequence for the Cases of Satellite Images