Soft ComputDOI 10.1007/s00500-014-1436-0

METHODOLOGIES AND APPLICATION

Image quantization using improved artificial fish swarm algorithm

Shaimaa Ahmed El-said

© Springer-Verlag Berlin Heidelberg 2014

Abstract Most image compression algorithms suffer fromseveral drawbacks: high-computational complexity, moder-ate reconstructed picture qualities, and a variable bit rate. Inthis paper, an efficient color image quantization techniquethat depends on an optimized Fuzzy C-means (OFCM) algo-rithm is proposed. It exploits the optimization capability ofthe improved artificial fish swarm algorithm to overcome theshortage of Fuzzy C-means algorithm. It uses error diffusionalgorithms to obtain perceptually better images after quan-tization. Experiments are carried out to estimate the perfor-mance of the proposed OFCM algorithm in image compres-sion using standard image set. The results indicate that thealgorithm can decrease effectively the mean square deviationof color quantization, keep overall arrangement of ideas andpart characteristic detail in image reconstruction. The per-formance efficiency of the proposed technique is comparedwith those of three other quantization algorithms. The Com-parative results confirmed that the OFCM has potential interms of both accuracy and perceptual quality as comparedto recent methods of the literature.

Keywords Image quantization · Compression · Dataclustering · FCM · Swarm intelligence · Artificial fish swarmalgorithm (AFSA)

Communicated by V. Loia.

S. A. El-said (B)Electronics and Communications Department, Faculty of Engineering,Zagazig University, P.O. Box 44519, Zagazig, Egypte-mail: eng.sahmed@windowslive.com

1 Introduction

Image compression plays a key role in the development ofvarious multimedia computer services and telecommunica-tion applications. There is a pressing need to limit the imagedata volume for efficient image storing and transmission. Thegoal of image quantization is to remove the data redundancyin a way that enables image reconstruction. The strong corre-lation between image data items enables data reduction with-out significant quality degradation (Vasmatkar et al. 2011).Color quantization (CQ) process is done in two steps (Say-ood 2006). In the first step, a codebook is constructed. Inthis step, it has to be determined how many colors have tobe decreased. In fact, the number of considered colors is thenumber of codewords in the codebook. Each codeword rep-resents a color and its index in the codebook that each ofthese codewords is representative of multiple colors in theoriginal image. In the second step, each image is decodedby its corresponding codebook. The main goal of CQ is toobtain an appropriate codebook (Sayood 2006). If the code-book is not proper, the resulted image from CQ has muchdisharmony with the original image and distortion increasesbetween the original image and decoded one.

One of the applied approaches for producing codebookbased on the color distribution is using clustering algorithmslike k-means (Celebi 2011) and Fuzzy C-means (FCM)(Schaefer and Zhou 2009). In clustering CQ algorithms, thecolor histogram is produced from the original image and afterthat, clustering according to the color distribution among pix-els is done. The number of cluster centers in clustering algo-rithms is determined equal to the number of decreased colorsin the codebook. In clustering process, cluster centers con-tain a smaller set of colors. Other colors with respect to thedifference between their color numbers and the numbers ofcluster center colors become a member of one of the clusters.

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That is, each of colors becomes a member of a cluster thatits center color is more similar than other cluster centers.

Using traditional FCM algorithm in Image quantizationbrings randomness in initial clustering centers’ selection andit is sensitive to the noise data and can easily converge to localextremism points (Yang et al. 2004). Thus, it can not achievethe global optimum, as to color quantification it appears sen-sitive to the initial clustering centers selecting of color sam-ples, and the local color rendering is not correct (Kim et al.2004). One of the applied methods for clustering enhance-ment is the use of the swarm intelligence algorithms.

Swarm Intelligence such as Particle Swarm Optimiza-tion (Tsai and Kao 2011), Genetic Algorithm (Surekha etal. 2010), Bee Colony Optimization (BCO) (Teodorovicand Dell’Orco 2005), or Ant Colony Optimization (Tonget al. 1999), and Artificial Fish Swarm algorithm (Yazdaniet al. 2010) is a modern method employed in optimizationproblems. It is based on the en masse movement of liv-ing animals like birds, fishes, ants and other social animals.Migration, seeking for food and fighting with enemies aresocial behaviors of animals. Artificial Fish Swarm algorithm(AFSA) method is one of the Swarm Intelligence approachesthat works based on the population and stochastic search.Fishes show very intelligently social behaviors. This algo-rithm is one of the best approaches of the Swarm Intelligencemethod with considerable advantages like high-convergencespeed, flexibility, error tolerance and high accuracy (Neshatet al. 2012). AFSA has many applications in various fieldslike optimization, control, image processing, data mining,improving neural networks, networks, scheduling, and sig-nal processing and so on.

1.1 Problem statements

A common approach in color image quantization is to usefuzzy iterative clustering algorithms that provide a partitionof the pixels into a given number of clusters. FCM alter-nately optimizes membership degrees and centroids until thebest clusters are found. However, most of these algorithmspresent several drawbacks: they are time consuming, sen-sitive to noise and rely on the initial conditions. The maincontribution of this paper is proposing a novel fuzzy clus-tering algorithm, which combines Improved AFSA and anefficient extension of FCM, to enhance the performance ofFCM in color image quantization.

The remainder of the paper is organized as follows: Sect.2 depicts the related work. In Sect. 3, a literature survey onFCM clustering algorithm and artificial fish swarm algorithmand a brief description of the two algorithms are provided, andin Sect. 4, the proposed optimized Fuzzy C-means (OFCM)algorithm is presented. Section 5 studies the experiments andanalyzes the results. Section 6 concludes the study and dis-cusses directions for future research.

2 Related work

The need for image compression arises for resourceful stor-age and transmission. A number of CQ algorithms, alongwith their modifications, have been developed during the pastseveral decades. Vector quantization (VQ) is one of the lossydata compression techniques (Gray 1984; Linde et al. 1980).It is a mapping function which maps k-dimensional vectorspace to a finite set CB = {C1, C2, C3, . . ., CN }. The set CBis called as codebook consisting of N number of code vec-tors and each code vector Ci = {ci1, ci2, ci3, . . ., cik} is ofdimension k. Good codebook design leads to reduced distor-tion in reconstructed image. Codebook can be designed inspatial domain by clustering algorithms (Gray 1984; Lindeet al. 1980; Kekre and Sarode 2009a, b, c, d). For encoding,image is split in blocks and each block is then converted tothe training vector Xi = (xi1, xi2, . . ., xik). The codebookis searched for the nearest code vector Cmin by computingsquared Euclidean distance.

Clustering quantization techniques are categorized intotwo families: preclustering methods and postclustering meth-ods (Brun and Trémeau 2002). Preclustering methods aremostly based on the statistical analysis of the color distri-bution of the images. Divisive preclustering methods startwith a single cluster that contains all N ′ image colors. Thisinitial cluster is recursively subdivided until C clusters areobtained. Well-known divisive methods include median cut(Heckbert 1982), octree (Gervautz and Purgathofer 1988),variance-based method (Wan et al. 1990), binary splittingmethod (Orchard and Bouman 1991), and greedy orthog-onal bipartitioning method (Wu 1991). On the other hand,agglomerative preclustering methods (Velho et al. 1997;Brun and Mokhtari 2000) start with N ′ singleton clusterseach of which contains one image color. These clusters arerepeatedly merged until C clusters remain. In contrast topreclustering methods that compute the palette only once,postclustering methods first determine an initial palette andthen improve it iteratively using any data clustering method.Since these methods involve iterative or stochastic optimiza-tion, they can obtain higher quality results when comparedto preclustering methods at the expense of increased compu-tational time. Clustering algorithms adapted to color quanti-zation include hard c-means (Celebi 2009, 2011), competi-tive learning (Celebi 2009; Celebi and Schaefer 2010), FCM(Ozdemir and Akarun 1999; Kim et al. 2004; Schaefer andZhou 2009), and self-organizing maps (Papamarkos et al.2002). Yu and Yang (2005) conduct a theoretical analysisof the Inter-cluster separation (ICS) fuzzy clustering algo-rithm. They establish the fixed-point property of ICS basedon the decomposition of the Hessian matrix and then analyzethe effect of the parameters. Finally, they propose a numeri-cal approach in choosing the appropriate parameters m andgamma for ICS.

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Recently, Yang and Zhang (2010) have proposed a tech-nique for image compression using modified FCM algorithmbased vector quantization (VQ). The VQ codebook is gener-ated by a modified FCM algorithm. In Boopathi and Arocki-asamy (2011), a clustering algorithm about FCM of weightedCharacteristic is proposed. It compensates the colors inhomo-geneous by interposing weighted matrix RGB color space,and a statistic clustering algorithm of minimal distance inclass to initiate clustering center.

Some stochastic optimization methods have been devotedrecently to efficiently resolve various optimization problemsin color image quantization. Particle swarm optimization(PSO) is an evolutionary computation technique developedthrough a simulation of simplified social models. It is basedon swarms such as fish schooling and bird flocking. In Kaur etal. (2011), a novel color image quantization technique usingPSO is proposed. The LAB color model based clustering isused. The selected images’ dataset is converted from RGBor CMYK spaces into LAB space then the color map is cre-ated where a small set of colors is chosen from all possi-ble combinations in Lab color space. Finally, the proposedPSO algorithm with the selected fitness function is appliedto explore optimal or near-optimal solutions among com-plex and huge searching spaces. Su et al. (2013) propose aDE-based color image quantization algorithm. In the per-formance of the DE-CIQ algorithm, a population includingNP candidate colormaps are randomly initialized in the colorspace [0,255]3. Then, the population is updated by the muta-tion, crossover and selection operations in DE. During theselection operation, some better colormaps are determinedby the values of the fitness function. The mutation, crossoverand selection operations are repeated until a specified maxi-mal number of iterations tmax. The optimal solution obtainedby DE is the optimal colormap.

3 Subjects and methods

3.1 Artificial fish swarm algorithm

Artificial fish swarm algorithm (AFSA) was presented by Liet al. (2002) and improved in Yazdani et al. (2010) and Heet al. (2009). This algorithm is a technique based on swarmbehaviors that were inspired from social behaviors of fishswarm in nature. AFSA works based on population, randomsearch, and behaviorism. This algorithm has been used inoptimization applications, such as clustering (Xiao 2010),machine learning (Yazdani et al. 2010), PID control (Luo etal. 2007), data mining (Zhang et al. 2006), and image seg-mentation (Li et al. 2010). The fish swims towards locationswhere food concentration is highest. As a typical applicationof behaviorism in artificial intelligence, AFSA can search forthe global optimum.

Fig. 1 Vision concept of the artificial fish

Artificial fish (AF) is a fictitious entity of true fish, which isused to carry on the analysis and explanation of problem. TheAF realizes external perception by its vision shown in Fig. 1.X is the current state of an AF, Visual is the visual distance,and Xv is the visual position at some moment. If the state atthe visual position is better than the current state, it goes for-ward a step in this direction and arrives the Xnext (X i(t + 1))state; otherwise, continues an inspecting tour in the vision.The AF model includes two parts (variables and functions).The variables include: X (Xi (t)) is the current position ofthe AF, step is the moving step length, visual represents thevisual distance, try_number is the try number and δ is thecrowd factor (0 < δ < 1). The functions include the behav-iors of the AF: AF_Prey, AF_Swarm, and AF_Follow. In eachstep of optimization process, AF looks for locations with bet-ter fitness values in problem search space by performing theprevious three behaviors based on algorithm procedure. Thebehaviors of artificial fish are assumed as follows:

– Prey behavior:This behavior is an individual behavior that each AF per-forms independently and performs a local search arounditself. Every AF by performing this behavior attemptstry-number times to move to a new position with betterfood concentration. Here, it is supposed that AFi is inposition Xi , then within the visual area S, it randomlychooses another position X j . AFi will swim towards X j

if Y j > Yi where Y represents the food concentration.

X j = Xi + visual × rand(−1, 1) (1)

if Y j > Yi where Y represents the food concentration,position of AFi is updated to the next step by Eq. (2).

Xi (t+1)= Xi (t)+ X j −Xi (t)

||X j −Xi (t)|| × step × Rand(0, 1).

(2)

The above two steps are repeated try-number times. IfAFi could not move toward better positions, it moves

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with a random step in its visual by means of Eq. (3):

Xi (t + 1) = Xi (t) + visual × rand(0, 1) (3)

– Swarm behavior:Fish usually assembles in groups to capture coloniesand/or to avoid dangers. There are n f neighbors withinthe visual area S. Xc is the center of those neighbors. AFi

will swim a random distance towards Xc. If (n f/n) < δ

(where n is the number of artificial fishes) and Yc > Yi

which means that the companion center has more food(higher concentration value) and is not very crowded,it goes forward a step to the companion center (Eq. 4).Otherwise, AFi will resume the behavior of Prey.

Xi (t + 1) = Xi (t) + Xc − Xi (t)

||Xc − Xi (t)|| × step

× Rand(0, 1) (4)

– Follow behavior:When some fishes find food through the moving processof swarm, their neighborhood partners tend to followthem to the best food concentration location. Let Xi bethe AF current state, and it explores the companion X j inthe neighborhood (||X j − Xi (t)|| < Visual), which hasthe greatest Y j . If Y j > Yi and (n f/n) < δ, which meansthat the companion X j state has higher food concentra-tion (higher fitness function value) and the surroundingis not very crowded, it goes forward a step to the com-panion X j . Otherwise, AFi will resume the behavior ofPrey.

Xi (t + 1) = Xi (t) + X j − Xi (t)

||X j − Xi (t)|| × step

× Rand(0, 1) (5)

The collective behaviors of Prey, Follow and Swarm of allfishes are simulated in each iteration. The fishes will choosethe behavior that has the best position (concentration). AFSAis independent on the initial condition. A termination crite-rion can be added for each specific problem. In basic AFSAmodels, the iterations terminate when either

1. The estimated standard deviation in two successive iter-ations is less than user-set delta, or

2. The maximum number of iterations has been reached,whichever occurs first.

3.2 Fuzzy C-means algorithm

Fast and robust clustering algorithms play an important rolein extracting useful information in large databases. The aimof cluster analysis is to partition a set of N object into Cclusters such that objects within cluster should be similar to

each other and objects in different clusters should be dis-similar with each other (Yang 1993). One of the most widelyused algorithms is fuzzy clustering algorithms which dependon Fuzzy set theory (Zadeh 1973, 2005). Fuzzy clusteringhas been widely studied and applied in a variety of sub-stantive areas (Yang et al. 2004; Kim et al. 2004). In thefuzzy clustering literature, the FCM clustering algorithm pro-posed by Dunn (1974) and extended by Bezdek (1987) isthe most used and discussed (Gath and Geva 1989; Yangand Ko 1997). It is widely used directly or indirectly inimage processing. It performs classification based on theiterative minimization of the following objective functionand constraints (Bezdek 1981; Ali et al. 2006; Ahmed et al.2002).

FCM is a method of clustering which allows one pieceof data to belong to two or more clusters. This method isfrequently used in pattern recognition. It is based on min-imization of the objective function. In the FCM algorithm,each data point belongs to a cluster with a degree specified bya membership grade between 0 and 1. Thus, the FCM algo-rithm partitions n vectors into c-fuzzy groups. The objectivefunction is defined as:

Jm =N∑

i=1

c∑

j=1

(ui j )m ||xi − c j ||2 (6)

where ui j is the degree of membership of xi in the clusterj . m is the fuzziness factor (real number greater than 1),xi is the i th of d-dimensional measured data. c j is the d-dimensional center of the cluster, it denotes the set of quan-tization colors, and || ∗ || is any norm expressing the simi-larity between any measured data and the center. Fuzzy par-titioning is carried out through an iterative optimization ofthe objective function shown above to produce locally opti-mal codebooks depending on initial code vector locations,with the update of membership ui j and the cluster centers c j

by:

ui j = 1c∑

k=1

( ||xi −c j ||||xi −ck ||

) 2m−1

c j =

N∑i=1

umi j .xi

N∑i−1

umi j

(7)

This iteration will stop when maxi j |u(k+1)i j −u(k)

i j | < ε, whereε is a termination criterion between 0 and 1, whereas k is theiteration step. This procedure converges to a local minimumor a saddle point of Jm . The algorithm is composed of thefollowing steps:

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4 The optimized Fuzzy C-means technique

In this section, a stable and accurate color quantization algo-rithm framework based on an efficient extension of FCMclustering (Ozdemir and Akarun 1999; Yu and Yang 2005)and an improved artificial fish swarm optimization (He et al.2009) algorithms is proposed. The color image usually rep-resents each color component with an 8-bit integer, and eachpixel requires 24 bits to completely and accurately specifyits color. Of the many color spaces in existence RGB, CMY,YIQ, YUV, CIE Lab, HSV, we employ the commonly usedRGB color space. OFCM is an image-dependent quantiza-tion technique in which every codebook is built based oncolor distribution in a specific image. Thereafter, to use itfor decoding the image, the codebook with encoded imageis transferred. Dependent image techniques are slower thanindependent image techniques, but obtained results from theformer have higher quality than the latter. The process ofcolor image quantization with OFCM as shown in Fig. 2 canbe summarized in the following steps:

1. A dataset has to be determined that clustering has to bedone on it. In this problem, dataset consists of all pixels’values. Determine the clusters number c1, c2, . . ., cM toform M clusters that contain all the image samples.

2. Randomly select Ki , i = 1, 2, . . ., M pixels in RGB colorspace as the initial clusters’ centroids.

3. For each pixel in the image, assign pixel to the groupthat has the closest centroid based on the predetermineddistance measure.

4. When all pixels have been assigned, recalculate the posi-tions of the current clustering Ki , i = 1, 2, . . ., M cen-troids.

5. Repeat steps (3) and (4) until the centroids no longermove, then go to step (6).

6. Repeat steps (2) to (5) until M clusters are completed.7. Apply the improved AFSA behaviors.

Input images dataset

Choose color space

FCM Clustering 1

FCM Clustering 2

FCM Clustering M

Improved AFSA

Codebook construction

………………..

Error Diffusion Dithering

Image quantization Using the codebook

Fig. 2 Framework of the proposed algorithm

8. Choose the behavior set that gives the minimum fitness;the optimized CHs set.

9. Construct Codebook with the selected clusters’ heads asquantization levels, and then perform image quantizationby mapping the original colors in the image to their near-est neighbors (CHs) in the codebook.

10. Improve the visual quality using Error Diffusion quality.

The following subsections describe the proposed OFCMalgorithm in detail.

4.1 Applying the inter-cluster separation FCM algorithm tothe input image

Ozdemir and Akarun (1999) proposed an efficient extensionof FCM algorithm. In this approach, an objective functionincluding an inter-cluster separation (ICS) term is minimized.The goal here is to move cluster centers apart from each othertowards the convex hull of the color space, hence obtaininga color palette which is more suitable for dithering, an oper-ation generally applied after the quantization of the images.The separation (si ) of a fuzzy cluster (i) is defined as the sumof the distances from its cluster center (yi ) to the center ofthe other (c,1) clusters.

si =c∑

t=1

||yi − yt ||2 (8)

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The idea is to obtain a wider color space by displacing thepairs of quantization centers opposite to each other, thereforecreating the illusion of more colors. For the same purpose,using si to give displacements to all quantization centerstowards the convex hull of the image color space yields alarger volume of colors in the convex hull of the quantiza-tion colors. We incorporate si into the objective function Jto minimize the fuzzy Euclidean distance and maximize theinter-cluster separation. We define the objective function:

J (u, y) = 1

n

n∑

j=1

c∑

i=1

(ui j )md(x j , yi ) − γ

c

c∑

t=1

d(yi , yt ). (9)

Again, we minimize J (u, y) using the Lagrange multipliersmethod. Because si is independent of u, the membershipfunction is the same as in the FCM. The update function yi

is obtained as follows:

∂ Ji

∂yi= −2

n

n∑

j=1

(ui j )m(x j −yi )− 2γ

c

c∑

t=1

(yi −yt )=0 (10)

1

n

n∑

j=1

(ui j )m x j + 1

n

n∑

j=1

(ui j )m yi − γ yi + γ

c

c∑

t=1

yt =0 (11)

yi =1n

n∑j=1

(ui j )m x j − γ

c

c∑t=1

yt

1n

n∑j=1

(ui j )m − γ

(12)

4.2 Applying improved AFSA with adaptive visual andadaptive step to the FCM-clustered data

After clustering the data, the improved AFSA (He et al. 2009)is applied. Every AF consists of K initial cluster centerswhich displace these cluster centers in the problem spaceby means of improved AFSA behaviors and their goal is todetermine cluster centers in a way that minimizes Eq. (16) asa fitness function. Improved AFSA accelerates convergencespeed enhances global stability and increases the precisionof the optimization process as follows:

– Adaptive visual:Adaptive Visual may increase convergence in laterrounds of iterations and thus enhances the precision.Some adaptive Visual methods are shown below:

(a) visualt+1 = α ∗ visualtwhere

α =(

visualNvisual1

) 1N

(13)

where t denotes the current iteration, N is total iteration,visual1 is the initial visual value, and visualN is the pre-defined value of visual at the last iteration.

(b) visualt+1 =(

β − t

N

)∗ visualfixed (14)

where β is a constant and it depends on the precision ofthe result. It is suggested β ∈ [1.1,1.5].Fixed visual is used in early iterations since it has a fasterconvergence in early iterations this increases the globalstability, but adaptive visual gives better precision andincreases the convergence speed in later iterations.

– Adaptive step: the standard AFSA can increase conver-gence speed by increasing step. However, if the stepincrease is beyond certain ranges, convergence speedcould be reduced. When Visual is smaller than fixed step,AF might go beyond the target. Improved AFSA proposesa method using adaptive step to overcome this problem:

Xi(t+1) = Xi(t) + step ∗ (X j (t) − Xi(t)) (15)

where t represents the current iteration. The current posi-tion of AFi is at Xi , and the target AF j is at X j . step isa positive constant (step ∈ (0,1)). This adaptive step canprevent unexpected vibrations.

4.3 Evaluating the AFSA behaviors

After performing the three behaviors on the clustered data,one of the most known clustering criteria called sum of intra-cluster distances is used to choose the optimized CHs set. Eq.(16) is a function which calculates the sum of intra-clusterdistances that according to it, the best clustering is the onewhen this function’s value is minimum. Eq. (16) shows thesum of differences between original image’s pixel color num-bers and decoded image’s.

J (C1, C2, . . . , CK ) =K∑

i=1

∑

X j ∈Ci

||zi − X j || (16)

This function is used as a fitness function for the improvedAFSA algorithm and is considered as a minimizing problem.In Eq. (16), the Euclidean distance between each data vectorin a cluster and the centroid of that cluster is calculated andsummed up. Here, we have K clusters Ci (1 ≤ i ≤ K ) thateach of N data vectors X j (1 ≤ j ≤ N ) are clustered onthe basis of distance from each of these cluster centers Zi

(1 ≤ i ≤ K ). Data vectors belong to a cluster that theirEuclidean distance from its cluster center is less than theirEuclidean distance from other cluster centers. Therefore, thegoal of the improved AFSA is to determine cluster centerswhich minimize Eq. (16), and consequently optimal clustercenters are determined.

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Image quantization using improved artificial fish swarm algorithm

4.4 Codebook construction

Codebook is determined using the final result of cluster-ing. Indeed, the codebook contains cluster centers and theirindices. Each cluster center is a one-dimensional vector thatdefines a quantization level. Hence, after completing thecodebook, every pixel of the original image is transformedinto a cluster center index in the codebook that it belongs to.To represent the image again (decoding), each pixel takes itscorresponding cluster center color values with respect to itsencoded value.

4.5 Applying dithering

Quantization of the images causes some visual artifacts suchas false edges and color streaks. For this reason, quantizationis generally followed by a dithering operation. The goal isto hide these defects and to achieve a more faithful repro-duction of colors using the averaging property of the humaneye. In a dithering technique called error diffusion (Marcu2000), this is achieved by spreading the quantization errorto neighboring pixels. Error Diffusion has the tendency toenhance edges in an image. This can make text in imagesmore readable than in other halftoning techniques.

Two-dimensional error diffusion algorithm reduces thevisual artifacts. The simplest algorithm scans the image onerow at a time and one pixel at a time. The current pixel iscompared to a half-gray value. If it is above the value, awhite pixel is generated in the resulting image. If the pixel isbelow the halfway brightness, a black pixel is generated. The

generated pixel is either fully bright or full black, so there isan error in the image. The error is added to the next pixel,and one quarter of the error is added to the pixel on the nextline below, and one quarter of the error is added to the pixelon the next line below and one pixel forward. The kernel is:

1

4

[# 21 1

]

where “#” denotes the pixel currently being processed. Fur-ther refinement can be had by dispersing the error furtheraway from the current pixel, as in the matrix given abovein Enter the digital era.

For Color error diffusion The same algorithm is appliedto each of the red, green, and blue (or cyan, magenta, yellow,black) channels of a color image to achieve a color effect onprinters such as color laser printers that can only print singlecolor values.

5 Results and discussions

The algorithm discussed above is implemented using MAT-LAB 7.0 on HP Core2Duo laptop, 2.5 GHZ, and 3 GB RAM.To evaluate its performance, it is compared with six otheralgorithms. To test the performance of these algorithms,eight color images belonging to different classes of size256 × 256 × 3 are used as shown in Fig. 3. The used imagesbelong to class portrait, collection of objects, Bird, Fruit,Flower, Monument, Place, Scenery etc.

The parameters for the OFCM algorithm are as follows:visual = 2.5, step = 0.1, maxgen = 200, trynumber = 50, δ =

Aishwariya Balls Bird Boat

Flower Ganesh Scenary Strawberry

Fig. 3 Eight training images

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S. A. El-said

0.6, stop_error = 10−4, and Fuzziness parameter (m) is setto 1.3.

5.1 Performance evaluation metrics

The most important measurement criteria for CQ algo-rithms efficiency include mean-squared error (MSE) andpeak signal-to-noise ratio (PSNR) (Celebi 2011). MSE isusually used for assessing distortion between the originalimage and resulted image from CQ and lesser value of itshows better efficiency of CQ algorithm. For the gray-scaleimages, MSE is computed by Eq. (17). For RGB images, Eq.(17) is extended to include the three components (R, G, B)as in Eq. (18). PSNR is a standard way for evaluating fidelitybetween the original image and the obtained image from CQ.PSNR is calculated by Eq. (19).

MSE = 1

n × m

m∑

j=1

n∑

i=1

[x(i, j) − y(i, j)]2 (17)

MSE = 1

n × m × p

p∑

k=1

m∑

j=1

n∑

i=1

[x(i, j, k) − y(i, j, k)]2.

(18)

where x , y are the original image and the quantized image,respectively. p is the number of the image components (p =3 for RGB), and n × m is the size of each component.

PSNR = 10 log10

(x2

o

MSE

)(19)

where, xo is the largest amount which a pixel can take. PSNRis measured in decibels (dB) and the larger value of it showsbetter efficiency of CQ method.

Due to their evident physical meanings and mathematicalconvenience, MSE and PSNR are widely used. But they arenot very well matched to human judgment of image qual-ity. As a more promising new paradigm of images’ qualitymeasurements, a universal image quality index was proposedin Wang and Bovik (2002). The average quality index UQIcoincides with the mean subjective ranks of observers. Thatgives to researchers a very powerful tool for images’ qualityestimation.

Q = 4σxy x y

(σ 2x + σ 2

y )[(x)2 + (y)2] (20)

where

x = 1

n × m × p

p∑

k=1

m∑

j=1

n∑

i=1

x(i, j, k),

y = 1

n × m × p

p∑

k=1

m∑

j=1

n∑

i=1

y(i, j, k)

σ 2x = 1

n × m × p − 1

p∑

k=1

m∑

j=1

n∑

i=1

[x(i, j, k) − x]2,

σ 2y = 1

n × m × p − 1

p∑

k=1

m∑

j=1

n∑

i=1

[y(i, j, k) − y]2

σxy = 1

n × m × p − 1

p∑

k=1

m∑

j=1

n∑

i=1

(x(i, j, k) − x)

×(y(i, j, k) − y)

where x , y, σx , σy, σxy are the mean value of x , the meanvalue of y, the variance of x , the variance of y, and thecovariance of x and y, respectively. UQI quality measure-ment method is applied to local regions using sliding windowapproach. For overall quality index to be obtained, averagevalue of local quality indexes Qi must be calculated:

Q = 1

M

M∑

i=1

Qi (21)

where M is the number of local windows in the image.The Structural Similarity Index (SSIM) that is proposed in

Wang et al. (2004) is a generalized form of a Universal Qual-ity Index Wang and Bovik (2002). As above, x and y arediscrete non-negative signals. As shown in Fig. 4, the lumi-nance (l), contrast (c), and structure comparison (s) measuresare used to calculate the SSIM as follows:

SSIM(x, y) = [l(x, y)]α.[c(x, y)]β.[s(x, y)]γ (22)

where

l(x, y) = 2x y + C1

x2 + y2 + C1

c(x, y) = 2σxσ y + C2

σ 2x + σ 2

y + C2

s(x, y) = σxy + C3

σxσy + C3

where α, β, and γ are parameters to define the relative impor-tance of the three components. If α = β = γ = 1, theresulting SSIM index is given by:

SSIM = (2x y + C1)(2σxy + C2)

(x2 + y2 + C1)(σx + σy + C2)(23)

Constants C1 and C2 are used in Eq. (23) to avoid unstabilitywhen denominators (x2 + y2)and(σx + σy) approach tozero. Where C1 = (K1xo)

2, C2 = (K2xo)2. K1 and K2 are

very small positive constants such that K << 1. In prac-tice, one usually requires a single overall quality measure ofthe entire image. We use a mean SSIM (MSSIM) index toevaluate the overall image quality:

MSSIM(x, y) = 1

M

M∑

j

SSIM(x j , y j ) (24)

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Image quantization using improved artificial fish swarm algorithm

Fig. 4 Structural Similarity Index

5.2 Experimental results

Figure 5 shows Flower and Balls images whose colors havebeen decreased and have been compacted by rate 2.67 usingthe proposed OFCM technique. As it is observed, OFCM hasachieved better results in all cases. It generates a codebook bydecreasing the sum of intra-cluster distances which decreasesdistortion in decoded image. Therefore, the obtained imagesfrom the proposed algorithm would have more fidelity withthe original image. On the whole, experimental results showthat compressed images by means of generated codebookby the proposed algorithm are of higher quality than otheralgorithms.

Table 1 shows the performance comparison among HT-HVS (Veeraswamy et al. 2007), FCM (Yang and Zhang2010), ICS (Yu and Yang 2005), and OFCM quantizationtechniques on four color images from different categories.According to results of Table 1, generally, standard FCMhas less efficiency than ICS. But OFCM has achieved betterefficiency than both FCM and ICS by optimizing the perfor-mance of the ICS using improved AFSA.

Figures 6 and 7 show the average MSE of HT-HVS, FCM,ICS, and OFCM algorithms on the eight color images fromdifferent categories of size 256×256×3 for codebook sizes1,024, 512, 256 and 128. Comparing the curves we can seethat under the same conditions, the proposed algorithm has

(a) Flower (b) MSE: 236.97 (c) MSE: 167.20 (d) MSE: 70.86

(a) Balls (b) MSE: 1801.6 (c) MSE: 507.33 (d) MSE: 465.95

Fig. 5 Results of FCM, ICS and OFCM algorithms from codebook size 1,024 on Flower and Tajmahal image; a Original Image, Reconstructedimages after compressing by b FCM, c ICS, and d OFCM

123

S. A. El-said

Table 1 Performancecomparison among HT-HVS,FCM, ICS, and OFCM on fourcolor images from differentcategories of size 256 × 256 × 3for codebook sizes 128, 256,512, and 1,024

Image Codebooksize

Criteria HT-HVS(Veeraswamyet al. 2007)

FCM (Yangand Zhang2010)

ICS (Yu andYang 2005)

OFCM

Bird 128 MSE 305.40 295.28 174.12 99.48

PSNR 23.98 26.16 28.72 32.61

256 MSE 302.11 320.79 148.98 85.27

PSNR 24.33 27.40 29.40 34.16

512 MSE 296.48 233.30 116.87 108.71

PSNR 25.41 28.45 31.45 35.80

1,024 MSE 285.04 164.40 90.08 74.31

PSNR 27.58 29.97 33.58 36.64

Boat 128 MSE 620.06 762.48 410.54 308.12

PSNR 20.21 19.31 22.00 25.31

256 MSE 614.73 685.49 344.17 257.75

PSNR 20.24 19.77 22.76 26.25

512 MSE 604.63 526.01 252.76 189.34

PSNR 20.32 20.92 23.80 28.05

1,024 MSE 583.59 417.39 222.50 142.68

PSNR 20.47 21.93 24.66 30.14

Scenary 128 MSE 355.95 453.20 291.29 199.09

PSNR 22.62 21.57 25.31 27.22

256 MSE 352.46 406.39 205.17 152.1

PSNR 22.66 22.04 26.28 29.12

512 MSE 346.55 296.83 119.19 96.54

PSNR 22.73 23.41 27.37 30.23

1,024 MSE 333.81 189.24 90.19 82.02

PSNR 22.90 25.36 28.58 32.43

Strawberry 128 MSE 933.50 393.16 266.80 154.87

PSNR 18.43 22.19 23.87 26.15

256 MSE 925.90 338.06 228.22 115.97

PSNR 18.47 22.84 24.55 27.81

512 MSE 912.77 277.28 186.65 102.66

PSNR 18.53 23.70 25.42 28.04

1,024 MSE 884.99 233.57 152.89 99.05

PSNR 18.66 24.45 26.29 29.22

a significant improvement, especially at the 16-level quanti-zation, the algorithm has a better quantified effect. Its valueof the mean square difference is equally smaller than thatof other techniques. The differences in reconstructed imagesare smaller, and the details are more ideal. From the curvesin Fig. 8, we can see that the average PSNR of the proposedtechnique is higher than that of the other techniques at allcodebook sizes; 1,024, 512, 256 and 128.

Table 2 shows a performance comparison among the pro-posed OFCM algorithm and other Some stochastic optimiza-tion methods (He et al. 2009; Kaur et al. 2011; Su et al. 2013).The four techniques are applied on the eight color test images

(Fig. 3) and the average values of PSNR, MSSIM, and Q arerecorded in Fig. 9. The parameters for both OFCM and He etal. (2009) algorithms are as follows: visual = 2.5, step = 0.1,maxgen = 200, trynumber = 50, δ = 0.6, stop_error = 10−4,and Fuzziness parameter (m) is set to 1.3. The parameters inthe DE-CIQ algorithm are set as the population size NP =100, the scaling factor F = 0.4, the crossover rate CR = 0.7and the maximal number of iteration tmax = 200. The PSO-CIQ algorithm parameters are set as follows: swarm size NP= 100, the inertia weight ω = 0.72, the acceleration constantsc1 = c2 = 1.49, the maximum velocity Vmax = 0.4, and themaximal number of iteration tmax = 200 . These parameters

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Image quantization using improved artificial fish swarm algorithm

100150200250300350400450500550600650

Codebook size

Ave

rag

e M

SE

HVS

FCM

ICS

OFCM

HVS

FCM

ICS

OFCM

650.85 469.72 302.15 259.48611.33 396.81 257.89 210.27520.38 292.12 185.33 118.71370.74 221.22 148.35 89.31

128 256 512 1024

Fig. 6 Average MSE of HT-HVS, FCM, ICS, and OFCM algorithmson the eight color images from different categories of size 256×256×3for codebook sizes 1,024, 512, 256 and 128

are the same as those in Su et al. (2013). It is clear that theperceptual quality of the proposed algorithm is higher thanother techniques.

Summarizing from the above experiments, we can con-clude that the proposed technique is stable and accurate.The OFCM algorithm not only maintains the whole colorlevel of the original image, but also avoids the color lost orstained improperly and thus can solve the contradiction bet-ter between the whole level and the local details in imagereconstruction.

6 Conclusions and future works

Using traditional FCM Clustering algorithm in Image quanti-zation brings randomness in initial clustering centers’ selec-tion and it is sensitive to the noise data and can easily convergeto local extremism points. Thus can not achieve the globaloptimum, as to color quantification it appears sensitive to theinitial clustering centers selecting of color samples, and thelocal color rendering is not correct, especially the images withvery uneven color distribution, its reconstruction effect wasnot ideal enough. One of the applied methods for overcoming

these drawbacks is the use of the swarm intelligence algo-rithms. In this paper, an efficient and stable color image quan-tization algorithm is proposed. It uses an improved AFSAto optimize the behavior of an efficient extension of FCM.The proposed algorithm exploits the search capability of theimproved AFSA to overcome the local optimal problem ofthe ICS–FCM algorithm. It uses error diffusion algorithms toobtain perceptually better images after quantization. To thebest of our knowledge, it is the first time that the improvedAFSA heuristics are applied as a method for solving colorimage quantization problems. Experimental results show thatthe proposed method outperformed both the conventionalFCM and the ICS algorithms using the test images’ dataset.These results also indicate that the proposed method is par-ticularly effective at high-compression ratio.

The future investigation will pay much attention to exploitthe robustness of intuitionistic Fuzzy C-means algorithm(IFCM) which is based upon intuitionistic fuzzy set the-ory Xu and Wu (2010); Chaira (2011); Ananthi et al. (2014)

15

20

25

30

35

40

Codebook size

Ave

rag

e P

SN

R

HVS

FCM

ICS

OFCM

HVS

FCM

ICS

OFCM

22.37 23.41 25.49 26.96

23.45 24.37 26.31 28.54

24.13 26.86 28.86 31.87

27.61 29.16 31.8 34.64

128 256 512 1024

Fig. 8 Average PSNR of HVS, FCM, ICS, and OFCM algorithms onthe eight color images from different categories of size 256 × 256 × 3for codebook sizes 1,024, 512, 256 and 128

Fig. 7 MSE Comparisonsamong FCM, ICS, and OFCMalgorithms on the eight colorimages from different categoriesof size 256 × 256 × 3 forvarying codebook sizes 128,256, 512 and 1,024

123

S. A. El-said

Table 2 Performance comparison between the proposed OFCM algo-rithm and other stochastic optimization methods (He et al. 2009; Kauret al. 2011; Su et al. 2013) at codebook size = 256

Technique Average Q AveragePSNR

AverageMSSIM

OFCM 0.869 34.7 0.947

Su et al. (2013) 0.845 33.1 0.938

Kaur et al. (2011) 0.771 31.15 0.922

He et al. (2009) 0.727 32.47 0.921

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

OFCM [49] [48] [44]

Quantization techniqueaverage Q average MSSIM

Fig. 9 Performance comparison between the proposed algorithm andother optimization metaheuristics

in improving the performance of the proposed OFCM tech-nique.

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