Image Clustering and Compression Using An Annealed Fuzzy

Abstract—This paper presents a new approach to image

compression based on clustering. This new approach includes new

objective function, and its minimization by Lyapunov energy

function based on unsupervised two dimensional an annealed fuzzy

Hopfield neural network. Combination of annealing strategy and

Hopfield neural network gives feasible online learning and hardware

implementation. New objective function consists combination of

classification entropy function and average distance between image

pixels and cluster centers. After applying new method on gray scale

sample images at different number of clusters, better compression

ratio and number of iteration was observed. The new method is also a

new clustering analysis method, and it provides more compact and

separate clustering, when examining cluster validity measure consists

geometrical features.

Keywords—Fuzzy clustering, Image compression, Image

segmentation, Simulated annealing, Validity function.

I. INTRODUCTION

owadays, importance of image compression increases

with advancing communication technology. Limited

hardware and budget is also important in sending of data fast.

The amount of data associated with visual information is so

large that its storage requires enormous storage capacity. The

storage and transmission of such data require large capacity

and bandwidth, which could be very expensive. Image data

compression techniques are concerned with reduction of the

number of bits required to store or transmit images without

any appreciable loss of information. Image transmission

applications are in broadcast television; remote sensing via

satellite, aircraft, radar, or sonar; teleconferencing; computer

communications; and facsimile transmission. Image storage is

required most commonly for educational and business

documents, medical images. Because of their wide range of

applications, data compression is of great importance in digital

image processing [1 - 3].

Clustering is useful in several exploratory pattern-

analysis, grouping, decision-making, and machine-learning

situations, including data mining, document retrieval, image

Manuscript received November 03, 2003.

Metin Kaya is with Türk Demirdöküm Fab. A. Ş., Bozüyük, Bilecik,

Türkiye (corresponding author to provide phone: +90-228-3145500;

email: [email protected]).

segmentation, and pattern classification. In image

segmentation coding techniques, image is segmented to

different regions separated with contours, and coded with

different coding techniques. Region growing, c-means, and

split and merge methods are used generally for image

segmentation. Beside of this crisp classical segmentation

methods, the fuzzy logic segmentation methods were also

seen very effective for coding [4 - 8].

During the past decade, the Hopfield neural network has

been studied extensively with its feature of simple

architecture and potential for parallel implementation [9 – 11].

The Hopfield neural network is a well-known technique used

for solving optimization problems based on Lyapunov energy

function. The application of a competitive Hopfield neural

network for medical image segmentation was described by

Cheng, Lin and Mao [12]. Polygonal approximation using a

competitive Hopfield neural network was demonstrated by

Chung et al. In Refs. 12 and 13, The winner-take-all rule has

been adopted in the two dimensional discrete Hopfield neural

network to eliminate the need for finding weighting factors in

the energy function. Lin, Cheng and Mao proposed the

segmentation of single and multispectral medical images using

a fuzzy Hopfield neural network [14, 15]. An edge detection

algorithms based on the Hopfield neural network were

proposed by Chao, Dhawan and Chang, Chung [16, 17].

Endocardial boundary detection using the Hopfield neural

network was described by Tsai et al [18]. Amatur, Piriano and

Takefuji used the two dimensional Hopfield neural network for

segmentation of multispectral MR images [19]. Fuzzy

possibilistic neural network to vector quantizer in frequency

domains was proposed by Lin [20]. Fuzzy Hopfield neural

network with fixed weight for medical image segmentation was

proposed by Chang and Ching [21]. Robust segmentation of

medical images using competitive Hopfield neural network as a

clustering tool was proposed by Roozbahani, Ghassemian and

Sharafat [22]. Medical image segmentation using a contextual-

constraint-based Hopfield neural cube was proposed by

Chang and Chung [23]. A new image clustering and

compression method based on fuzzy Hopfield neural network

was proposed by Kaya [24].

The simulated annealing strategy was first proposed in 1953

by Metropolis et al [25] to simulate molecular processes.

Kirkpatrick, Gelatt, and Vecchi [26] used the idea as a method

to resolve minimizing functions of many variables, such as NP-

Image Clustering and Compression Using An

Annealed Fuzzy Hopfield Neural Network

Metin Kaya

N

INTERNATIONAL JOURNAL OF SIGNAL PROCESSING VOLUME 1 NUMBER 1 2004 ISSN:1304-4478

IJSP VOLUME 1 NUMBER 1 2004 ISSN:1304-4478 80 COPYRIGHT © 2004 ENFORMATIKA

hard problems. Simulated annealing derives its name from an

analogy between its behavior and that of a physical process

of thermodynamics and metallurgy in which a metal is first

melted at a very high temperature and then slowly cooled until

it solidifies in a structure of minimum energy. At the

beginning, the temperature T, used to control the

probability of accepting a worsening perturbation over

time, is set to a very high value; later it is multiplied by a factor

Trate ( 0 < Trate < 1 ), called the annealing factor or cooling rate,

after every iteration [11].

Simulated annealing is a stochastic relaxation algorithm used

successfully to resolve optimization problems including

computer network topology [27], circuit routing [26], image

processing [28, 29] and clustering problems [30]. Unlike other

optimization methods such as the steepest descent approach

used in the Hopfield neural network, the simulated annealing

technique, which allows the search to move away from a local

minimum, seeks the global or near global minimum of an energy

function without getting trapped in a local minimum. The

simulated annealing technique has nonzero probability to go

from one state to another and moves temporarily toward a

worse state so as to escape from local traps. The probability

function depends on the temperature and the energy difference

between the two states. With the probabilistic hill-climbing

search approach, the simulated annealing technique has a

promising probability to go to a higher energy state at a higher

temperature [11].

In this study, a new image clustering and compression

method based on an annealed fuzzy Hopfield neural network

(ICC-AFHNN) was introduced for gray scale images. This new

approach includes new objective function, and its minimization

by Lyapunov energy function based on unsupervised two

dimensional fuzzy Hopfield neural network incorporates the

characteristics of the annealing strategy. After applying

new method on gray scale sample images at different number

of clusters, better compression ratio, validity value and number

of iteration was observed.

The rest of this paper is organized as follows. Section 2

reviews the fuzzy clustering; Section 3 reviews validity

measure; Section 4 proposes a clustering algorithm using

annealed fuzzy Hopfield neural network; Section 5 presents

experimental results; and finally, Section 6 gives the

conclusions.

II. FUZZY CLUSTERING

Clustering analysis is based on partitioning a collection of

data points into a number of subgroups, where the objects

inside a cluster (a subgroup) show a certain degree of

closeness or similarity. It has been playing an important role in

solving many problems in pattern recognition and image

processing. Clustering methods can be considered as either

hard (crisp) or fuzzy depending on whether a pattern data

belongs exclusively to a single cluster or to several clusters

with different degrees. In hard clustering, a membership value

of zero or one is assigned to each pattern data (feature vector),

whereas in fuzzy clustering, a value between zero and one is

assigned to each pattern by a membership function. In general,

fuzzy clustering methods can be considered to be superior to

that of its hard counterparts since they can represent the

relationship between the input pattern data and clusters more

naturally. Clustering algorithms such as hard c-means (HCM)

and fuzzy c-means (FCM) are based on the “sum of intracluster

distances” criterion. The fuzzy c-means clustering algorithm

was first introduced by Dunn, and the related formulation and

algorithm was extended by Bezdek [31 – 36].

The fuzzy c-means algorithm is based on minimization of the

following objective function, with respect to m, a fuzzy c-

partition of the data set, and to v, a set of c prototypes:

Where mx , i (x=1,2,...,N, i=1,2, ...,c) is membership value, it

denotes fuzzy membership of data point x belonging to class

i, vi (i=1, 2, ..., c) is centroid of each cluster and zx (x=1,2,..., N) is

data set (pixel values in image), m is fuzzification parameter,

d2(zx, vi) is Euclidean distance between zx and vi , N is the

number of data points, c is number of clusters.

Fuzzy partition is carried out through an iterative

optimization of equation (1) according to [5]:

1) Choose primary centroids vi (prototypes).

2) Compute the degree of membership of all data set in all the

clusters:

3) Compute new centroids vıi :

and update the degree of memberships, mx,i to mıx,i according to

equation (2).

4)

if max x,i [ | mx,i - mıx,i | ] < e stop, otherwise go to step 3 (4)

Where e is a termination criterion between 0 and 1, and

d2(zx, vi) is given by

III. VALIDITY MEASURE

Quality of clustering is important together with increasing

of importance of clustering. So validity criterion was created,

(1))v,(zd2

1J ix

2N

1x

c

1i

m

ix,FCM µåå= =

=

(2)1)1/(m

c

1i ix

2

2

1)1/(m

ix,

)v,(z

1

ix

d

d )v,(z

1

µ -

=

-

å

÷÷ø

öççè

æ

÷÷

ø

ö

çç

è

æ=

)3(

µ

µ

N

1x

m

ix,

N

1xx

m

ix,

i

zv

å

å

=

==ı

(5)2

ixix

2 v-z)v,(zd =



and based on a validity function which identifies overall

compact and separated clustering. Several validity functions

such as partition coefficient (PC), classification entropy (CE),

partition exponent (PE), csc (compact and separate clustering)

index (S) and so on, have been used for measuring validity

mathematically [37 – 40]. PC and CE have slightly larger

domains than PE, and in this sense are more general. But PC,

CE and PE validity measures are the lack of direct connection

to geometrical property. S validity function also includes

geometrical properties [34, 35] and it is proportion of

compactness to separation. A smaller S indicates a partition in

which all the clusters are overall compact and separate to each

other. S is given as

The compactness of fuzzy cluster ci is computed as

The variation of fuzzy cluster i is defined as

dx,i is called the fuzzy deviation of zx from class i.

s is separation of the fuzzy c-partition, where dmin is minimum

distance between cluster centroids.

The compactness and separation validity function S is

defined as the ratio of compactness to separation, and partition

index is obtained by summing up this ratio over all clusters

Also another important validity functions are classification

entropy (CE), and partition coefficient (PC). Minimum value of

classification entropy and closing value of PC to one is shows

better data classified. Classification entropy (CE) function is

given by

Partition coefficient (PC) is given by

IV. A CLUSTERING ALGORITHM USING ANNEALED

FUZZY HOPFIELD NEURAL NETWORK

In this section, the new image clustering and compression

method based on fuzzy Hopfield neural network was

introduced. The Hopfield neural network is a well-known

technique used for solving optimization problems based on the

Lyapunov energy function. In this method, two dimensional

Hopfield neural network consists of Nxc neurons which are

fully interconnected neurons. The total weighed input for

neuron (x,i) is given [15] as

Where N is the number of data points, c is number of

clusters, Vy, j denotes the binary state of neuron (y,j), Wx,i;y,j is

interconnection weight between neuron (x,i) and neuron (y,j),

Ix,i is external bias vector for neuron (x,i). Lyapunov energy

function of two dimensional Hopfield neural network is also

given [15] as

The neural network reaches a stable state, when the

Lyapunov energy function is minimized. Using the within-

class scatter matrix criteria, the optimization problem can be

mapped into a two dimensional fully interconnected Hopfield

neural network with the fuzzy reasoning strategy. Instead of

using the competitive learning strategy, the fuzzy Hopfield

neural network use the fuzzy reasoning algorithm to eliminate

the need for finding weighting factors in the energy function.

The total weighed input for neuron (x,i) and Lyapunov energy

can be modified [41] as

Where is the total weighed input

received from the neuron (y,i), m is the fuzzification parameter

and membership value mx,i is the output state at neuron (x,i), zx

is x. pixel value of image. A neuron (x,i) in a maximum

membership state indicates that zx pixel belongs to class i.

Each column of this modified Hopfield neural network

represents a cluster centroids, and each row represents an

image pixel in a proper class. The neural network reaches a

stable state when the modified Lyapunov energy function is

minimized.

In order to generate an adequate classification with the

constraints, we define Lyapunov energy function as follows:

(6)s

πS =

å === (7)c1,...,i,σσ,N

σπ i

(8)N1,...,x,)(dσ 2

ix,i ==å

(9)xiix,ix, zvµd -=

( ) (10)2

minds =

(11)titi,min vvmind -=

( )(12)

d

N

σ

s

πS

2

min

÷ø

öçè

æ

==

(13)||vv||minN

||zv||µ

S2

titi,

c

1i

N

1x

2

xi

2

ix,

-

-=åå= =

(14)N

lnµµ

c),CE(µ

ix,

N

1x

c

1i

ix,åå= =-=

(15)N

µ

c),PC(µ

N

1x

c

1i

2

ix,åå= ==

(16)ix,jy,

N

1y

c

1j

jy,i;x,ix, IVWNet += åå= =

)17(ix,

c

1i

ix,

N

1x

N

1x

N

1y

c

1i

c

1j

jy,jy,i;x,ix, VIVWV2

1E åååååå

=== = = =

--=

(18)ix,

2

m

iy,

N

1y

iy,i;x,xix, IµWzNet +-= å=

(19)µIµWzµ2

1E

N

1x

c

1i

N

1x

m

ix,

c

1i

ix,

2

m

iy,

N

1y

iy,i;x,x

m

ix,å å ååå= = = ==

--=

m

iy,

N

1y

iy,i;x, µWå=

2

jx,

c

1i

ix,

c

1j

N

1x

2

m

iy,

N

1y

yN

1h

m

ih,

x

c

1i

m

ix,

N

1x

µµ2

Bµz

µ

1zµ

2

AE ú

û

ùêë

é+-= å ååå

ååå

= ===

=

==

)20(Nµ2

C2

c

1i

ix,

N

1x

úû

ùêë

é-÷ø

öçè

æ+ åå

==



Where E is the total intra-class scatter energy that accounts

for the scattered energies distributed by all pixels in same

class.

The first term in energy function (20) is the within-class

scatter energy that is average distortion between image pixels

to the cluster centroids over c clusters. The second term

attempts to ensure that any image pixel zx doesn’t show up on

the final solution in two classes i and j. While the third term

guarantees those number of data point N in image can only be

distributed among these c classes. More specifically, the last

two terms, which are the penalty terms, impose constraints on

the energy function and the first term minimizes the within-

class Euclidean distance from cluster centroids to image pixels

in any given cluster. These terms are combined into a weight

sum using three coefficients determined by the designer. As

mentioned in Refs. 42 and 43, the quality of classification result

is very sensitive to the weighting factors and good values for

them are difficult to find when even a moderate number of

training samples are considered. Searching for optimal values

for these weighting factors is expected to be time consuming

and laborious. In Ref. 43, Van Den Bout and Miller indicated

that good values for penalty terms can easily determine using a

trial and zero approach or analytical techniques in a TSP

problem on the order of 10 cities. Unfortunately, these terms do

not scale even as the problem grows modestly to 30 cities.

Therefore, the problem of finding feasible cluster centroids

from N data point has been replaced with the problem of

finding the best value of A, B, and C. A new energy function

is developed that doesn’t require any weighting factors. Based

on Ref. 43, each state mx,i is looked upon as the probability of

finding image pixel zx closing to class i undergo random thermal

perturbations. The probability of the image pixel zx occupied by

class i at a given temperature T conforms to a Boltzmann

distribution

Because each image pixel can only be occupied by one

class, every row can have at most 1. In other words, the

summation of states in the same row equals 1. This also

ensures that only N data points will be classified into these c

clusters [11].

That is, the network must match the following constraints

and

Therefore, the energy function can be further simplified as

Then the mean field Ex,i can be calculated from Eq. (24) to be

The probability that image pixel zx occupied by class i can

then be normalized as follows [11]:

The normalization operation in Eq. (26) guarantees that each

image pixel will be absorbed on several classes with certain

probability degrees so there will be N data points assigned

among c clusters. By using Eg. (24) , which is modified from Eq.

(20) , the minimization of energy E is greatly simplified because

it contains only one term and hence the requirement of having

to determine the weighting factors A, B, and C vanishes.

Comparing Eq. (24) with the modified energy function Eq. (19) ,

the synaptic interconnection weights and the bias input can be

obtained as

and

By introducing equations (27) and (28) into (18), the input to

neuron (x,i) can be expressed as

Consequently, membership function for x-th pixel in clusters

is given as

This membership function is effective to minimize new

objective function in iteration. New objective function consists

equal weighted combination of classification entropy function

and average distance between image pixels and cluster

centroids for separate and compact clustering. New objective

function is given as

Minimization of classification entropy is effective to take

separate and compact clustering and minimization of

classification entropy shows minimum uncertainty.

(21)T/E∆

ix,ix,eαµ

-

(22)1µc

1i

ix,å=

=

(23)NµN

1x

c

1i

ix,å å= =

=

(24)

2

m

iy,

N

1y

yN

1h

m

ih,

x

c

1i

m

ix,

N

1x

µz

µ

1zµ

2

1E å

ååå

=

=

==

-=

(25)

2

m

iy,

N

1y

yN

1h

m

ih,

xix, µz

µ

1zE å

å=

=

-=

(26)c

1j

T/E

T/E

ix,

jx,

ix,

e

eµ

å=

-

-

=

)27(N

1h

m

ih,

y

i,y;i,

µ

zW

å=

=x

(28)0I ix, =

(29)

2

m

iy,

N

1y

yN

1h

m

ih,

xix, µz

µ

1zNet å

å=

=

-=

(30)i.allfor

1

c

1j

11/m

jx,

ix,

ix,Net

Netµ

-

=

-

úú

û

ù

êê

ë

é

÷÷ø

öççè

æ= å

(31)N

1x

ix,

c

1i

ix,ix

N

1x

c

1i

m

ix, lnµµN

1)v,d(zµ

N

1J åååå

= == =

-=



Classification entropy is given as

Iterative minimization of new objective function (18) consists

of the following steps:

1) Choose number of cluster c, iteration criteria Î,

fuzzification parameter m chosen to be 2, and primary

centroids v0 .

2) Compute initial membership values [5]:

3) Compute new membership values:

4) If mx,i membership values is out of 0 £ mx,i £ 1 range, new

mx,i values are given as

5) Compute new cluster centroids [5]:

6) Compute Jk :

7)

If | Jk+1

- Jk| £ Î stop, otherwise go to step 3 (38)

After obtaining the membership values and the cluster

centroids, segmented image is created and it is coded by run-

length coding.

V. EXPERIMENTAL RESULTS

This new method (ICC-AFHNN) was applied to 128x128

dimensional five sample gray scale images and compared with

results of fuzzy c-means (FCM) and hard c-means (HCM)

algorithms. Comparing parameters are compression ratio, csc

index (S) validity measure and number of iterations.

Comparison results are given at table 1 according to different

number of clusters (c). Original images and segmented images

by ICC-AFHNN, FCM and HCM methods were given in figures

1 – 15 respectively according to different number of clusters.

ICC-AFHNN method provides better image compression

than other methods according to experimental results. It

preserves intelligibility of images together with this high

compression ratio. The images obtained by new method has

less smaller clusters, that generates noise effect, than images

obtained by FCM and HCM methods. There aren’t also any

block effects in segmented images.

The number of iteration of new method is less than other

methods. So it takes a little time to reach to the result. When

examining clustering quality, validity measures (S) of ICC-

AFHNN method are better than other methods.

VI. CONCLUSIONS

Importance of image clustering and compression methods

increases nowadays. New image clustering and compression

method using annealed fuzzy Hopfield neural network provides

better compression ratio, performing time and validity measure.

This method can be used for pattern recognition additionally,

because it provides good validity measure. There isn’t

derivative minimizing algorithm for new objective function, so

there isn’t possibility to reach incorrect results. But some of

methods as FCM and HCM has high possibility to go to a local

minimum according to selection of initial values and may not

give correct results. Combination of annealing strategy and the

fuzzy Hopfield neural network can also provide feasible online

learning, and a more efficient mechanism and a greater

potential to multispectral image segmentation in parallel

processing using the hardware implementation. Because of

these advantages, this new method is a good alternative

method for image clustering and compression.

(32)N

1x

ix,

c

1i

ix, lnµµN

1c),CE(µ åå

= =

-=

(33)c1,2,...,iN;1,2,..., x,

c

1k

1)1/(m

20

kx

1)1/(m

20

ix

ix,

vz

1

v-z

1

µ ==

å=

-

-

÷÷÷

ø

ö

ççç

è

æ

-

÷÷÷

ø

ö

ççç

è

æ

=

(34)i.and x allfor

1

c

1j

11/m

jx,

ix,

ix,Net

Netµ

-

=

-

úú

û

ù

êê

ë

é

÷÷ø

öççè

æ= å

(35)i.andxallfor

ix,ix,

ix,

ix, µmax

µµ =

(36)

µ

µ

N

1x

m

ix,

N

1xx

m

ix,

i

zv

å

å

=

==

(37)N

1x

ix,

c

1i

ix,ix

N

1x

c

1i

m

ix,

k lnµµN

1)v,d(zµ

N

1J åååå

= == =

-=



Segmented image samples “Cameraman”, “Lena”, “Pepper”,

“Brain Tomography”, a nd “Test Image” by ICC-AFHNN are

shown at figure 1–5 according to different number of clusters

(c).

Figure 1. Cameraman image: a) original image; b) segmented image

according to c=4; c) segmented image according to c=5; d) segmented

image according to c=6.

Figure 2. Lena image: a) original image; b) segmented image according to

c=4; c) segmented image according to c=5; d) segmented image according

to c=6.

Figure 3. Pepper image: a) original image; b) segmented image according

to c=4; c) segmented image according to c=5; d) segmented image

according to c=6.

Figure 4. The image of computer aided brain tomography: a) original

image; b) segmented image according to c=4; c) segmented image

according to c=5; d) segmented image according to c=6.

TABLE I

EXPERIMENTAL RESULTS

a b

c d

a b

c d

a b

c d

a b

c d



Figure 5. Test image: a) original image; b) segmented image according to


to c=6.


“Brain Tomography”, and “Test Image” by FCM are shown at

figure 6–10 according to different number of clusters (c), and

fixed termination criterion e (e = 0.1) and fuzzification parameter

m (m=2).




Figure 7. Lena image: a) original image; b) segmented image according to


to c=6.

Figure 8. Pepper image: a) original image; b) segmented image according


according to c=6.




Figure 10. Test image: a) original image; b) segmented image according


according to c=6.

a b

c d

a b

c d

a b

c d

a b

c d

a b

c d

a b

c d




“Brain Tomography”, and “Test Image” by HCM are shown at

figure 11–15 according to different number of clusters (c), and

fixed termination criterion e (e = 0.1).




Figure 12. Lena image: a) original image; b) segmented image according


according to c=6.

Figure 13. Pepper image: a) original image; b) segmented image






Figure 15. Test image: a) original image; b) segmented image according


according to c=6.

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c d

a b

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a b

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a b

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Image Clustering and Compression Using An Annealed Fuzzy

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