image clustering and compression using an annealed fuzzy
TRANSCRIPT
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
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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 憆= =
=
(2)1)1/(m
c
1i ix
2
2
1)1/(m
ix,
)v,(z
1
ix
d
d )v,(z
1
µ -
=
-
å
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öççè
æ
÷÷
ø
ö
çç
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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 =
INTERNATIONAL JOURNAL OF SIGNAL PROCESSING VOLUME 1 NUMBER 1 2004 ISSN:1304-4478
IJSP VOLUME 1 NUMBER 1 2004 ISSN:1304-4478 81 COPYRIGHT © 2004 ENFORMATIKA
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 -=
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minds =
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-
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(15)N
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=== = = =
--=
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(19)µIµWzµ2
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--=
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2
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ix,
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1y
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µ
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2
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= ===
=
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==
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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=
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(23)NµN
1x
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=
(24)
2
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2
1E å
ååå
=
=
==
-=
(25)
2
m
iy,
N
1y
yN
1h
m
ih,
xix, µz
µ
1zE å
å=
=
-=
(26)c
1j
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T/E
ix,
jx,
ix,
e
eµ
å=
-
-
=
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m
ih,
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i,y;i,
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å=
=x
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(29)
2
m
iy,
N
1y
yN
1h
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ih,
xix, µz
µ
1zNet å
å=
=
-=
(30)i.allfor
1
c
1j
11/m
jx,
ix,
ix,Net
Netµ
-
=
-
úú
û
ù
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ë
é
÷÷ø
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æ= å
(31)N
1x
ix,
c
1i
ix,ix
N
1x
c
1i
m
ix, lnµµN
1)v,d(zµ
N
1J åååå
= == =
-=
INTERNATIONAL JOURNAL OF SIGNAL PROCESSING VOLUME 1 NUMBER 1 2004 ISSN:1304-4478
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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
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(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
= == =
-=
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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
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Figure 5. Test 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.
Segmented image samples “Cameraman”, “Lena”, “Pepper”,
“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 6. 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 7. 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 8. 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 9. 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.
Figure 10. Test 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.
a b
c d
a b
c d
a b
c d
a b
c d
a b
c d
a b
c d
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Segmented image samples “Cameraman”, “Lena”, “Pepper”,
“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 11. 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 12. 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 13. 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 14. 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.
Figure 15. Test 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.
REFERENCES
[1] Rafael C. Gonzalez and Richard E. Woods, Digital image
processing. Addison-Wesley Publishing, 1993.
[2] M. Kunt, A. Ikonomopoulos and M. Kocher, “Second generation
image coding techniques,” Proceeding IEEE 73 (4), 1985, pp. 549-
575.
[3] M. Kaya, “The development of image compression techniques
using fuzzy image segmentation,” Ph. D. Thesis, Osmangazi
University Electrical and Electronics Engineering Department,
Eskisehir, 2001.
[4] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, pp.
338-353, 1965.
[5] James C. Bezdek, “A convergence theorem for the fuzzy
ISODATA clustering algorithms,” IEEE Transactions on Pattern
Analysis And Machine Intelligence, vol. pami-2 (1), pp. 1-8, 1980.
[6] Mahmoud R. El-Sakka, Mohamed S. Kamel, “Adaptive image
compression based on segmentation and blok classification,”
International Journal of Imaging Systems and Technology, 10 :
(1), pp. 33-46, 1999.
a b
c d
a b
c d
a b
c d
a b
c d
a b
c d
INTERNATIONAL JOURNAL OF SIGNAL PROCESSING VOLUME 1 NUMBER 1 2004 ISSN:1304-4478
IJSP VOLUME 1 NUMBER 1 2004 ISSN:1304-4478 87 COPYRIGHT © 2004 ENFORMATIKA
[7] Seong-Gon Kong and Bart Kosko, “Image coding with fuzzy
image segmentation,” IEEE International Conference On Fuzzy
Systems, SanDiego - USA, pp. 213-220, 1992.
[8] A.K. Jain, M.N. Murty and P.J. Flynn, “Data clustering: a review,”
ACM Computing Surveys, vol. 31, no. 3, pp. 264-323, 1999.
[9] J. J. Hopfield and D. W. Tank, “Neural computation of decisions in
optimization problems,” Biol. Cybern.52, pp. 141-152, 1985.
[10] J. J. Hopfield, “Neural network and physical systems with
emergent collective computational abilities,” Proc. Ntl. Acad. Sci.,
USA 79, pp. 2554-2558, 1982.
[11] Jzau-Seng Li, “Image vector quantization using an annealed
Hopfield neural network,” Optical Engineering, 38 (4), pp. 599-
605, 1999.
[12] K.S.Cheng, J.S. Lin and C.W. Mao, “The application of
competitive Hopfield neural network to medical image
segmentation,” IEEE Transactions on Medical Imaging, 15, pp.
560-567, 1996.
[13] P. C. Chung, C. T. Tsai, E. L. Cheng and Y. N. Sun, “Polygonal
approximation using a competitive Hopfield neural network,”
Pattern Recognition, 27, pp. 1505-1512, 1994.
[14] J. S. Lin, K. S. Cheng and C. W. Mao, “A fuzzy Hopfield neural
network for medical image segmentation,” IEEE Transactions on
Nucl. Science 43, pp. 2389-2398, 1996.
[15] Jzau-Sheng Lin, Kuo-Sheng Cheng and Chi-Wu Mao,
“Multispectral magnetic resonance image segmentation using fuzzy
Hopfield neural network,” International Journal of Bio-Medical
Computing 42, pp. 205-214, 1996.
[16] C. H. Chao and A. P. Dhawan, “Edge detection using a Hopfield
neural network,” Optical Engineering 33, pp. 3739-3747, 1994.
[17] C.Y. Chang and P.C. Chung, “Two-layer competitive based
Hopfield neural network for medical image edge detection,” Optical
Engineering, vol. 39, pp. 695-703, 2000.
[18] C. T. Tsai, Y. N. Sun, P. C. Chung and J. S. Lee, “Endocardial
boundary detection using a neural network,” Pattern Recognition
26 (7), pp. 1057-1068, 1993.
[19] S. C. Amatur, D. Piriano and Y. Takefuji, “Optimization neural
network for the segmentation of magnetic resonance images,”
IEEE Transactions on Medical Imaging 11, pp. 215-220, 1992.
[20] J.S. Lin, “Fuzzy possibilistic neural network to vector quantization
in frequency domains,” Optical Engineering, vol. 41, pp. 839-847,
2002.
[21] C.L. Chang and Y.T. Ching, “Fuzzy Hopfield neural network with
fixed weight for medical image segmentation,” Optical Engineering,
vol. 41, pp. 351-358, 2002.
[22] R.G. Roozbahani, M.H. Ghassemian, A.R. Sharafat, “Robust
segmentation of medical images using competitive Hopfield neural
network as a clustering tool,” Iranian Journal of Science and
Technology, vol. 25, pp. 427-439, 2001.
[23] C.Y. Chang and P.C. Chung, “Medical image segmentation using a
contextual-constraint-based Hopfield neural cube,” Image and
Vision Computing, vol. 19, pp. 669-678, 2001.
[24] M. Kaya, “A new image clustering and compression method based
on fuzzy Hopfield neural network,” IJCI Proceedings of
International Conference on Signal Processing, Canakkale-
Turkiye, vol. 1, no. 2, pp. 11-16, 2003.
[25] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller
and E. Teller, “Equations of state calculations by fast computing
machines,” J. Chem. Phys. 21, pp. 1087-1092, 1953.
[26] S. Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi, “Optimization
by simulated annealing,” Science 220, pp. 671-680, 1983.
[27] S. Pierre, M. A. Hyppolite, J. M. Bourjolly, and O. Dioume,
“Topological design of computer communication networks using
simulated annealing,” Engineering Applications of Artificial
Intelligence 8, pp. 61-69, 1995.
[28] H. L. Tan., S. B. Gelfand and E. J. Delp, “A cost minimization
approach to edge detection using simulated annealing,” IEEE
Transactions on Pattern Analysis and Machine Intelligence 14, pp.
3-18, 1991.
[29] S. Jalali and J. F. Boyce, “Determination of optimal general edge
detectors by global minimization of a cost function,” Image Vis.
Comput. 13, pp. 683-693, 1995.
[30] R. W. Klein and R. C. Dubes, “Experiments in projection and
clustering by simulated annealing,” Pattern Recognition 22, pp.
213-220, 1989.
[31] Hyun-Sook Rhee and Kyung-Whan Oh, “A validity measure for
fuzzy clustering and its use in selecting optimal number of
cluster,” IEEE International Conference on Fuzzy Systems, New
Orleans – USA, vol.2, pp. 1020-1025, 1996.
[32] J. C. Dunn, “Well separated clusters and optimal fuzzy partitions,”
J. Cybern., vol. 4, no. 3, pp. 95-104, 1974.
[33] J.C.Bezdek, Pattern recognition with fuzzy objective function
algorithms. Plenum Press, New York, 1981.
[34] I. Gath and A. B. Geva, “Unsupervised optimal fuzzy clustering,”
IEEE Transaction on Pattern Analysis and Machine Intelligence,
vol. 11, no. 7, pp. 773-781, 1989.
[35] II Hong Suh, Jae-Hyun Kim, and Frank Chung-Hoon Rhee,
“Convex-set-based fuzzy clustering,” IEEE Transaction on Fuzzy
Systems, vol. 7, no. 3, pp. 271-285, 1999.
[36] R. Krishnapuram and J. Kim, “Clustering algorithm based on
volume criteria,” IEEE Transactions on Fuzzy Systems, vol. 8, no.
2, pp. 228-236, 2000.
[37] M. P. Windham, “Cluster validity for fuzzy clustering algorithms,”
Journal of Fuzzy Sets and Systems, vol. 5, pp. 177-185, 1981.
[38] Xuanli Lisa Xie and Gerardo Beni, “A validity measure for
fuzzy clustering,” IEEE Transactions on Pattern Analysis and
Machine Intelligence, vol. 13, no.8, pp. 841-847, 1991.
[39] M. Sugeno and T. Yasukawa, “A fuzzy-logic-based approach to
qualitative modeling,” IEEE Transactions on Fuzzy Systems, vol.
1, pp. 7-31, 1993.
[40] Amine M. Bensaid, Lawrence O. Hall, James C. Bezdek, Laurence
P. Clarke, Martin L. Silbiger, John A. Arrington and Reed F.
Murtagth, “Validity-Guided (re)clustering with application to image
segmentation,” IEEE Transactions on Fuzzy Systems, vol. 4, pp.
112-123, 1996.
[41] J.S. Lin, S.H. Liu and C.Y.Lin, “The application of fuzzy Hopfield
neural network to design better codebook for image vector
quantization,” IEICE Transactions on Fundamentals, vol. E81-A,
No. 8, pp. 1645-1651, 1998.
[42] P. C. Chung, C. T. Tsai, E. L. Cheng and Y. N. Sun, “Polygonal
approximation using a competitive Hopfield neural network,”
Pattern Recognition 27, pp. 1505-1512, 1994.
[43] D. E. Van Den Bout and T. K. Miller III, “A traveling salesman
objective function that works,” International Conference on Neural
Network II, pp. 299-303, 1988.
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