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An improved harmony search algorithm for reducing computational time of fractal image coding

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An improved harmony search algorithm for reducing computational time of fractal image coding

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Fractal image coding (FIC) based on the inverse problem of an iterated function system plays an essential role in several areas of computer graphics and in many other interesting applications. FIC received considerable attention because of its high resolution, fast decoding, and many other advantages. However, the method has not been used widely because it required high computation time in the encoding process, which is one of its drawbacks. Many optimization methods are introduced to serve in solving this drawback and reducing the searching time for optimal solution. The approach that based on meta-heuristic methods is promising, which employ some degree of randomness to search for an optimal solution. This study introduces the harmony search algorithm to improve the FIC technique. The algorithm searches for an optimum solution while playing music. Finding music harmony has been proven to solve optimization problems by searching for an optimal solution. This algorithm is naturally inspired and is currently in demand. The experiments show that, compared with other techniques, the proposed method has excellent performance in image quality and reduces the computation time and storage space.
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Journal of Theoretical and Applied Information Technology
30
th
April 2017. Vol.95. No 8
© 2005 – ongoing JATIT & LLS
ISSN:
1992-8645
www.jatit.org E-ISSN:
1817-3195
1669
AN IMPROVED HARMONY SEARCH ALGORITHM
FOR REDUCING COMPUTATIONAL TIME OF FRACTAL
IMAGE CODING
NADIA M. G. AL-SAIDI
1
*, SHAIMAA S. AL-BUNDI
2
,
AND NASEIF J. AL-JAWARI
3
1
Applied Sciences Department- University of Technology
2
Department of Mathematics-College of Education for pure Sciences- Ibn Al-Haitham-Baghdad University
2,*
Department of Mathematics-College of Sciences-Al-Mustansiriya University-Iraq
E-mails: nadiamg08@gmail.com, shaimaaalbundi@gmail.com and njaljawari@hotmail.com
ABSTRACT
Fractal image coding (FIC) based on the inverse problem of an iterated function system plays an essential
role in several areas of computer graphics and in many other interesting applications. FIC received
considerable attention because of its high resolution, fast decoding, and many other advantages. However,
the method has not been used widely because it required high computation time in the encoding process,
which is one of its drawbacks.
Many optimization methods are introduced to serve in solving this
drawback and reducing the searching time for optimal solution.
The approach that based on meta-
heuristic methods is promising, which employ some degree of randomness to search for an optimal
solution. This study introduces the harmony search algorithm to improve the FIC technique. The algorithm
searches for an optimum solution while playing music. Finding music harmony has been proven to solve
optimization problems by searching for an optimal solution. This algorithm is naturally inspired and is
currently in demand. The experiments show that, compared with other techniques, the proposed method has
excellent performance in image quality and reduces the computation time and storage space.
Keywords— Fractal, iterated function system (IFS), fractal inverse problem, fractal image compression
(FIC), harmony search algorithm (HAS).
1. I
NTRODUCTION
In 1988, the concept of fractal coding was
firstly introduced by Barnsley et al. [1], when they
were tried to find the IFS of an image, such that if
these system of function are iterated they
approximate the given image as an attractor. This
theory is then improved in 1992 by Barnsley's
student, Jacquin [2]. He introduced the concept of
fractal image coding and interested in studying of
partitioned iterated function system. Jacquin
approach has been popularized practically and
theoretically by several researchers, as soon as he
published his technique. Nevertheless, none of
these attempts was proven to be efficient.
Therefore, many efforts are highlighted towards
employing of evaluative algorithms. The
optimization models have been proposed to
represent a normal evolution mechanism. Several
real world problems are considered as optimization
problems, therefore, a developed method is
required to increase their efficiency and
productivity in order to search for optimal solution.
Some degree of randomness should be added and
this is possible by using meta-heuristic methods.
They are considered as attempts to approximate
best solution and increase the efficiency. Some of
these methods are nature inventor [3]. The harmony
search algorithm (HSA) is a metaheuristic
algorithm used to solve optimization problems [4].
The method has been applied to solve different
kinds of problems in the past years and provided
effective results compared with other metaheuristic
algorithms and conventional techniques, which are
computationally costly. The remainder of this paper
is organized as follows: Some literatures review is
presented in Section 2. Section 3 introduced some
of the important fractal terminologies to understand
the FIC technique. Section 4 presents some of the
metaheuristic optimization algorithms to improve
the proposed technique. Section 5 discusses the
implementation of the proposed algorithm with
some experiments and comparisons. Section 6
summarizes the conclusions.
Journal of Theoretical and Applied Information Technology
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2. LITERATURE
REVIEW
After introducing of genetic algorithm
(GA) approach by Goldberg [5], it is used to solve
some of the optimization problem in a large search
space with different optima, and hence, several
interesting applications are based on. The use of
this approach in fractal image coding started in
1995, when Jacques et al. [6] introduced a genetic
programming method which is investigated in
solving the general inverse problem and perform at
the same time a numeric and symbolic
optimization. In 1997, Vences and Rudomin [7]
used genetic algorithms (GAs) to find a partial
iterated function system (PIFS) which encodes a
single image. They did this work by reducing the
needed time to achieve process in about 30%
compared with Barnesly's [8]. In 2000, Dasgupta et
al. [9] introduced the effectiveness of an
evolutionary algorithm to obtain the IFS code in
black and white images. In their work, the IFS is a
set of maps that can be represented as a set of real
parameters. In 2000, Wang, Zhang, and Yu [10]
used the fractal technique to encode a part of the
image and employed other algorithms to the
remaining part. They proposed a new image space
mapping, which is called the partial fractal
mapping. In 2006, Mohamed and Aoued [11]
presented the solution to the fractal inverse problem
using GA. Moreover, they designed a fractal
compression algorithm to search in the domain
block based on GA. They applied the standard
Barnsley algorithm [8] and the Y. Fisher algorithm
[12], and the results were compared with the results
of their work to prove the efficiency of their
method. In 2006, Bouboulis et al. [13] introduced
an image compression program employing fractal
interpolation surfaces that are attractors of some
local iterated function systems (LIFS). They
attempted to improve the fractal image compression
by replacing the contraction constant by the
contraction function that provides a flexible
construction. Therefore, the number of regions and
domains in the image coding can be reduced, and
the quality of the decoded image can be improved.
In 2010, Lon Lin [14] introduced similar measures
for fractal image compression that are resistant to
noises. He proposed the integration of the robust
estimation technique from statistics into the
encoding process of the fractal inverse problem to
obtain the parameters. However, the main
drawback of the robust FIC is high cost. He tried to
overcome this drawback by using the particle
swarm optimization (PSO) technique that is used to
decrease the search time. In 2014, Nadira et al. [15]
improved the iterated fractal algorithm by
designing an efficient search of the domain pools
for color image compression using GA. They
obtained a decreased coding time and intensive
computation works. In 2016,
Al-Bundi
et al. [16]
proposed the crowding optimization method to
improve the FIC technique. They proved that their
method performed better than the other
evolutionary optimization methods.
HSA is employed in the present study. The
algorithm was proposed in 2001 by Geem et al.
[17] as one of the optimization algorithms. They
noticed the resemblances among the music
improvisation methods and formed an optimal
solution to difficult problems. The better harmony
vector replaces the worst harmony vector in a
harmony memory (HM). HSA proved its efficiency
and effectiveness in various studies [3, 18]. In
2012, El-Satawy and Ahmed [19] introduced a new
multi-objective evolutionary method. The new
technique merges harmony search optimization
with chaos search. In 2013, Jiaqi Di and Nihong
Wang [20] proposed a new HSA with chaos. The
proposed algorithm initialized and developed HSA
with the chaos optimization algorithm depending
on the secondary carrier wave, which improved the
optimization accuracy and convergence rate. In
2014, Osama Abdel-Raoufet et al. [21] improved
the HSA to solve linear assignment problems. Their
proposed method, which was based on the chaotic
behavior of a caustic monophony, was used to
generate solutions. In 2014, Osama Abdel-Raouf et
al. [22] proposed a novel hybrid optimization
technique called the improved flower pollination
algorithm using chaotic harmony search. In 2015,
Gao et al. [23] described the classical harmony
algorithm and its basic applications. They
presented, discussed, and applied the use of the
modified algorithm to improve the performance of
wind generators. In the present work, HSA is used
to improve the encoding time of the FIC technique.
This improvement is used in one of the important
application, image retrieval [24], the result was
promising. In 2017, Al-Saidi et al. is also
improved fractal image coding based in their
approach on block complexity [25].
3. FRACTALS
IMAGE
CODING
Barnsley [26] encoded computer graphics
by using the IFS and obtained a 1,000:1
compression ratio. This result encouraged his
student Jacquin to propose a new coding method by
partitioning the given image into blocks. This
approach gained extensive interest because of its
Journal of Theoretical and Applied Information Technology
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novelty, high compression ratio with good
resolution, and fast decoding.
3.1 Basic Terminology
The IFS theory is important in the fractal
field and is a powerful tool to search for fractals.
The theory is applied in generating and modeling
irregular patterns and can be viewed as image
compression when associated with the automatic
generation of fractal and irregular forms. In both
applications, the generated information is called the
attractor of the IFS. Works on contractive mappings
to produce fractal sets have been considered by
many researchers. Moreover, most fractal image
compression methods are based on the IFS.
The IFS was developed by Hutchison [27]
and Barnsley et al. [1, 7,26, 28]. These systems of
mapping were widely discussed and used in many
applications, such as in the image compression
method. The mathematical definition of the IFS is
expressed as:
In the collage theorem, a set of
transformations
 must be obtained
so that the union or collage of the transformations is
close to the given set L to obtain an IFS with
attractor that is “close to” a set.
With the rapid development of multimedia
technology, the need for digital images is rapidly
growing. Managing these resources effectively has
become a focus of many studies. Considerable
attention was focused on fractal geometry because
of its ability to describe natural phenomena
qualitatively and quantitatively. Fractal is an image
that represents the attractor generated by the
interpolation method, that is, IFS. Some natural
images are not globally self-similar; they contain
local self-similarity. In this case, the LIFS is
introduced to describe such types of images.
Iterated Function System (IFS) is developed by
Hutchison (1981) [27], Barnsley and others (1985,
1986, 1988) [1-7-26-28]. These systems of
mapping were discussed widely and used in many
applications, such as image compression method.
The mathematical definition of the IFS is given as
follows:
Definition 1: Let (X, d) be a complete metric space.
a finite set of contractive mappings
with
contractivity factors
for  such that

 is called the IFS on X.
IFS is based on the affine transformation given by:-

!"
#
$
"
$

"
$
!"
$
%
&
'
(
…. (1)
Sometimes, the world (hyperbolic) is used
with the iterated function systems, and it sometimes
drops in practices. The following theorem will
show why we use the concept of hyperbolic IFS.
Theorem 1: (Fixed point theorem) Let ) be a
metric space, let 
$
*
be a hyperbolic
iterated function system where
 is a
contraction mapping. Let +,
,is
defined by: +-.
/
-
…. (2)
Then + is a contraction mapping with
contractivity factor on the complete metric space
,0).
That is mean 0+-+12
30-1 for all -1
,.
Let A
H(X) be defined as;
4+4+
4
Then4 is a unique fixed point such that;
456
7
+
- for
any -
,
Definition 2: In IFS, any compact subset (fixed
point) 4
, is called attractor for IFS if;
4.
/
4
The fixed point observes existence and
uniqueness by the contraction mapping theorem.
An important property of any IFS is the attractor,
that means the attractor4 is fully known by the set
of coefficients of +, and can be generated by
applying the IFS to any starting point several times.
Theorem 2: The Collage Theorem [26]
Let ) be a complete metric space. Let
8
999 be an IFS with
contractivity factor , :2
<
 and let ; be a
closed subset of such that:
0;.
;<=
*
/
…. (3)
for some =>:, where 0 is the Hausdorff distance.
Then 0;42
?
@A
for the attractor 4 of the IFS's.
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In the collage theorem, a set of
transformations
 must be obtained
so that the union or collage of the transformations is
close to the given set L to obtain an IFS with
attractor that is “close to” a set.
With the rapid development of multimedia
technology, the need for digital images is rapidly
growing. Managing these resources effectively has
become a focus of many studies. Considerable
attention was focused on fractal geometry because
of its ability to describe natural phenomena
qualitatively and quantitatively. Fractal is an image
that represents the attractor generated by the
interpolation method, that is, IFS. Some natural
images are not globally self-similar; they contain
local self-similarity. In this case, the LIFS is
introduced to describe such types of images. A
collection of all local transformations w
i
is known
as a PIFS w
i
, i = 1, 2, …, N. Each w
i
can be written
as:
3.2 Jacquin Method for FIC [2]
The FIC is an important search area with a
large number of possible application fields. The
FIC determines the representative codes of any
given object that is self-similar or contains different
types of self-similarities. Barnsley [26] introduced
this concept with the collage theorem. When the
considered object is an image, fractal image
compression is applied. A method has been
proposed by Jacquin [2] to solve this kind of
problem, which has been investigated by many
authors. Compared with other methods, the FIC is
time consuming in searching similar domain
blocks. Therefore, the demand for a new technique
to solve this problem and improve the speed of this
method arises.
The problem can be easily solved when
the given set is self-similar. In this case, the IFS can
be easily found by transforming the self-similarity
property. Barnsley [26] stated an approximated
solution to the inverse problem of fractals using the
IFS. This solution was verified in the collage
theorem, which supplied the first step in solving
fractal inverse problems.
This approach reduces the redundant area
in the image. The process of transforming images to
codes is complicated, but the reversing task is
simple. The Jacquin approach is dependent on the
IFS attractor, and obtaining the PIFS parameter is
regarded as the main problem in fractal coding. The
following is an explanation of the main processes
of Jacquin technique:
1. A given image M is partitioned into non-
overlapping blocks R={R
1
,R
2
,…,R
m
} of size
×
called range blocks, where BCD
$
, and
overlapping blocks D={D
1
,D
2
,…,D
n
} of size 2r×2r
called domain blocks, where C#&
$
as
shown in Figure 1.
a- Range Blocks b- Domain block
Figure 1(a,b): Domain and range blocks of the PIFS
2. For each range block R
i
, choose its
approximate domain D
k,
, k=1,…,n and a
suitable contractive affine transformation w
ik
that satisfied the following:
)EF
G
H
G
I6J)F
K
EH
K
I
where
G
is the contractive affine transformation
from H
K
to F
. This can be represented by
)F
K
EH
K
I where ) refers to the mean square
error (MSE) between F
and
K
that consist of two
mappings L
K
and "
K
such that:
K
"
K
ML
K
, and
hence, L
K
represents the contraction transformation
that transformed H
size into F
size. This
transformation is described as follows:
The domain block D
j
is divided into non-
overlapping blocks of size 2×2, where the pixel
value of the transformed block T
j
(D
j
) is computed
from the average of the four pixels in the block of
D
j
, as shown in Figure 2.
Figure 2: The contraction of a domain block.
The mapping "
K
is performed in two
steps; firstly, the block L
K
H
K
is transformed using
eight isometries given in Figure 3.
Journal of Theoretical and Applied Information Technology
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Figure 3: The eight isometries
Second step, the pixel values of the
resulting block from first step is transformed by
+
K
, where +
K
is defined as follows:
+
K
N
N&!
….(4)
Here z represents the pixel value that
obtained from the first step, and the scaling
and
the offset !
defined in (5) and (6) respectively, are
calculated by the mean square error (MSE) of the
pixel values of the range block F
.
…. (5)
=
==
n
ii
n
ii
asb
n
o
11
1
…. (6)
The MSE difference is calculated using:
+
+
=
=
====
n
ii
n
ii
n
iii
n
ii
n
ii
bnoo
aobaassb
n
MSE
1
111
2
1
2
2
22
1
…. (7)
This representation contains two parts, namely,
the massive part to represent the image pixels and
the geometric part to represent the isometric and
scaling.
4. M
ETAHEURISTIC
M
ETHOD TO
S
OLVE
F
RACTAL
I
MAGE
C
ODING
4.1 Swarm Method
Swarm algorithm is an optimization
algorithm that generates the optimal or near-
optimal solutions. Swarm algorithm was developed
by Kemedy and Eberhert in 1995 [29]. The
mechanism of this algorithm is based on the
behavior of birds flocking at the same time seeking
for food. The birds connect with each other when
searching for food and reach their aim with
minimal time. The method can be easily
implemented and considered to be simpler than the
other metaheuristic methods, such as the ant colony
optimization algorithm. Compared with the GA, the
size of the population in the swarm method is
small, which leads to the initialization of the
populations in the simplest manner. Thus, the
swarm method is preferred over the other
optimization algorithms. Swarm algorithm
improves the general purpose and employs the
concept of fitness.
The swarm population is initialized with
random solutions called particles. Every particle
can freely fly through the search process. Each
particle is considered a point in k-dimensional
space. The following equations represent the
stoning step for the swarm algorithm.
O
G
PO
G
P#&Q
R
G
#
G
P#&
Q
$
$
R
G
#
9G
P# …(8)
G
P
G
P#&O
G
P …(9)
where Q
Q
$
>: and
$
S: are random
numbers, P is time, 

$
G
are the
particles R

R
$
R
G
are the best positions
(best fitness) of kth particles up to time P#, and
R
G
is the best position among all populations (or
among all swarms) up to time P#.
First, the population size should be
determined, and the position and velocity of the
particles should be initialized. The movement of
each particle depends on Eqs. (8) and (9), and the
fitness function is calculated. Afterward, the best
position of each particle and the swarm is record.
Finally, the work ends when the criterion is
satisfied. The best position of the swarm is the final
solution [30].
Fractal Image Compression using Swarm
Method
The FIC determines the best domain block
identical to each rang block, but it takes a long
time. The swarm method can provide faster
encoding of the scale blocks. The method is based
on the population to search for global optimum
[20]. The PSO method is used to screen and remove
the trivial domain blocks to reduce the amount of
data in the encoding step. In 2013, D.
Venkatasekhar et al. [31] used the PSO method to
reduce the search for the best domain block of each
range block in the FIC method. The similarity
between these blocks cannot be calculated unless
they have the same type. The blocks in the domain
and range sets are partitioned into three classes
according to the coefficients of the third-level
wavelet. For each range block, the similarity is
=
==
===
2
11
2
111
n
i
i
n
i
i
n
i
i
n
i
i
n
i
ii
aan
baban
s
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measured only in the domain block from the same
class. Only four transformations are required to
determine the similarity in the dihedral
transformation [30]. The steps of the modified
fractal encoding according to the swarm method
[32] is as follows:
Algorithm for the fractal image compression
Step1. Set the size of the swarm to be proportional
to (M-2r+1) which represent the maximum
number of iteration, the position and the
particles is initialized randomly.
Step2. Finding MSE between the particles position
(domain block) and range block as a fitness
value.
Step3. If the fitness of the new best position is
better, then the swarm best position is updated.
Step4. The algorithm is stopped if after some
maximum iteration the best position is not
changed.
Step5. Using (7) and (8) to update the particle best
position, and go to step 2.
4.2 Harmony Search
The relationship between playing music
and finding an optimal solution leads to the
development (creation) of the HSA. Finding
harmony in music is analogous to finding an
optimal solution in an optimization method. After
the introduction of the HSA music optimization
algorithm by Geem et al. [17], its efficiency and
effectiveness has been developed and improved by
various researchers (for example, see [3, 18, 33]).
Obtaining the optimal harmony is the goal
of HSA. Therefore, the following three potential
methods should be implemented to satisfy this goal:
1. Playing for memory;
2. Pitch amendment;
3. Randomization.
In 2001, Geem et al. [17] recognized the
resemblance between the music improvisation
methods and finding the optimal solution to
difficult problems. The researcher formalized three
methods as part of the optimization algorithm
developed. The following are the main steps of the
HSA:
1. HM search;
2. Pitch amendment;
3. Randomization.
These parts are also considered the main
parameters of the HSA [3, 34], which are described
as follows:
1. Initialization: The program parameters are
defined, and the HM is created with
random solutions. All the solutions are
evaluated by an evaluation or objective
function.
2. Harmony improvisation: A new solution is
formed. The three parts of the HSA are
utilized to determine which value will be
assigned to each variable in the solution.
3. Selection: The best solution (harmony) is
chosen when the termination condition is
achieved.
Classical Harmony Search Algorithm
Define objective function
T
$
U
V
Define harmony memory consideration rate
(r
accept
)
Define pitch adjustment rate
W
X
. Harmony in
addition to other harmony parameters is
generated randomly.
While (P< max C) Do YC is
the max number of iteration
While (2
) Do
Y is the number of variables
If (rand <
W
X
) certain amount is
added to adjust the
W
X
value
else
Choose another random value
end if
end while
Accept the new solution
end while
Determine the best solution
end
The aforementioned algorithms with some
parameters are reviewed as follows:
1. The maximum number of iterations is used as a
terminator.
2. The size of the HM is used to determine the
number of solutions to be saved in the HM.
3. The memory consideration rate represents the
new solution from the memory solutions.
4. The pitch adjustment rate is used to add a
specific rate to the modified rate of the
memory.
The Proposed Algorithm
Step1. Initialize the harmony memory
a. A given image M is partitioned into
non-overlapping blocks
R={R
1
,R
2
,…,R
m
} of size
×
called
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© 2005 – ongoing JATIT & LLS
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1817-3195
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range blocks, where
B
C
D
$
, and
overlapping blocks D={D
1
,D
2
,…,D
n
} of
size 2r×2r called domain blocks, where
C#&
$
.
b. For each range block R
i
, choose domain
D
k,
, k=1,…,n and a suitable contractive
affine transformation w
ik
that satisfied
)EF
G
H
G
IB)F
K
EH
K
I
where
G
is a contractive affine transformation,
G
ZH
K
F
, d refers to the mean square error
(MSE) between F
and
G
where
K
"
K
M
L
K
, L
K
represented the contraction
transformation L
K
ZH
F
, and can be
described as follows:
D
j
is divided into non-overlapping blocks of size
2×2, where the pixel value of the transformed
block T
j
(D
j
) is computed from the average of the
four pixels in the block of D
j
, the mapping "
K
is
performed in two steps;
The block L
K
H
K
is carrying out using
eight isometries which was given in
Figure 4.
The pixel values of the output block
from first step is transformed by +
K
,
such that, +
K
N
N&!
, where the
and !
as in equations (4) and (5).
This comparison resulting L
K
[
\
T
K
K
!
K
]
Step2. The form of the harmony memory (HM)
is as follows:
,C^ L
L
$
_
L
`ab
c
^
\
T

!
$
\
$
T
$

$
!
$
_d_
`ab
\
`ab
T
`ab

`ab
!
`ab
c
where HMS should be between 50 and 100.
Recognize the worst member L
efgAh
,
such that L
efgAh
S,Ci
Step 3. Suppose a random solution L
g
, L
g
[
g
\
g
T
g
!
g

g
]
Step 4.Now, choosing any value from HM
(random value) by HMCR (harmony
memory consideration rate) where 0 <
HMCR < 1. Let this random value be
L
[
\
T

!
], L
S,C.
Step 5. Let L
j
be a new harmony vector,
L
j
k
If
l
:
2
,
C1F
for all
,Ci
where
l
is a uniform
random number generator
Step 6. Choose a new solution
L
j
L
$j
L
`ab
S,C
Let L

j
SL
j
L
$j
L
`ab

If l:2m4F where PAR is
adjusting, such that, L

j
n
G
, and n
G
is the pitch adjusting value.
Step 7. Now, evaluation the new harmony vector
L

j
by comparing L

j
with the worst
value L
efgAh
S,C, if TL

j
<
TL
efgAh
where T is the fitness
function, this comparison resulting a new
harmony vector L
j
okL
j
S,C, and
then
L
efgAh
is excluded from HM i.e.
L
efgAh
HM.
5. IMPLEMENTATION
AND
ANALYSIS
The improved HSA is implemented using
MATLAB, and the code is encoded in a PC with
the specification Intel 2.5 GHz Core i7, with 8 MB
memory. A sample of five gray scale images with a
size of 512 × 512 is tested to show that, for the 4
×
4 range block size, a good compression ratio is
obtained, but the computation time is increased. By
contrast, for the 2
×
2 range block size, the
compression ratio is decreased and the computation
time is insignificantly improved. This inversely
proportional relationship concludes that the 4
×
4
range block size is preferable for many applications
because it satisfies the compromise between the
computation time and the compression ratio. This
result is depicted in Table 1.
The MSE is an evaluator used in
measuring the average of the squares of the errors
or deviations. The MSE strongly depends on the
image intensity. In the case of image compression,
this measure is used between original and
compressed image values. The MSE is also
considered one of the indications of image quality.
The results of the experiments shown in Table 2
reveal that the range blocks with size 4 × 4 has
better results than the range blocks with size 2 × 2,
where the MSE is less and the peak signal-to-noise
ratio (PSNR) is high. The PSNR is used to indicate
the image quality, where its high value is mostly an
indication of a high-quality compressed image. The
ratio is defined via MSE and is inversely
proportional to the MSE. Finally, the improved
HSA is compared with the original FIC technique
and the improved crowding method proposed in
Journal of Theoretical and Applied Information Technology
30
th
April 2017. Vol.95. No 8
© 2005 – ongoing JATIT & LLS
ISSN:
1992-8645
www.jatit.org E-ISSN:
1817-3195
1676
[15] to prove its efficiency. The comparison was in
terms of encoding time, MSE, and compression
ratio. Table 3 shows that the proposed method
performed better in terms of these measures given
the same samples of five images
Table 1: Harmony Search Algorithm For Different Range Size In Terms Of Coding, Decoding Time And Compression
Ratio
Images Range Initial
Time
Coding Time Decoding Time Compression Ratio
2
×
2 0.3758 0.0201 0.5936 3.6
4
×
4 0.1101 0.0097 0.2591 8.5
2
×
2 0.3752 0.0178 0.5288 2.7
4
×
4 0.0985 0.0098 0.2564 5.5
2
×
2 0.3237 0.0178 0.5621 4
4
×
4 0.0978 0.0098 0.2595 7.1
2
×
2 0.3289 0.0173 0.5622 2.7
4
×
4 0.0981 0.0094 0.2584 6.7
2
×
2 0.3260 0.0168 0.5618 3.8
4
×
4 0.0978 0.0099 0.2635 8.7
Table 2: PSNR And MSE For Different Range Block Based On HAS
Images Range PSNR MSE
2
×
2 12.11 0.07
4
×
4 14.50 0.03
2
×
2 7.13 0.38
4
×
4 12.04 0.06
2
×
2 7.03 0.113
4
×
4 10.53 0.08
2
×
2 9.06 0.039
4
×
4 15.8 0.02
2
×
2 10.01 0.08
4
×
4 15.14 0.03
Journal of Theoretical and Applied Information Technology
30
th
April 2017. Vol.95. No 8
© 2005 – ongoing JATIT & LLS
ISSN:
1992-8645
www.jatit.org E-ISSN:
1817-3195
1677
Table 3: Comparison Between Jacquin Approach, Genetic And The Proposed Harmony For Range Block Of Size 4
Images
Fractal image
compression based
on Jacquin
Approach
Coding Time 4.01 5.32 5.32 6.91 6.20
MSE 0.81 0.89 0.94 0.91 0.83
Compression
Ratio
11.6 9.21 10.82 9.41 12.1
Fractal Image
Compression
based on
Crowding method
[]
Coding Time 2.13 2.13 2.13 2.13 2.13
MSE 0.026 0.026 0.026 0.026 0.026
Compression
Ratio
7.04 4.33 5.72 6.03 7.16
Fractal Image
Compression
based on
Harmony Search
Algorithm
Coding Time 0.11 0.108 0.107 0.107 0.107
MSE 0.035 0.062 0.088 0.026 0.030
Compression
Ratio
8.5 5.5 7.1 6.7 8.7
6. C
ONCLUSION
The properties of the fractal function
have been investigated over the years, and their
inherent complexity, from the extreme sensitivity
of the system to the initial conditions, was
derived. These functions are used to model many
real-life problems. Fractal image compression is
one of the important applications. However, this
approach has an optimization problem. Several
optimization methods have been used, but the
naturally inspired algorithm is preferred. The
phenomena that mimic the musical process to
search for the perfect state of harmony is used in
this study to ensure a short coding time, a large
storage area, and a high quality with less error
ratio. In such methods, some degree of
randomness should be added, which is an attempt
to approximate the best solution and increase the
efficiency. The combination between
randomness and rules shows a promising result.
The results of the experiments show that the
image quality, coding time, and compression
ratio are improved when using the proposed
approach compared with the other optimization
approaches. For more enhancements of this
approach, we suggest utilizing of hybridizing
between genetic algorithms and harmony search
method to solve fractal inverse problem.
REFERENCES
[1] Barnesly, M.F., Sloan, A.D., “A Better Way
to Compress Images”, Byte Mag., 1988,
pp.215–223.
[2] Jacquin, A. E., “Image coding based on a
fractal theory of iterated contractive image
transformations”. Image Proc., IEEE Trans.,
1992, Vol.1, pp.18-30.
[3] Geem, Z. W., “Music-Inspired Harmony
Search Algorithm Theory and Application”,
Studies on Computational Intelligence,
2009, Vol. 191, pp. 2-13.
[4] Geem, Z. W., “Music-Inspired Harmony
Search Algorithm Theory and Application”,
Studies on Computational Intelligence,
2009, Vol. 191, pp. 2-13.
[5] Goldberg, D. E., "Genetic Algorithm in
search, optimization and Machine
Learning", Addison – Wesley, reading, MA,
1989.
[6] Jacquin, A.E., Lutton, E., L´evy-V´ehel J.,
Cretin, G., Glevarec, P., Roll, C., “Mixed
IFS : Resolution of the Inverse Problem
Using Genetic Programming” Complex
Systems, 1995, Vol. 9, No. 5, pp. 375–398.
[7] Levey Vehel, J. and Rudomin, I., “Genetic
Algorithms for Fractal Image and Image
Sequences Compression”, Comptacion
Visual, 1997.
[8] Barnsley, M.F., Demko, S., Iterated
function systems and the global construction
of fractals”. In Proceedings of the Royal
Society of London A399, 1 985, pp. 243 -
275.
[9] Dasgupta, D., Hernande, G., Nino, F., An
Evalutionary Algorithm for Fractal Coding
of Binary Images”, IEEE Transactions on
Evolutionary Computation, 2000, Vol. 4,
No.2.
Journal of Theoretical and Applied Information Technology
30
th
April 2017. Vol.95. No 8
© 2005 – ongoing JATIT & LLS
ISSN:
1992-8645
www.jatit.org E-ISSN:
1817-3195
1678
[10] Wang, Z., Zhang, D., Yu, Y., “Hybrid Image
Coding Based on Partial Fractal Mapping”,
Single Processing: Image Communication
15, 2000.
[11] Mohamed, F. K., Aoued, B., “Speeding Up
Fractal Image Compression by Genetic
Algorithms”, Multidimensional Systems and
Signal Processing, Springer
ScienceþBusiness Media, Inc.,
Manufactured in The Netherlands, 2006,
Vol. 16, pp. 217–236.
[12] Fisher, Y., “Fractal Image Compression:
Theory and Application to Digital Images”,
Springer Verlag, 1995.
[13] Bouboulis, P., Dalla, P. L., Drakopoulos, V.,
“Image Compression Using Recurrent
Bivariate Fractal Interpolation Surfaces”,
Internat J. Bifur. Chaos, 2006, Vol. 16, No.
7, pp. 2063-2071.
[14] Lon Lin, Y., “Robust Estimation of
Parameters for Fractal Inverse Problem”,
Computers and Mathematics with
Applications, 2010, Vol. 60, pp. 2099–2108.
[15] Nadira, A. R., Kamal, B., Priyanga, P.,
“Iteration Free Fractal Compression Using
Genetic Algorithm for Still Color Image”,
Ictal Journal on Image and Video
Processing, 2014, Vol. 4, No. 03.
[16] Al-Bundi S. S., Al-Saidi N. M. G., and Al-
Jawari N., J.: “Crowding Optimization
Method to Improve Fractal Image
Compressions Based Iterated Function
Systems”, (IJACSA) International Journal of
Advanced Computer Science and
Applications, 2016, Vol. 7, No. 7.
[17] Geem, Z. W., Kim, J. H., Loganathan, G.,
"A New Heuristic Optimization Algorithm:
Harmony Search" Simulation, 2001, Vol.
76, pp. 60-68.
[18] Geem, Z. W., “Recent Advances in
Harmony Search Algorithm”, Studies on
Computational Intelligence, 2009, Vol. 270,
pp. 1-9.
[19] El-Satawy, M. F., Ahmed, A. N., “A Multi-
Objective Chaotic Harmony Search
Technique for Structural Optimization”,
International Journal of Computing Science,
2012, Vol. 1, No. 3.
[20] Di, J., Wang, N., “Harmony Search
Algorithm with Chaos for Training
RBFNN”, 2013, pp. 2231-2237.
[21] del-Raoufet, O., Abdel-Baset, M., El-
Henawy, I., “Improved Harmony Search
with Chaos for Solving Linear Assignment
Problems”, I.J. Intelligent Systems and
Applications, 2014, pp. 55-61.
[22] Abdel-Raoufet, O., Abdel-Baset, M., El-
Henawy, I., “A Novel Hybrid Flower
Pollination Algorithm with Chaotic
Harmony Search for Solving Sudoku
Puzzles”, I.J. Modern Education and
Computer Science, 2014, pp. 38-44.
[23] Gao, X. Z., Govindasamy, V., Xu, H.,
Wang, X., Zenger, K. “Review Article:
Harmony Search Method: Theory and
Applications”, Hindawi Publishing
Corporation Computational Intelligence and
Neuroscience, 2015, Vol. 2015.
[24] Al-Saidi N. M. G., Ali A. H.: Towards
Enhancing of Fractal Image Compression
Performance via Block Complexity. IEEE
Conference- NTICT 2017, 7-9 March, 2017.
[25] Al-Saidi N. M. G., Al-Bundi S. S. and Al-
Jawari1 N. J., A Hybrid of Fractal Image
Coding and Fractal Dimension for Efficient
Retrieval Method. The Computational and
Applied Mathematics, Springer 2016.
[26] Barnesly, M. F., "Fractal Everywhere",
Second edition, Academic Press, 1988.
[27] Hutchinson, J.E., “Fractals and Self-
similarity”, Indiana University journal of
mathematics, 1981, Vol. 30. No.5, pp. 713-
747.
[28] Barnsley, M. F., Ervin, V., Hardin, D.,
Lancaster, J., “Solution of an Inverse
Problem of Fractals and Other Sets”,
Proceeding of the National Academy of
Science, 1986, Vol. 83, pp. 1975-1977.
[29] Kennedy, J., Eberhert, R., “Particle Swarm
Optimization”, Proc. IEEE International
Conf. on Neural Networks (Perth,
Australia), IEEE Service Center,
Piscataway, 1995.
[30] Lin, Y., Chen, W., “Fast Search Strategies
for Fractal Image Compression”, Journal of
Information’s science and Engineering,
2012, Vol. 28, pp. 17-30.
[31] Venkatasekhar, D., Aruna, P., Parthiban, B.,
“Fast Search Strategies Using Optimization
for Fractal Image Compression”,
International Journal of Computer and
Information Technology, 2013 Vol.2 No. 3.
[32] Tseng , C., Hsieh, J., Jeng, J., “Fractal Image
Compression Using Visual-based Particle
Swarm Optimization”, Image and Vision
Computing, 2008, Vol. 26, pp. 1154–1162.
[33] Romero, V., Tomes, L., Yusiong, J., “Tetris
Agent Optimization Using Harmony Search
Algorithm”, International Journal of
Journal of Theoretical and Applied Information Technology
30
th
April 2017. Vol.95. No 8
© 2005 – ongoing JATIT & LLS
ISSN:
1992-8645
www.jatit.org E-ISSN:
1817-3195
1679
Computer Science Issues, 2011,Vol. 8, No.
1, pp. 22-31.
[34] Kennedy, J., Eberhert, R., “Particle Swarm
Optimization”, Proc. IEEE International
Conf. on Neural Networks (Perth,
Australia), IEEE Service Center,
Piscataway, 1995.
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The storage requirements for images can be excessive, if true color and a high-perceived image quality are desired. An RGB image may be viewed as a stack of three gray-scale images that when fed into the red, green and blue inputs of a color monitor, produce a color image on the screen. The abnormal size of many images leads to long, costly, transmission times. Hence, an iteration free fractal algorithm is proposed in this research paper to design an efficient search of the domain pools for colour image compression using Genetic Algorithm (GA). The proposed methodology reduces the coding process time and intensive computation tasks. Parameters such as image quality, compression ratio and coding time are analyzed. It is observed that the proposed method achieves excellent performance in image quality with reduction in storage space.
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The author proposes an independent and novel approach to image coding, based on a fractal theory of iterated transformations. The main characteristics of this approach are that i) it relies on the assumption that image redundancy can be efficiently exploited through self-transformability on a block-wise basis, and ii) it approximates an original image by a fractal image. The author refers to the approach as fractal block coding. The coding-decoding system is based on the construction, for an original image to encode, of a specific image transformation--a fractal code--which, when iterated on any initial image, produces a sequence of images that converges to a fractal approximation of the original. It is shown how to design such a system for the coding of monochrome digital images at rates in the range of 0.5-1.0 b/pixel. The fractal block coder has performance comparable to state-of-the-art vector quantizers.