A local search operator in Quantum Evolutionary Algorithm and its application in Fractal Image Compression

Abstract

Fractal Image Compression is an optimization problem in the class of NP-Hard problems. Quantum Evolutionary Algorithm is a novel optimization algorithm proposed for class of combinatorial problems like Knapsack problem. While QEA is highly suitable for NP-Hard problems, QEA is not widely used in Fractal Image Compression. In order to improve the performance of QEA in Fractal Image Compression, this paper proposes a local search operator for QEA. The proposed algorithm uses Simulated Annealing algorithm in its search process. The SA is performed on observed possible solution to help the algorithm escaping from local optima. The proposed Simulated Annealing Quantum Evolutionary Algorithm (SAQEA) for fractal image compression is tested on several images like Lena, Pepper and Baboon for several times and is compared with QEA and GA. Experimental results show better performance for the proposed algorithm than QEA and GA and in comparison with full search, the proposed algorithm reaches suitable solutions with much less computation complexity.

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A Local Search Operator in Quantum Evolutionary Algorithm and its Application
in Fractal Image Compression

Hossein Mohamadi
Islamic Azad University
Azadshahr Branch
H_mohamadi@hotmail.com
Ali Nodehi
Islamic Azad University
Gorgan Branch
alinodehi@gorganiau.ir


Mohamad Tayarani
Islamic Azad University
Mashhad Branch
tayarani@ieee.org
Abstract—Fractal Image Compression is an optimization
problem in the class of NP-Hard problems. Quantum
Evolutionary Algorithm is a novel optimization algorithm
proposed for class of combinatorial problems like Knapsack
problem. While QEA is highly suitable for NP-Hard problems,
QEA is not widely used in Fractal Image Compression. In
order to improve the performance of QEA in Fractal Image
Compression, this paper proposes a local search operator for
QEA. The proposed algorithm uses Simulated Annealing
algorithm in its search process. The SA is performed on
observed possible solution to help the algorithm escaping from
local optima. The proposed Simulated Annealing Quantum
Evolutionary Algorithm (SAQEA)
compression is tested on several images like Lena, Pepper and
Baboon for several times and is compared with QEA and GA.
Experimental results show better performance for the
proposed algorithm than QEA and GA and in comparison with
full search, the proposed algorithm reaches suitable solutions
with much less computation complexity.
for fractal image
Keywords- Fractal Image Compression, Quantum Evolutionary
Algorithm, Simulated Annealing.
I.
INTRODUCTION
QUANTUM Evolutionary Algorithm is a novel
optimization algorithm proposed
optimization problems. The probabilistic representation of
possible solutions in QEA makes the quantum individuals
able to represent all the search space simultaneously.
Nevertheless, like other evolutionary algorithms, QEA
suffers from trapping in local optima. There are many
approaches in evolutionary algorithms, trying to maintain the
diversity in evolutionary algorithms. In order to maintain the
diversity in QEA [1] proposes a diversity preserving operator
for QEA which is called DPCQEA. Reference [2] uses a
sinusoid sized population for QEA which preserves the
diversity of the population and improves the performance of
QEA. Reference [3] uses a diversity controlled GA for
frequency-response masking (FRM) FIR digital filters over
the double base number system multiplier coefficient space.
In order to overcome premature convergence in GA, [4]
proposes a novel adaptive genetic algorithm based on
diversity maintaining.
Several works try to improve the algorithm of fractal
image compression using Genetic algorithm. In [5] a new
method for finding the IFS code of fractal image is
for combinatorial
developed and the influence of mutation and the crossover is
discussed. The low speed of fractal image compression
blocks its way to practical application. In [6] a genetic
algorithm approach is used to improve the speed of searching
in fractal image compression. A new method for genetic
fractal image compression based on an elitist model in
proposed in [7]. In the proposed approach the search space
for finding the best self similarity is greatly decreased.
Reference [8] makes an improvement on the fractal image
coding algorithm by applying genetic algorithm. Many
researches increase the speed of fractal image compression
but the quality of the image will decrease. In [9] the speed of
fractal image compression is improved without significant
loss of image quality. Reference [10] proposes a genetic
algorithm approach which increases the speed of the fractal
image compression without decreasing of the quality of the
image. In the proposed approach a standard Barnsley
algorithm, the Y. Fisher based in classification and the
genetic compression algorithm with quad-tree partitioning
are compared. In GA based algorithm a population of
transformations is evolved for each range block. In order to
prevent the premature convergence of GA in fractal image
compression a new approach is proposed in [11], which
controls the parameters of GA adaptively. A spatial
correlation genetic algorithm is proposed in [12], which
speeds up the fractal image compression algorithm. In the
proposed algorithm there are two stages, first the spatial
correlations in image for both the domain pool and the range
pool is performed to exploit local optima. In the second stage
if the local optima were not certifiable, the whole of image is
searched to find the best self similarity. A schema genetic
algorithm for fractal image compression is proposed in
[13,17] to find the best self similarity in fractal image
compression.
In order to help QEA escaping from local optima, this
paper proposes a local search algorithm for QEA. In the
proposed algorithm, after convergence of q-individuals,
when the algorithm is trapped in local optima, a Simulated
Annealing algorithm is performed on possible solution, to
help the algorithm escaping from local optima. The SA
algorithm is performed on all the possible solutions, until the
algorithm escapes from the local optima. Then QEA starts its
searches until its convergence. The proposed algorithm is
used in fractal image compression and several experimental
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results show better performance for the proposed algorithm
than QEA and GA.
This paper is organized as follows: Section II discusses
about QEA, in Section III the SAQEA is proposed. In
Section IV, the proposed algorithm is applied on Fractal
Image Compression problem. Section V discusses about the
experimental results and finally section VI concludes the
paper.
II.
QEA
QEA is inspired from the principles of quantum
computation, and its superposition of states is based on
qubits, the smallest unit of information stored in a two-state
quantum computer. A qubit could be either in state “0” or
“1”, or in any superposition of the two as described below:
10
βαψ+=
(1)
Where ? and ? are complex number, which denote the
corresponding state appearance probability, following below
constraint:
1
= + βα
(2)
This probabilistic representation implies that if there is a
system of m qubits, the system can represent 2m states
simultaneously. At each observation, a qubits quantum state
collapses to a single state as determined by its corresponding
probabilities.
A. Representation
QEA uses a novel representation based on the above
concept of qubits. Consider i-th individual in t-th generation
defined as an m-qubit as below:
?
=
t
ii
i
i
βββ
21
Where
1||||
=+
ijij
βα
, j=1,2,…,m , m is the number
of qubits, i.e., the string length of the qubit individual,
i=1,2,…,n , n is the number of possible solution in
population and t is generation number of the evolution. Since
a qubit is a probabilistic representation, any superposition of
states is simultaneously represented. If there is, for instance,
a three-qubits (m = 3) individual such as (4):
?
?
=
2
3
2
Or alternatively, the possible states of the individual can
be represented as:
001 000
2262
++
22
?
?
?
?
?
?
?
t
im
t
im
t
ii
t
t
i
t
t
i
t
iq
β
αααα
... ...
21
(3)
22
tt
?
?
?
?
?
?
3
2
1
2
3
1
1
2
1
t
iq
(4)
111
100101 100
011010
2
1
32
1
22
1
62
1
2
1
32
111
+
+
+++=
t
iq
(5)
Note that the square of above numbers are true
probabilities, i.e. the above result means that the probabilities
to represent the state
001, 000
1/24 and 1/12 respectively. Consequently, the three-qubits
system of (4) has all eight states information at the same time.
010, 100,
are 1/24, 1/8,
Evolutionary computing with the qubit representation has
a better characteristic of diversity than classical approaches
since it can represent superposition of states. Only one qubit
individual such as (4) is enough to represent eight states,
whereas in classical representation eight individuals are
needed. Additionally, along with the convergence of the
quantum individuals, the diversity will gradually fade away
and the algorithm converges.
B. QEA Structure
In the initialization step of QEA,
initialized with
2
q0
with equal probability. The next step makes a set of binary
instants; xt
(
tQ
( )
},...,,...,,{
21
ni
xxxxtX
=
at generation t is a random instant
of qubit population. Each binary instant, xt
formed by selecting each bit using the probability of qubit,
either
|
ij
α
or
|
ij
β
give some measure of its fitness. The initial best solution
n
i
=
binary instants of X(t). Then, in ‘update Q(t),’ quantum gates
U update this set of qubit individuals Q(t) as discussed below.
This process is repeated in a while loop until convergence is
achieved. The appropriate quantum gate is usually designed
in accordance with problems under consideration.
The pseudo-code of QEA algorithm is defined as [4]:
Procedure QEA
begin
t=0
1. initialize Q(0).
2. make X(0) by observing the states of
Q(0).
3. evaluate X(0).
4. Store X(0) into B(0). Store the best
solution among X(0) into b.
5. while not termination condition do
Tt
ij
t
ij
][
βα
of all q0
i are
1
. This implies that each qubit individual
i represents the linear superposition of all possible states
i by observing
tt
}
,...,,
{
)
21
t
n
tt
qqq
=
states, where
tt
i of length m, is
2|
t
2|
t
of qt
i. Each instant xt
i is evaluated to
)}({ max
1
t
i
xfb
=
is then selected and stored from among the
begin
t=t+1
6.
make X(t) by observing the states of
Q(t-1)
evaluate X(t)
update Q(t) using Q-gates
store the best solutions among B(t-1)
and X(t) into B(t)
store the best solution among B(t)
into b
if global migration condition
then migrate b to B(t) globally
else if local migration condition
then migrate
locally
end
end

has a population
7.
8.
9.
10.
11.
12.
t
k b in B(t) to B(t)
QEA of quantum
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individuals ( )
n is the size of population.
A comprehensive description of QEA can be found in [4].
The QEA procedure is described as:
1. In the initialization step all qubits ?0
i=1,2,…,n and j=1,2,…,m are initialized with
the probability of observing "0" and "1" for all qubits is
equal.
2. In this step the binary solutions
at generation t=0 are creating by observing Q(0). Observing
xt
ijij
][
βα
?
<
=
otherwise1
Where
) , ( ⋅ ⋅
R
, is a uniform random number generator.
3. All solutions in X(t) are evaluated with fitness function.
4. Store X(0) into B(0). Select best solution among X(0)
and store it to b.
5. The while loop is running until termination condition is
satisfied. Termination condition can be considered as
maximum generation condition or convergence condition.
6. Observing X(t) from Q(t-1).
7. Evaluate X(t) by fitness function
8. Update Q(t)
9, 10. Store the best solutions among B(t-1) and X(t) to
B(t). If the fittest solution among B(t) is fitter than b then
store the best solution into b.
11, 12. If global migration condition is satisfied copy b to
all the solutions in B(t). If local migration condition is
satisfied replace some of solutions in B(t) with best one of
them.
C. Quantum Gates Assignment
The common mutation is a random disturbance of each
individual, promoting exploration while also slowing
convergence. Here, the quantum bit representation can be
simply interpreted as a biased mutation operator. Therefore,
the current best individual can be used to steer the direction
of this mutation operator, which will speed up the
convergence. The evolutionary process of quantum
individual is completed through the step of “update Q(t).” A
crossover operator, quantum rotation gate, is described
below. Specifically, a qubit individual qt
the rotation gate U(?) in this algorithm. The j-th qubit value
of i-th quantum individual in generation t
updated as:
?
ΔΔ
?
?
?
?
Where ?? is rotation angle and controls the speed of
convergence and determined from Table I. Reference [14]
shows that these values for ?? have better performance.
},...,,{
21
t
n
tt
qqqtQ
=
, where t is generation step and
ij and ?0
2
. It means
ij,
1
} ,...,,{ ) 0 (
X
00
2
0
1
n xxx
=

ij from qubit
Ttt
is performed as below:
2
t
ij
(6)
??
??
) 1 , 0 (R0
t
ij
if
x
α
i is updated by using
Tt
ij
t
ij
][
βα
is
?
?
?
?
?
?
?
?
??
?
??
Δ
θ
−
cos(
Δ
=
?
?
?
?
+
+
t
ij
t
ij
t
ij
t
ij
β
αθ
)
θ
θ
β
α
)sin(
)sin(
)cos(
1
1
(7)
III. SIMULATED ANNEALING QEA
Evolutionary algorithms suffer from trapping in local
optima and early convergence and Quantum Evolutionary
Algorithm is not an exception. In QEA, q-individuals search
the search space and after some iterations trap in true states
of [0 1]T
much chance to escape from the local optima. Several works
have tried to proposed new methods to help the algorithm
searching other parts of the search space. This paper
proposes a novel local search algorithm based on Simulated
Annealing for Quantum Evolutionary Algorithms. In the
proposed algorithm in each iteration, the convergence of the
population is checked, and if the population converged, a
Simulated Annealing algorithm is performed on possible
solutions. The procedure of the proposed algorithm is as
follows:
The pseudo-code of SAQEA algorithm is defined as [4]:
Procedure SAQEA
begin
t=0
1. initialize Q(0).
2. make X(0) by observing the states of Q(0).
3. evaluate X(0).
4. Store X(0) into B(0). Store the best
solution among X(0) into b.
5. while not termination condition do
begin
or [1 0]T. In this condition, q-individuals have not
t=t+1
6.
make X(t) by observing the states of
Q(t-1)
evaluate X(t)
update Q(t) using Q-gates
store the best solutions among B(t-1)
and X(t) into B(t)
store the best solution among B(t)
into b
if global migration condition
then migrate b to B(t) globally
else if local migration condition
7.
8.
9.
10.
11.
12.
TABLE I.
LOOKUP TABLE OF θΔ
.
ix
ib
)()(
bfxf
≥

θ
Δ

0 0
false
0
0 0
true
0
0 1
false
π
01 . 0

0 1
true
0
1 0
false
π
01. 0
−
1 0
true
0
1 1
false
0
1 1
true
0
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13.
then migrate
locally
if the population is converged
perform SA on all the possible
solutions until the population
leaves the converged status
end
end
The proposed SAQEA is like QEA except in line 14. In
this step the convergence status of the population is
calculated. The convergence of the population is calculated
as [16]:
t
k b in B(t) to B(t)
14.
??
=
i
1
=
−
×
=
mn
j
ij
mn
C
1
2
21
1
α
(8)
Where C is the convergence of the population, m is the
size of population (the number of the q-individuals in the
population) and n is the number of q-bits in the q-individuals
(the dimension of the problem).
The population is converged if satisfy the below
constraint:
TABLE II.
EXPERIMENTAL RESULTS ON LENA, PEPPER AND BABOON
Population
Size
Full Search -
30
25
20
15
30
25
20
15
30
25
20
15
Full Search
-
30
25
20
15
30
25
20
15
30
25
20
15
Full Search
-
30
25
20
15
30
25
20
15
30
25
20
15

Picture Method
MSE
computations
59,474,944
6,144,000
5,120,000
4,096,000
3,072,000
6,144,000
5,120,000
4,096,000
3,072,000
6,144,000
5,120,000
4,096,000
3,072,000
59,474,944
6,144,000
5,120,000
4,096,000
3,072,000
6,144,000
5,120,000
4,096,000
3,072,000
6,144,000
5,120,000
4,096,000
3,072,000
59,474,944
6,144,000
5,120,000
4,096,000
3,072,000
6,144,000
5,120,000
4,096,000
3,072,000
6,144,000
5,120,000
4,096,000
3,072,000
PSNR
28.85
28.49
28.28
27.95
27.43
28.56
28.35
28.16
27.53
28.11
28.04
27.55
27.27
29.85
29.55
29.09
28.87
28.12
29.63
29.21
28.98
28.51
29.14
28.92
28.64
28.11
20.04
19.28
19.18
18.95
18.62
19.62
19.31
19.08
18.56
19.17
19.02
18.65
18.41
Lena
QEA
SAQEA
GA
Pepper
QEA
SAQEA
GA
Baboon
QEA
SAQEA
GA
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γα
2
>−
×
=
??
=
i
1
=
mn
j
ij
mn
C
1
2
1
1
(9)
If the population is converged the Simulated Annealing
algorithm is performed on all the possible solutions. The SA
algorithm is as follows:
Procedure Local search SA
begin
1.
T=
0
T
2.
t ? 0
3.
while not (termination condition) do
begin
4.
for all possible solutions xt
i in X(t)
begin
i)
i) and store the result in yt
i)
5.
6.
7.
8.
9.
10.
11.
12.
13.
1
E =evaluation (xt
perform change(xt
E =evaluation (yt
E
Δ =
2
E -
if E
Δ >0 then
xt
else
ep
=
if ) 0 , 1 (
Up <
i
2
1
E
i = yt
i
T
E
Δ
−

then xt
i = yt
end
i
14.
T=?×T
end
end
The termination condition in step 3, is when the
population escapes from convergence status. The termination
condition is as follows:
??
==
ij
11
Which means the population has escaped from local
optimum.
The step 5 performs a change on possible solutions. The
change is like mutation in Genetic Algorithms and is defined
as:
Procedure Change
begin
1. for all alleles xt
2.
if R(0,1)<mutation rate
3.
xt
end

γα
2
<−
×
=
mn
ij
mn
C
2
1
1
(10)
i,k in xt
i do
i,k=R(lk,uk)
IV. SAQEA FOR FRACTAL IMAGE COMPRESSION
The proposed SAQEA for fractal image compression
searches among the domain blocks to find the best domain
block and the best transformation for each range block. For
each range block, SAQEA searches among all the domain
pool to find the best domain block and the best
transformation. The coding method for the q-individuals in
the proposed method is as below:

px
py
pt

In the proposed approach each q-individual, has three
parts: px shows the horizontal position of domain block, py
shows the vertical position of the domain block and pt shows
the transformation. The transformations are the 8 ordinary
transformations: rotate 0°, 90°, 180°, 270°, flip vertically,
horizontally, flip relative to 45°, and relative to 135°. Each
part of each solution is a real number and is converted to an
integer number before evaluation process.
V.
EXPERIMENTAL RESULTS
This section experiments the proposed algorithm and
compares the proposed algorithm with the performance of
GA and QEA in fractal image compression. The proposed
algorithm is examined on images Lena, Pepper and Baboon
with the size of 256×256 and gray scale. The size of range
blocks is considered as 8×8 and the size of domain blocks is
considered as 16×16. In order to compare the quality of
results, the PSNR test is performed:
?
()()()?
?
?
?
?
?
?
?
?
?
?
?
?
−
×=
??
=
i
1
=
×
nm
j
nm
jigjif
1
2
1
2
,,
255
log10 PSNR

Where m×n is the size of image.
The crossover rate in GA is 0.8 and the probability of
mutation in GA and SAQEA is 0.003 for each allele. The
parameter ? in (9) and (10) is considered as 0.9, and ? is
considered as 0.95. Table II shows the experimental results
on the proposed algorithm and GA. The number of iterations
for GA, QEA and SAQEA for all the experiments is 200.
According to table II the proposed algorithm improves the
performance of fractal image compression for all the
experimental results.
VI. CONCLUSION
This paper proposes a Simulated Annealing QEA for
fractal image compression. In evolutionary algorithms during
the search process the diversity in the population is
decreased, and the algorithm is trapped in the local optima.
This is true for QEA which uses a probabilistic
representation for the individuals. Here we propose a novel
local reach operator for QEA to help the algorithm escaping
from the local optima. In the proposed algorithm, after the
convergence of population, Simulated Annealing algorithm
is performed on possible solutions until algorithm escapes
from local optima. The proposed SAQEA has some
parameters and this paper finds the best parameters for the
proposed algorithm. There are some other questions that
should be focused in our future works. The main question is
the best parameters of the proposed algorithm. As it is seen
the proposed algorithm has some parameters: ? and ?.
Finding the best parameters for the proposed algorithm will
be answered in our future researches. Finally experimental
results on Lena, Pepper, and Baboon picture show an
improvement on fractal image compression. The time
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