A multilevel thresholding algorithm for histogram-based image segmentation is presented in this paper. The proposed algorithm introduces an adaptive adjustment strategy of the rotation angle and a cooperative learning strategy into quantum genetic algorithm (called IQGA). An adaptive adjustment strategy of the quantum rotation which is introduced in this study helps improving the convergence speed, search ability, and stability. Cooperative learning enhances the search ability in the high-dimensional solution space by splitting a high-dimensional vector into several one-dimensional vectors. The experimental results demonstrate good performance of the IQGA in solving multilevel thresholding segmentation problem by compared with QGA, GA and PSO.
1. Introduction
Image segmentation which is considered as an important basic operation for computer vision is a low-level image processing task, and its result could be presented as input to higher-level processing tasks such as pattern recognition, object tracking, and scene analysis. Image segmentation is also a classical problem of image processing. Various techniques for image segmentation have been proposed and improved so far, but most of the existing segmentation methods are designed for specific applications. There’s neither a unified theoretical system for image segmentation nor a kind of common and effective approach to all types of images. For the advantages of simplicity of implementation, small amount of calculation, and stable performance, image thresholding technique becomes an effective and widely used tool in image segmentation [1, 2]. It is useful to separate objects from background or discriminate objects from objects that have distinct grey levels. Thresholding can be divided into bilevel thresholding and multilevel thresholding. Bilevel thresholding classifies the pixels into two groups (object and background), one including those pixels with gray levels above a certain threshold and the other including the rest. Multilevel thresholding divides the pixels into several classes. The pixels belonging to the same class have gray levels within a specific range defined by several thresholds [3]. Especially for the images of which target and background occupy different range of gray, the target and background which are with different gray levels can be separated by thresholding techniques, such as extraction of the logos, printed characters, or images in the document images [4]; segmentation of blood cell image and CT image in medical applications [5, 6]; and segmentation of infrared thermal image in infrared nondestructive detection [7]. The use of thresholding techniques has been a very crucial step for fast and accurate segmentation.
Thresholding methods are distinguished as six categories, based on the exploitation of histogram shape information, measurement space clustering, histogram entropy information, image attribute information, spatial information, and local characteristics, respectively, and then spread out dozens of objective functions [7]. Generally, existing image thresholding segmentation problem is transformed into an optimization problem: to propose an objective function which is based on the different information considered or different theory and then to obtain the optimal thresholds by getting the maximum or minimum value of the objective function. Among all the objective functions existing, the thresholding technique, the objective functions based on the entropy (Kapur method) [8], and the between-class variance (Otsu method) [9] are the most popular ways [10]. Otsu method can obtain a good segmentation result to the images with obvious bimodal histogram; Kapur method can also achieve a desired result to the images with no obvious bimodal or no bimodal histogram. However, due to the exhaustive searching mechanism, the increasing number of the thresholds leads computation time to grow exponentially which makes the above two methods become time-consuming issues, and it will greatly limit the application of the multilevel thresholding segmentation algorithm.
Image thresholding segmentation is also a task to find the optimal parameters in the complex parameter space, and the global optimal parameters of the space can be found through optimization algorithms in an acceptable period of time for their fast computational ability. Many algorithms are performed to multilevel thresholding segmentation to improve the computational efficiency [11–13].
Genetic algorithm (GA) provides a common system framework to solve complex optimization problems, and independent on the specific problem areas. GA has been applied to select the parameters in segmentation problem successfully and plays a very important role in image segmentation [14–18]. However, it falls into local optimum easily for the decrease of the diversity of the population in the later period of evolution [19].
Quantum-inspired genetic algorithm (QGA) is a new optimization algorithm which combines the concept of quantum computing and classical GA. QGA which was firstly introduced by Narayanan and Moore [20] is a new and promising branch of evolutionary algorithms. The experiment in [20] has demonstrated that QGA has a faster convergence speed than classical GA. In QGA, quantum-inspired bit (Q-bit) and quantum-inspired gate (Q-gate) are applied to represent genotype individuals and update Q-bits to generate offspring, and the genotypes and phenotypes are linked by a probabilistic observation process. The Q-bit encoding of chromosome of QGA has a better diversity of population, and the Q-gate which is defined as an update operator of QGA to drive the individuals toward better solutions can treat the balance between exploration and exploitation and avoid premature convergence and escape local optimal efficiently. Nowadays, the research on QGA has attracted the interest of many researchers of various research areas. QGA has been introduced to combinatorial optimization problems [21–23], numerical optimization problems [24], face detection [25], disk allocation [26], image segmentation [27, 28], feature selection [29], multiobjective optimization problems [30, 31], digital filter design [32], power dispatch [33], and so forth. The research on QGA has become a rapidly expanding field. QGA was introduced to image segmentation and achieved a faster speed of convergence than GA [34]. QGA was applied in [35] to determine the parameters and got better segmentation results than traditional fuzzy entropy-based method.
The convergence speed of QGA is dependent on the rotation angle greatly. An appropriate angle can help to accelerate the convergence speed and improve the searching ability of the algorithm. Many studies are interested in the research of the rotation angles and the update strategies [36]. Fixed rotation angles in lookup table are proposed in [21]; a Hε gate is employed in [22] to avoid premature convergence; the adjustments of the rotation angles which decrease gradually with the iterations are introduced in [37, 38]. Additionally, for high-dimensional optimization problem, the individual is an n-dimensional vector. In the processing of evolution, the update operation of Q-gate is only guided by the current best individual which has the maximum fitness but not always has a reasonable structure of the solution, and each update step is performed on a full n-dimensional vector. This leads to the possibility of some components of the n-dimensional vector having been moved closer to the best solution, while others have actually been moved away from the best solution [39]. This kind of holistic information exchange may cause the sharp decline of the search ability with the increase of the dimension. So it still can potentially get trapped in suboptimal locations in search space. In order to solve the above problems and improve the performance of the QGA-based multilevel thresholding segmentation, an improved QGA (IQGA) with adaptive rotation angle and cooperative learning for maximum entropy based multilevel thresholding segmentation is proposed in this paper. The experimental results show that the convergence speed, accuracy, and stability of the proposed algorithm are superior to traditional GA and QGA.
This paper is arranged as follows. Section 2 describes basic concept of the QGA and the improved QGA which introduced the adaptive adjustment strategy and cooperative learning strategy. Section 3 presents the maximum entropy criterion to multilevel thresholding segmentation. The detailed algorithm for IQGA-based image multilevel thresholding segmentation is presented in Section 4. The experimental results are given in Section 5. The conclusions are drawn in Section 6.
2. Improved Quantum Genetic Algorithm2.1. QGA
QGA is an attractive tool to provide efficient solutions for most complex optimization problems. It is a probabilistic optimization algorithm which introduces the concept and theory of quantum computing into classical GA. Like classical GA, QGA is characterized by the representation of individuals, population diversity, and the use of a fitness evaluation mechanism. However, a quantum-inspired bit (Q-bit) representation is used in QGA instead of binary, numeric, or symbolic representation which is usually used in GA. Since the Q-bit representation can achieve a linear superposition of states given its probabilistic approach, it is conductive to population diversity. A Q-bit is the basic computing unit in a QGA and is defined as a column vector: (αβ)T (a Q-bit is often represented as |ψ〉=α|0〉+β|1〉 in quantum mechanical ket-notation), where the numbers of α and β satisfy the normalization condition |α|2+|β|2=1. The values of |α|2 and |β|2 denote the probabilities that the Q-bit will be found in the states of “0” and “1,” respectively. By a process of probabilistic observation, each Q-bit can be rendered into one binary bit. A multi-Q-bit system can be extended naturally. For example, a m-Q-bits system can be described as follows:
(1)qi=(α1α2⋯αmβ1β2⋯βm).
This representation has the advantage that it is able to represent any superposition of states. The combination of the m-Q-bits is called quantum chromosome. GA uses the operations such as selection, crossover, and mutation, to maintain the diversity of the population and to obtain the global optimal solution. However, QGA applies quantum-inspired rotation gate (Q-gate) U(Δθi) to update the state of Q-bits only. The Q-gate is represented as follows:
(2)U(Δθi)=(cos(Δθi)-sin(Δθi)sin(Δθi)cos(Δθi)).
The Q-gate U(Δθi) is a unitary operation which is used to change the phase of the Q-bit and does not change the length of the Q-bit. The Q-gate is employed to update the state of a Q-bit as follows:
(3)(αi′βi′)=U(Δθi)(αiβi)=(cos(Δθi)-sin(Δθi)sin(Δθi)cos(Δθi))(αiβi),
where (αi,βi)Tand (αi′,βi′)T are the ith Q-bits of the chromosome before and after the updating, respectively. Δθi represents the rotation angle of each Q-bit whose value and direction can be adjusted by some strategies.
2.2. Adaptive Adjustment Strategy of the Q-Gate
In QGA, the evolution of the individual is guided by the current best individual and adjusted by the Q-gate, so that the individual approximates and converges to the global optimal solution ultimately. A Q-gate is used to change the state of the quantum chromosome to evolve the individual. In the processing of evolution, the choice of the quantum rotation angles is very important, and a suitable choice of the rotation angles can help improving the search ability of the algorithm. Different design of the quantum rotation angle is suitable for different problem. Although the designs of the quantum rotation angle are different, the core idea of the designs is the same—to make the current individual evolve to a higher fitness solution.
Considered the feature of the problem and the relation between the current individual and the current best individual, an adaptive rotation angle is proposed in this study, and the rotation angle is defined as follows:
(4)Δθi=g·Δθ,
where the value of Δθi can be looked up in Table 1.
Lookup table of θi.
xi
bi
f(xi)≥f(bi)
Δθ
0
0
False
0
0
0
True
0
0
1
False
0.01π
0
1
True
0
1
0
False
-0.01π
1
0
True
0
1
1
False
0
1
1
True
0
g is an adjustment function which is defined as
(5)g=exp(fbest-fifbest-fworst),
where fi, fbest, and fworst represent the fitness of the current individual, the current best individual, and the current worst individual, respectively. The value of f reduces when fi approach to fbest in order to decrease the search step length and vice versa. The rotation angle is updated by
(6)θi′=θi+Δθi.
2.3. Cooperative Learning Strategy
The classical analysis on GA considers that the population represents a set of competing structures that explore various parts of the search space in parallel. Instead of emphasizing the notion of competition, Cobb regarded population as a set of cooperative agents, and the members are cooperative in that they share information; that is, they communicate partial solutions to one another [40]. Without sharing information, the time required for an agent to find a solution may be very long. It is believed that a group of cooperating agents engaged in problem solving can solve a task faster than either a single agent or the same group of agents working in isolation from each other [41]. A general model was presented for coevolution of cooperating species to improve the performance of GA [42]. A cooperative method was introduced to particle swarm optimization (PSO) to help the particle contribute its merits to the population in each dimension and to promote the cooperation between particles in each dimension [39].
As represented by the quantum probability model in QGA, each individual of the population, regardless of its fitness value being high or low, represents a potential solution which evolves towards the global optimum guided by the current best individual. The relationship among individuals is not only competition but also cooperation. That is to say, the information sharing and exchange exist in QGA also, but the information sharing and exchange are between the individual and the current best individual. The current best individual has the maximum fitness value, but, for high-dimensional optimization problem, the structure of the solution of the current best individual is not always reasonable for each individual is usually represented as an n-dimensional vector. This means that the same update operation performed on each dimensional of an individual may not always evolve the individual. Therefore, it is important to take into account not only the whole adjustment of the individual, but also the rationality of the structure of the solution. And it is an effective way to improve the global and local optimization ability of QGA to reinforce the holistic and local information exchange and promote the information sharing among individuals, especially for the high-dimensional optimization. In other words, updating a part of an individual instead of updating it as a whole item sometimes may help accelerate the evolution processing and improve the search capability of QGA.
Image multilevel thresholding segmentation is a high-dimensional optimization problem, and it is time-consuming. Applying QGA to multilevel thresholding can reduce computational time greatly. The individual in QGA-based method is an n-dimensional vectors, and the performance of QGA declines with the increase in the dimensions. In order to improve the performance of QGA, a cooperative learning strategy is proposed in this study. The n-dimensional vector is split into n one-dimensional vectors, and then each one-dimensional vector is updated in the direction of the current best individual. This mean is called cooperative learning in this study. The cooperative learning strategy is presented as follows.
Assuming an n-dimensional unconstrained optimization which needs to maximize the objective function f(x),
(7)X*=(x1*,x2*,…,xn*)=argmax(X),
where n represents the dimension of the parameters, X*=(x1*,x2*,…,xn*) is the global optimal solution, and X is the collection of the candidate solutions.
The QGA which introduced cooperative learning strategy is performed to this optimization problem. Assuming qbest=(qb1,…,qbj,…,qbn) is the current optimal n-dimensional individual with fitness fbest, some n-dimensional individuals Q={q1,…,qi,…,qm} are selected based on the fitness proportional model. For every individual qi=(qi1,…,qij,…,qin) with fitness fi, each dimensional component of qi is replaced by the corresponding dimensional component of qbest, and its fitness value fij is computed. The best individual with the maximum fitness fibest′=max{fi1,…,fij,…,fin} is selected, and then its fitness fibest′ is selected to be compared with fi, if fibest′>fi, qi is replaced by qibest′; otherwise, qi is retained. By means of this way, the structures of the individuals are readjusted. This method accelerates the global convergence. The pseudocode of the cooperative learning strategy is shown in Pseudocode 1.
<bold>Pseudocode 1: </bold>Pseudocode for cooperative learning strategy.
Choose mn-dimensional individuals
Repeat:
For each individual qi with fitness fii∈[1,m]
For each dimension j∈[1,n]
qij=(qi1,…,qbj,…,qin)
Evaluate qij and store its fitness fij
Endfor
fbest′=max(fij)
qibset′=arg max(fij)
Iffbest′>fi
qi=qibest′
Endif
Endfor
Until termination criterion is satisfied
3. Entropy Criterion Based Measure
An image which contains N pixels and G gray levels is divided into M categories (C1,C2,…,Ck,…,CM) by T(T={t1,t2,…,tk,…,tM-1}, t1<t2<⋯<tk<⋯<tM-1) thresholds. For convenience, the lowest gray gmin and the highest gray gmax are set as the lowest threshold t0 and the highest threshold tM, respectively. If a pixel with gray g meets the condition tk-1<g≤tk+1, the pixel is classified under the category Ck. Searching for the optimal threshold which met the given classification requirements in the histogram, we can maximize or minimize an objective function which chooses the threshold as its parameter. That is to say, thresholding problem can be seen as a global optimization problem to choose a collection of thresholds T* by optimization of an objective function f(T). This can be described as follows:
(8)T*=argmax0≤T≤G-1f(T).
Based on the entropy criteria, Kapur method looks the object and background of the image as two different sources, and when the total entropy of the two categories is maximum, the best segmentation and its corresponding optimal threshold are considered to have been obtained. Kapur method was originally used to solve the bilevel thresholding segmentation and later extended to the multithresholding segmentation. Kapur method works as follows.
For an image with G gray levels, the number of the pixels with gray g is h(g), and the total number of the pixels of the image is N=∑g=0G-1h(g); the normalized probability of each gray level is defined as p(g)=h(g)/N. Determining M-1 thresholds for a given image, the Kapur entropy can be defined as:
(9)f(t1,…,tk,…,tM-1)=H0+H1+⋯+HM-1,
where
(10)ω0=∑g=0t1-1pg,H0=-∑g=0t1-1pgω0lnpgω0ω1=∑g=0t2-1pg,H0=-∑g=t1t2-1pgω1lnpgω1⋮ωM-1=∑g=tM-1G-1pg,HM-1=-∑g=tM-1G-1pgωM-1lnpgωM-1,
where H0,H1,…,HM-1 represent the histogram entropy of each category. The best thresholds T*={t1*,…,tk*,…,tM-1*} based on maximum entropy are satisfied with
(11)T*=argmaxTf(t1,…,tk,…,tM-1).
Kapur method can obtain the best thresholds T with the maximum entropy by computing and comparing the entropy values of all categories. Its time complexity is O(GM-1). Obviously, the computation complexity and calculation time will increase greatly with the increase in the threshold numbers that limits the implementation and generalization of the Kapur method. IQGA proposed in this study can solve this problem in an acceptable time.
4. IQGA for Image Multilevel Thresholding Segmentation
The proposed algorithm uses IQGA to solve image multilevel thresholding segmentation problem, and the details are described in this section.
4.1. Quantum Angle Encoding
A Q-bit can be represented as |ψ〉=α|0〉+β|1〉,where the numbers of α and β satisfy the normalization condition |α|2+|β|2=1. Make α=cosθ and β=sinθ, so a Q-bit can be represented by a phase angle θ in a two-dimensional space. The phase angle θ which is called quantum angle is represented by (θ). (θ) is equivalent to |ψ〉=cos(θ)|0〉+sin(θ)|1〉 or (cos(θ),sin(θ))T. A quantum chromosome with length of m can be represented as q=(θ1∣θ2∣⋯∣θm). The update of the quantum chromosome can be simplified to the following expression: θ′=θi+Δθ.
A gray image includes 256=28 gray levels in the range of [0,255], and each gray level can be coded by an 8-Q-bits quantum angle qi=(θi1,θi2,…,θij,…,θi8),θij∈[0,2π]. If an image is divided into M classes by M-1 thresholds, the corresponding chromosome Qi is composed by L=8×(M-1) bits quantum genes:
(12)Qi=(q1,q2,…,qi,…,q(M-1))=(θ11,θ12,…,θ1j,…,θ18,…,θi1,θi2,…,θij,…,θi8,…,θ(M-1)1,θ(M-1)2,…,θ(M-1)j,…,θ(M-1)8),
where cosθij,sinθij represent the probability amplitude of 0 and 1. The binary number of the gray value can be obtained by quantum measure.
4.2. Crossover
It is necessary to introduce crossover in the algorithm to increase the diversity of the population and avoid premature convergence. Two-point crossover with the crossover probability Pc is adopted as follows: choose two chromosomes with length L and select l(l∈[1,L]) cross-bits. For example, select a pair of chromosomes qi,qj randomly:
(13)qi=(θi1,θi2,…,θi(m-1),θim,…,θi(n-1),θin,…,θ8×(M-1)-1,θ8×(M-1)),qj=(θj1,θj2,…,θj(m-1),θjm,…,θj(n-1),θjn,…,θ8×(M-1)-1,θ8×(M-1)),
and generate cross-bits m,n(0<m,n≤8×(M-1)); the crossed chromosomes are presented as
(14)qi′=(θi1,θi2,…,θi(m-1),∣θjm,…,θj(n-1),∣θin,…,θi[8×(M-1)-1],θi[8×(M-1)]),qj′=(θj1,θj2,…,θj(m-1),∣θim,…,θi(n-1),∣θjn,…,θj[8×(M-1)-1],θj[8×(M-1)]).
4.3. Mutation
Mutation imitates the phenomenon of a certain gene mutation on chromosome in the processing of evolution. Mutation changes the structure and the physical properties of the chromosome. The operation of mutation in QGA can increase the diversity of the population effectively. This study introduces the NOR-gate and Hadamard-gate to the mutation operation. The proposed mutation is performed as follows: select one or several gene bits randomly on a chromosome with mutation probability Pm, and update the gene bits by NOR-gate or Hadamard-gate randomly. The operation is performed by
(15)θi′={π2-θi,NOR-gate
π4-θi,Hdamard-gate.
5. Experimental Results
In this section, the performance of the proposed algorithm is evaluated by comparing its results with QGA which using a fixed look-up table and full cross-interference scheme [43], PSO [13] and GA [15] which using a fixed lookup table and full cross-interference scheme [43]. In order to compare the performances of the algorithms, all algorithms (PSO, GA, QGA, and IQGA) adopted the same objective function which is based on maximum entropy criterion. The parameters are set as follows: population=20, crossover probability Pc=0.2, and mutation probability Pm=0.1. The pseudocode for the proposed algorithm is shown in Pseudocode 2.
<bold>Pseudocode 2: </bold>Pseudocode for the IQGA.
Begin
g←0
Initialize quantum chromosome population Q(g)
Make P(g) by observing the states of Q(g)
Evaluate P(g)
Store the best solution among P(g) into B(g)
While termination criteria are not satisfied do
g←g+1
Make P(g) by observing the states of Q(g)
Evaluate P(g)
Store the best solution among P(g) into B(g)
Update Q(g) using Q-gate
Choose m individuals and perform cooperative learning strategy
Crossover
Mutation
Endwhile
End
Some popular images are selected in the experiment. The original test images along with their corresponding gray level histogram are shown in Figures 1 and 2.
Gray level histogram of the test images. (a) Cameraman; (b) Hunter; (c) Baboon; (d) Scene; (e) Tree; (f) F16; (g) Lena; (h) Peppers.
Table 2 shows the optimal segmentation thresholds, the maximum values of the objective function, and the average computation times obtained by standard Kapur method. Since the standard Kapur method is an exhaustive algorithm, the results of the method can be used to be the standard evaluation criteria, and the results obtained by other algorithms can be evaluated by this criterion. The letters “T” and “f” represent the optimal objective function values and the corresponding segmentation thresholds, respectively. As can be seen from Table 2, the average computational time increases from 9.5236 seconds to 19428 seconds exponentially when C changes from 2 to 4. The exhaustive method is a time-consuming task, and it is almost difficult to apply the method to obtain the optimal thresholds while increasing the number of the thresholds. Otherwise, the QGA-based multilevel thresholding method can reduce the computational time to several seconds.
Optimal segmentation thresholds, objective function values, and the average computation times of the standard ES-Kapur method with C=2,3,4.
Image
Segmentation results
C
2
3
4
Cameraman
T
128-193
44-104-193
44-97-146-197
f
12.1687
15.2274
18.3955
Hunter
T
91-172
62-117-174
62-117-174-230
f
12.6357
15.8115
18.7441
Baboon
T
79-143
44-98-152
33-74-114-159
f
12.2168
15.2801
18.1304
Scene
T
92-163
73-120-170
71-113-157-196
f
12.5254
15.5654
18.3681
Tree
T
86-160
79-129-178
76-118-157-196
f
12.1677
15.3271
18.0325
F16
T
71-173
69-127-183
67-106-145-185
f
12.2115
15.5038
18.3120
Lena
T
97-164
82-126-175
44-98-152
f
12.3471
15.3183
18.0126
Peppers
T
93-179
73-126-178
46-84-130-179
f
12.5532
15.8258
18.7338
Average computation times (s)
9.5236
354.3866
19428.6
5.1. Comparison of the Computation Speed
An experiment is performed to contrast the convergence speed and the average computational time of the GA, QGA, particle swarm optimization (PSO), and the proposed algorithms. Computation ceased when the value of the objective fi meets the termination condition: Dis=|fi-fmax|/fmax≤ε. Every algorithm is executed 100 times and yielded the following results presented in Table 3. The letters “t” and “g” in Table 3 represent the mean computational time and the average convergence generation, respectively. Data in this table is the average of 100 times repeated experiments of the test images. The results show that the convergence generation of the proposed algorithm is much smaller than other methods, and the computational speed of the proposed algorithm is faster than other methods.
Comparison of the mean computational time and the average convergence generation.
ε
C
GA
QGA
IQGA
PSO
t (s)
g
t (s)
g
t (s)
g
t (s)
g
1%
2
0.1248
2
0.2405
3.3
0.1831
1.6
0.1955
3.1
3
0.1456
5.3
0.5722
10.2
0.2627
2.2
0.3588
8.7
4
2.2828
31
1.074
16.6
0.5541
6.4
2.7550
19.1
0.5%
2
0.2092
5.0
0.3291
5.8
0.2524
2.6
0.2241
3.4
3
0.8652
18.5
0.6489
12.3
0.3335
4.8
0.5647
19.7
4
4.2961
37.2
1.4944
24.6
0.9776
11.4
3.4690
33.4
5.2. Comparison of the Search Ability and Stability
In order to test the search ability and stability of the proposed algorithm, some experiments have been implemented as follows. The terminal condition of the algorithms is changed to compare the stability and the search ability of the algorithm. Iteration will be terminated when |f-t-f-t-1|<ε (f-t presents the average of the fitness at generation t, ε=0.0001) or reach to the maximum number of iterations (max_gen=500). Since all the compared methods except ES-Kupar in this study are based on stochastic search mechanism, every experiment except ES-Kupar runs 100 times and the performance will be evaluated by the results of the 100-times experiments.
The maximum objective function values and the optimal thresholds with C=2,3,4 are shown in Tables 4 and 5, respectively. As can be seen from the tables, QGA and IQGA have obtained all optimal solutions when C=2, while PSO and GA have obtained most optimal solutions; with the increasing numbers of the segmentation threshold, the ability of the PSO and GA decreased sharply, QGA can search most of the global optimal solutions, and IQGA can obtain all the global optimal solutions. It can be seen that the search ability outperforms the PSO, GA, and QGA. The optimal thresholded images which are obtained by the proposed algorithm are shown in Figure 3.
Comparison of the objective function values with C = 2, 3, 4.
Image
C
Objective function values
GA
QGA
IQGA
PSO
Cameraman
2
12.1687
12.1687
12.1687
12.1687
3
15.2174
15.2273
15.2273
15.1733
4
18.3864
18.3949
18.3955
18.3902
Hunter
2
12.6333
12.6356
12.6356
12.6333
3
15.8115
15.8020
15.8115
15.8020
4
18.7101
18.7421
18.7441
18.7374
Baboon
2
12.2167
12.2167
12.2167
12.2167
3
15.2772
15.2789
15.2801
15.2782
4
18.1263
18.1278
18.1304
18.1294
Scene
2
12.5244
12.5253
12.5253
12.5248
3
15.5618
15.5654
15.5654
15.5253
4
18.3606
18.3631
18.3681
18.3647
Tree
2
12.1671
12.1676
12.1676
12.1676
3
15.3270
15.3270
15.3270
15.3270
4
18.0272
18.0300
18.0325
18.0246
F16
2
12.2114
12.2114
12.2114
12.2114
3
15.5022
15.5038
15.5038
15.5028
4
18.3082
18.3104
18.3120
18.2972
Lena
2
12.3469
12.3471
12.3471
12.3471
3
15.3174
15.3174
15.3183
15.3168
4
17.9775
18.0079
18.0126
18.0061
Peppers
2
12.5311
12.5532
12.5532
12.5532
3
15.7133
15.8116
15.8258
15.8205
4
18.7338
18.7338
18.7338
18.7334
Comparison of the optimal thresholds with C = 2, 3, 4.
Image
C
Optimal thresholds
GA
QGA
IQGA
PSO
Cameraman
2
128-193
128-193
128-193
128-193
3
65-129-193
44-104-193
44-104-193
48-115-193
4
42-96-143-197
44-96-146-197
44-97-146-197
43-96-146-197
Hunter
2
91-174
92-172
91-172
91-174
3
62-117-174
62-118-173
62-117-174
62-118-174
4
68-126-177-230
64-120-174-230
62-117-174-230
67-127-173-230
Baboon
2
79-143
79-143
79-143
79-143
3
44-100-152
44-95-150
44-98-152
44-96-150
4
33-71-113-160
33-74-115-162
33-74-114-159
33-74-113-159
Scene
2
94-164
92-163
92-163
94-163
3
76-124-172
73-120-170
73-120-170
76-120-172
4
72-115-161-198
69-111-160-197
71-113-157-196
69-113-159-197
Tree
2
86-159
86-160
86-160
86-160
3
79-129-178
79-129-178
79-129-178
79-129-178
4
75-115-153-191
76-120-160-197
76-118-157-196
76-120-157-193
F16
2
71-173
71-173
71-173
71-173
3
69-127-184
69-127-183
69-127-183
69-125-183
4
67-108-148-186
64-103-143-186
67-106-145-185
64-106-147-186
Lena
2
98-164
97-164
97-164
97-164
3
78-123-165
83-128-176
82-126-175
82-128-176
4
65-97-137-185
63-96-135-179
64-97-138-179
64-96-135-180
Peppers
2
92-177
93-177
93-179
92-177
3
73-126-178
74-125-181
73-126-178
73-125-177
4
46-84-129-179
46-84-130-179
46-84-130-179
45-83-130-179
Thresholded images. Images from top to bottom rows are thresholded images with C=2,3,4, respectively. (a) Cameraman; (b) Hunter; (c) Baboon; (d) Scene; (e) Tree; (f) F16; (g) Lena; (h) Peppers.
Table 6 represents the variance and the mean values of the objective functions. It can be seen that the variance values increase with the increasing numbers of the threshold; that is to say, the stability of the algorithms declines with the increasing number of the threshold. In the cases of the variance, IQGA is smaller than other algorithms by more than two orders of magnitude. Utilizing the quantum superposition and coherence, QGA maintains the diversity of the population better than classical GA and PSO, so QGA has a stronger performance than GA. The strategy which introduces cooperative learning method to QGA promotes the local information exchanges among the individuals and enhances the ergodicity of the solutions. Therefore, IQGA has better search performance and stronger stability than QGA.
Comparison of the value of mean and variance with C = 2, 3, 4.
Image
C
GA
QGA
IQGA
PSO
Mean
Var.
Mean
Var.
Mean
Var.
Mean
Var.
Cameraman
2
12.1141
1.1630e-02
12.1513
2.0682e-03
12.1684
1.5723e-7
12.1214
1.1307e-02
3
15.0477
3.8718e-02
15.1642
1.1989e-03
15.2254
1.0641e-5
15.0725
2.1288e-02
4
18.1732
5.3940e-02
18.2751
1.1967e-02
18.3881
2.9388e-5
18.2372
3.9943e-02
Hunter
2
12.5949
5.4531e-03
12.6115
5.8455e-04
12.6355
1.5758e-7
12.6019
4.3691e-03
3
15.6759
1.6188e-02
15.7445
1.2617e-03
15.8088
9.0123e-6
15.7206
1.8004e-02
4
18.5733
1.4035e-02
18.5996
4.7036e-03
18.7372
7.3949e-5
18.5823
9.9633e-03
Baboon
2
12.1810
7.7419e-04
12.2109
4.2185e-05
12.2165
2.9464e-8
12.1907
5.9346e-04
3
15.2305
2.8331e-03
15.2550
5.4095e-04
15.2785
2.5557e-6
15.2394
2.8012e-03
4
17.9545
2.6872e-02
18.0479
4.8961e-03
18.1219
2.8246e-5
17.8392
2.9964e-02
Scene
2
12.5163
1.7481e-05
12.5214
1.7481e-05
12.5252
2.8823e-8
12.5170
1.6997e-05
3
15.4991
5.0432e-03
15.4989
1.8712e-03
15.5629
5.0827e-5
15.4889
4.3875e-03
4
18.2521
1.0527e-02
18.2742
3.3150e-03
18.3587
1.6565e-4
18.2604
9.4887e-03
Tree
2
12.1499
1.1541e-03
12.1644
6.4341e-06
12.1674
3.7896e-8
12.1532
7.8016e-04
3
15.2076
1.8796e-02
15.2626
2.3912e-03
15.3252
2.3026e-6
15.2360
1.3356e-02
4
17.8916
1.0394e-02
17.9567
2.9546e-03
18.0293
4.7901e-6
17.9001
1.0041e-02
F16
2
12.1925
5.9577e-04
12.2027
5.0603e-05
12.2113
1.3833e-7
12.1977
3.6732e-04
3
15.3711
1.9190e-02
15.4379
2.1738e-03
15.5019
3.0649e-6
15.4136
2.0084e-03
4
18.2093
5.3186e-03
18.2459
2.1708e-03
18.3039
6.6394e-5
18.2209
4.3377e-03
Lena
2
12.3411
1.1129e-04
12.3352
1.0222e-04
12.3467
1.1837e-06
12.3397
1.0966e-04
3
15.2064
1.5275e-03
15.2950
3.5576e-04
15.3166
3.4162e-06
15.2399
1.0054e-03
4
17.9582
3.6579e-02
17.8461
1.4690e-02
17.9949
2.7414e-04
17.8327
2.1767e-02
Peppers
2
12.5311
1.2613e-03
12.5410
5.4765e-04
12.5521
3.4546e-06
12.5395
9.0061e-04
3
15.7133
1.2829e-02
15.7728
2.3938e-03
15.8259
5.8941e-05
15.7420
1.0766e-02
4
18. 5535
1.3629e-02
18. 5664
1.2948e-02
18.7338
1.0004e-04
18. 5591
1.3310e-02
6. Conclusion
A multilevel thresholding image segmentation method based on IQGA has been proposed in this paper. Combining the merits of QGA and the cooperative learning strategy and adjusting the quantum rotation angle adaptively, the proposed algorithm is employed for several test images. The significant improvement performance of the IQGA has been demonstrated by comparing it with QGA, GA, and PSO for several test images.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by the project of Sichuan Provincial Education Department under Grant 13za0151.
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