Computational Intelligence and Neuroscience
Volume 2016 (2016), Article ID 2450431, 12 pages
http://dx.doi.org/10.1155/2016/2450431

Research Article

Multiobjective Image Color Quantization Algorithm Based on Self-Adaptive Hybrid Differential Evolution

Zhongbo Hu,¹ Qinghua Su,¹ and Xuewen Xia²

¹School of Information and Mathematics, Yangtze University, Jingzhou, Hubei 434023, China
²School of Software, East China Jiaotong University, Nanchang 330013, China

Received 19 July 2016; Revised 24 August 2016; Accepted 4 September 2016

Academic Editor: Manuel Graña

Copyright © 2016 Zhongbo Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In recent years, some researchers considered image color quantization as a single-objective problem and applied heuristic algorithms to solve it. This paper establishes a multiobjective image color quantization model with intracluster distance and intercluster separation as its objectives. Inspired by a multipopulation idea, a multiobjective image color quantization algorithm based on self-adaptive hybrid differential evolution (MoDE-CIQ) is then proposed to solve this model. Two numerical experiments on four common test images are conducted to analyze the effectiveness and competitiveness of the multiobjective model and the proposed algorithm.

1. Introduction

Image color quantization is one of the common image processing techniques. It is the process of reducing the number of colors presented in a color image with less distortion [1]. Most of the image color quantization methods [2–12] are essentially based on data clustering algorithms. Recently, some heuristic methods, such as genetic algorithm (GA) [13, 14], particle swarm optimization algorithm (PSO) [15–17], and differential evolution (DE) [18–21], have been employed to solve the image color quantization problems which are considered as optimization problems. Evaluation criteria, which are used as objective functions of optimization problems, often incorporate mean square-error (MSE) [22–24], intracluster distance (), and intercluster separation () [25–28].

Most of the image color quantization algorithms based on heuristic methods are single-objective methods; that is, only one evaluation criterion is used. References [26–28] have used three evaluation criteria, but their three criteria have been merged to get a linear weighting objective function. In general, the objective function in any of the above algorithms holds only one evaluation criterion or a linear combination of several evaluation criteria. This paper presents the following two aspects:

(i)Develop multiobjective model for image color quantization problems. Based on the model, we can obtain a quantized image with the smallest color distortion among those images which meet a trade-off between the optimal color gradation and the optimal color details.(ii)Propose a multiobjective algorithm based on a self-adaptive DE for solving the multiobjective image color quantization model.

The rest of the paper is organized as follows. Section 2 establishes a multiobjective image color quantization model. Section 3 presents a multiobjective image color quantization algorithm based on self-adaptive hybrid DE (MoDE-CIQ). Experimental results and discussion on four test images are provided in Section 4. Conclusions are given in Section 5.

2. Establishment of a Multiobjective Image Color Quantization Model

2.1. Multiobjective Image Color Quantization Model

In single-objective models, mean square-error (MSE) (1) is the most popular evaluation criterion for color image quantization [29]. Intracluster distance () (2) and intercluster separation () (3) come next in importance to MSE. Smaller MSE means smaller color distortion. Smaller means smoother gradation of similar colors. Larger means more color details to be preserved. The three evaluation criteria are expressed in the following formulas [28]:

Here, is the size of a color image . is a pixel in . is a given color number of a colormap. is the sequence number of the colors in the colormap. is the th color of the colormap. and are two different sequence numbers of the colors in the colormap. is the cluster of all pixels in with similar color to . is the number of all pixels in . is the color of a pixel in . represents Euclidean distance.

This paper proposes a multiobjective image color quantization model which uses two evaluation criteria, and , as its subobjective functions. The model can be formulized as follows:

Here is decision space. Decision vector is a colormap consisting of randomly selected color triples in the color space. Let

be the th color of the colormap. Then

is the objective function with the following two subobjectives:

This model aims to make a trade-off between minimum and maximum. The solution set of this multiobjective model is called Pareto set, the solutions of which could balance color gradation and color details.

Obviously, the solution with the smallest MSE in the Pareto set of the above multiobjective model corresponds to a quantized image, which holds the smallest color distortion among those images with a balance between the optimal color gradation and the optimal color details.

2.2. Conflict Detection of the Subobjective Functions

As we all know, the subobjective functions of a multiobjective model should be conflicting. This means, as two subobjectives in the above model, and should not become better simultaneously. Namely, when becomes better (smaller), should not also become better (larger). In this part, several experiments are conducted to show that the subobjective functions, and , in the above model are obviously conflicting.

Figure 1 shows four common test images (Peppers, Baboon, Lena, and Airplane) with size pixels. Reference [15] presented a color image quantization algorithm based on self-adaptive hybrid DE (SaDE-CIQ), in which the objective function is MSE. We, respectively, replace its objective with and to obtain two algorithms, named SaDE-CIQ1 and SaDE-CIQ2. SaDE-CIQ, SaDE-CIQ1, and SaDE-CIQ2 are implemented to quantize all test images into the quantized images with 16 colors. Each algorithm is run 10 times on each test image. In the three algorithms, there are two parameters, a maximum iteration and a mixed probability . Here, . For showing the same relation of MSE, and for the different values of , we let take three different values, 0.1, 0.05, and 0.01 in the three algorithms.

(a) Peppers

(b) Baboon

(c) Lena

(d) Airplane

Figure 1: Test images.

For the three algorithms with different , we can get the similar relation of MSE, and . So, we only use the part results of SaDE-CIQ1 with as an example to analyze the relation of MSE, and . By any image and its quantized image, we can calculate the values of MSE, and . Table 1 gives all the objective values of SaDE-CIQ1 in 10 runs and the corresponding values of MSE and . Figure 2 shows the changes of these values in 10 runs. We include the curves of Peppers from first run to second run as an example of how to illustrate the conflicts of MSE, and . When becomes better (smaller), does not become better (larger). When MSE becomes better (smaller), does not become better (larger). When becomes better (smaller), MSE also becomes better (smaller). These mean and are conflicting, MSE and are conflicting, and and MSE are not conflicting. According to the statistical analysis for all test images, there are 15 conflicts between and , 16 between MSE and , and 11 between and MSE. These statistical data show that any two of MSE, and , are in conflict.

Table 1: The results of 10 runs for SaDE-CIQ1 ().

Test image

Test serial number

1
2
3
4
5
6
7
8
9
10

Peppers

27.8885
26.9858
26.0681
28.1934
25.7472
32.8054
26.8729
32.0317
25.5597
28.0979

41.5508
29.3541
25.0297
37.4886
40.54
33.6902
35.1127
36.4135
38.3825
36.1761

MSE
26.8221
26.2667
25.1661
26.8304
25.5318
31.5104
24.1938
29.6623
23.7019
27.8777

Baboon

26.6224
27.6260
28.3537
26.8452
26.6689
30.5386
27.6907
28.4376
26.8122
28.7434

36.2255
32.8375
34.9105
24.7064
36.5621
30.9652
26.8745
25.9984
33.4011
38.2255

MSE
20.3766
20.8404
20.1511
19.6093
20.9290
19.4481
21.6021
20.2362
19.4163
20.5033

Lena

27.2745
34.0579
28.5068
26.6540
26.6780
37.2558
12.6332
34.9201
28.0219
33.1166

21.9009
36.3725
33.3204
37.1832
37.1524
26.1205
37.3509
28.9622
24.5176
29.3508

MSE
8.3868
15.5077
28.0535
5.6724
15.9792
40.6224
9.5826
38.4419
17.6261
13.4949

Airplane

21.9009
36.3725
33.3204
37.1832
37.1524
26.1205
37.3509
28.9622
24.5176
29.3508

8.3868
15.5077
28.0535
5.6724
15.9792
40.6224
9.5826
38.4419
17.6261
13.4949

MSE
15.5626
25.9173
26.8238
22.2865
26.5673
20.2143
29.551
25.9917
21.0685
24.9274

Figure 2: The curves of , , and MSE obtained by SaDE-CIQ1 ().

In summary, for the conflict of and , it is appropriate to select them as the subobjective functions in the above multiobjective image color model. Meanwhile, for the conflicts of MSE with and , there does not exist preference when MSE is applied to select the solution in the Pareto set of the above multiobjective model.

3. Multiobjective Image Color Quantization Algorithm Based on Self-Adaptive Hybrid DE

For solving the above multiobjective image color quantization model, this section proposes a multiobjective image color quantization algorithm based on self-adaptive hybrid DE (MoDE-CIQ). This algorithm merges the ideas of SaDE-CIQ in [19] and a multipopulation DE algorithm VEDE [30], which is a Pareto-based multiobjective DE algorithm. The main steps of the proposed MoDE-CIQ algorithm are described as below.

Step 1 (initialize populations). Two initial populations including NP individuals are randomly selected separately. Here, each individual is a colormap with colors from an image . The initial populations are denoted by

Step 2 (optimize populations). The population is updated by SaDE-CIQ with as its objective. The population is updated by SaDE-CIQ with as its objective. Then, the best individuals of the two populations are exchanged. The update and exchange operations are repeated to achieve a predetermined iteration number . The set of th generation individuals of the two populations is denoted by

Step 3 (reserve nondominated solutions). All nondominated solutions in are recorded in a set POS.
(Note: for an individual , if there is no another one such that and , that is, , it is a nondominated solution. Otherwise, it is a dominated solution.)

Step 4 (obtain an approximative Pareto solution set). Steps 2 and 3 are repeated to achieve a predetermined iteration number Loop. The final set POS is recorded as an approximative Pareto solution set.

Step 5 (determine an optimal solution). In the set POS, the solution with the smallest values of MSE is finally reserved as an optimal solution of an image color quantization problem.

The pseudocode of MoDE-CIQ is shown as Pseudocode 1.

Input color image ;

Set parameters , , , , , ;

Initialize two populations , ;

Initialize Pareto-optimal solutions set , and all individuals in POS are set to

be enough large initial values of MSE;

for to

for to

      Update the first population by SaDE-CIQ (the pseudo-code is shown in [15]) with

          as its objective;

              Update the second population by SaDE-CIQ with as its objective;

              Exchange the best solutions of the two populations;

     for to

      If is a non-dominated solution and its MSE value is less than POS,

              then POS;

              If is a non-dominated solution and its MSE value is less than POS[],

              then POS[];

Output all the nonzero individuals in Pareto-optimal solutions set POS, their values of MSE, the

quantized image with the smallest value of MSE.

Pseudocode 1: The pseudocode of MoDE-CIQ.

4. Numerical Experiments

In this section, two sets of experiments are conducted to illustrate the effectiveness of MoDE-CIQ algorithm and the advantage of the multiobjective model, respectively.

4.1. Experiments for Showing the Multiobjective Algorithmic Superiority

4.1.1. Experimental Background

Currently, the heuristic algorithms employed to solve the image color quantization problem have mainly GA, PSO, and DE. Reference [16] indicated that PSO is superior to GA. In [31], DE and PSO show similar performance on image color quantization. However, due to simple operation, litter parameters, and fast convergence, DE is the better choice to use than PSO. These mean that DE is a competitive image color quantization in the heuristic algorithms for image color quantization. Reference [19] proposed a color image quantization algorithm based on self-adaptive hybrid DE (SaDE-CIQ), in which the parameters of DE are automatically adjusted by a self-adaptive mechanic. Then, SaDE-CIQ is compared with -means and the color image quantization algorithm using PSO (PSO-CIQ). Table 2 shows the smallest and the largest objective values for the three algorithms over 10 runs obtained in [19]. The results show that SaDE-CIQ is an effective color image quantization algorithm, and SaDE-CIQ has better quantization quality than -means and PSO-CIQ. It is naturally to be thought that SaDE-CIQ is the best one of the image color quantization algorithms based on heuristic algorithms.

Table 2: The MSE values resulting from SaDE-CIQ, -means, and PSO-CIQ.

Alg.
Peppers
Baboon
Lena
Airplane

Min
Max
Min
Max
Min
Max
Min
Max

SaDE-CIQ
17.4682
18.7266
22.7496
23.3382
12.9709
13.8055
8.2482
8.9740

-means
18.1086
21.2676
22.9532
24.9563
15.6401
19.1314
9.1141
10.4430

PSO-CIQ
36.3436
40.9532
35.8892
41.9940
29.6644
34.5867
21.3540
24.3200

Reference [28] presented a linear weighting objective function of and and MSE below:

where , , and are the user-defined weights of the subobjectives. The linear weighting objective function (10) is the only one, including the three evaluation criteria of MoDE-CIQ, in existing references. So in this section, we will compare MoDE-CIQ, SaDE-CIQ, and SaDE-CIQ3 obtained by replacing the objective function MSE with the linear weighting objective function (10) in SaDE-CIQ.

4.1.2. Experimental Design

MoDE-CIQ, SaDE-CIQ, and SaDE-CIQ3 are implemented to quantize the four test images in Figure 1 into the quantized images with 16 colors. Each algorithm is run 10 times. The parameters of algorithms are set as follows:

 , , , . Mixed probability takes three different values, 0.1, 0.05, and 0.01. , , and take the same values as those in [28].

4.1.3. Experimental Results

For MoDE-CIQ, Table 3 reports the best MSE values and the corresponding objective values , in 10 runs. In fact, smaller is better, larger is better, and smaller MSE is better. As shown in Table 3, the following conclusions are obtained. (i) For Peppers, only MSE is best as . and are best as . As , , , and MSE are all medians, and and MSE are similar to their corresponding best values. (ii) For Baboon, as , and MSE are all best. (iii) For Lena, and MSE are all best as . (iv) For Airplane, as , is best, is a median, and MSE is similar to the other two values.

Table 3: The best MSE values and the corresponding objective values of MoDE-CIQ.

Image
values

MSE

Peppers
0.1
25.6127
28.2967
19.1029

0.05
28.2967
31.9070
18.8444

0.01
24.8917
38.4062
19.5632

Baboon
0.1
27.8841
45.8284
22.9602

0.05
27.8083
45.5262
22.9887

0.01
27.9030
44.7175
22.9654

Lena
0.1
20.2311
26.4388
14.2847

0.05
20.2849
32.0907
15.6655

0.01
21.1913
32.9181
15.5229

Airplane
0.1
22.0570
24.1028
10.7517

0.05
22.0105
29.6160
11.2520

0.01
20.9759
26.9999
10.9591

According to the above conclusions, we will take as for Peppers, Baboon, and Lena in the following comparing experiments. However, there are few and extremely unequally distributed base colors in Airplane. For preserving main color gradations and richer color levers of original images, should be as small as possible. So we will take as for Airplane in the following comparing experiments.

For comparing MoDE-CIQ, SaDE-CIQ, and SaDE-CIQ3, Table 4 reports , , and MSE of their quantized images, MSE values of which are the smallest in their 10 runs. SaDE-CIQ aims to minimize its objective MSE, so its values of MSE are surely the best than those of other two algorithms. But the values of and by MoDE-CIQ are all better than those of SaDE-CIQ. The values of and obtained by SaDE-CIQ3 for Peppers and Baboon are also better than those of SaDE-CIQ. The values of , , and MSE obtained by MoDE-CIQ are better than those of SaDE-CIQ3, except for their similar values of , , and MSE for Baboon, and the values of MSE for Lena. Figures 3, 4, 5, and 6 show all quantized images of the four common test images in Figure 1. In Figures 3–6, all subfigures (a) are the original test images. Subfigures (b), subfigures (c), and subfigures (d) are the quantized images separately obtained by MoDE-CIQ, SaDE-CIQ3, and SaDE-CIQ. The visual effects of the quantized images are compared as follows. (i) For Peppers (shown in Figure 3), there are contrasting and equally distributed main base colors, so the quantized images obtained by three algorithms visually have similar color distortions. The differences in the quantization quality of these quantized images depend on their color gradations of larger regions with similar colors. The quantized images of MoDE-CIQ and SaDE-CIQ have the more rich color levers than the one of the SaDE-CIQ3. (ii) For Baboon (shown in Figure 4), there are also contrasting and equally distributed main base colors, but there are little larger regions with similar colors. So the quantized images of three methods have similar effects. (iii) For Lena (shown in Figure 5), there are many shaded regions in it. So differences in the quantization quality of the corresponding quantized images depend on the transition from shaded regions to highlights. MoDE-CIQ obtains the quantized image with more natural transition than SaDE-CIQ and SaDE-CIQ3. (iv) For Airplane (shown in Figure 6), there are extremely unequally distributed base colors. Obviously, the quantized image of SaDE-CIQ3 has the largest color distortion. Although the quantized image of SaDE-CIQ has a little better color distortion than that of the multiobjective algorithm, the former loses some detail colors, such as the cloud in the sky.

Table 4: , , and MSE of the quantized images with 16 colors by three algorithms.

Image
values
Algorithm

MSE

Peppers
0.1
MoDE-CIQ
25.6127
28.2967
19.1029

SaDE-CIQ3
34.2489
45.8673
20.3563

SaDE-CIQ
37.2450
22.2473
17.4577

Baboon
0.1
MoDE-CIQ
27.8841
45.8284
22.9602

SaDE-CIQ3
27.8122
45.8426
22.9592

SaDE-CIQ
28.1805
36.4773
22.7644

Lena
0.1
MoDE-CIQ
20.2311
26.4388
14.2847

SaDE-CIQ3
22.8824
27.5461
13.5264

SaDE-CIQ
22.2973
19.0143
12.9641

Airplane
0.05
MoDE-CIQ
22.0105
29.6160
11.2520

SaDE-CIQ3
113.2050
34.8630
17.4217

SaDE-CIQ
23.7529
8.2540
8.0544

(a) Peppers

(b) MoDE-CIQ

(c) SaDE-CIQ3

(d) SaDE-CIQ

Figure 3: The quantized images of Peppers with 16 colors obtained by three algorithms.

(a) Baboon

(b) MoDE-CIQ

(c) SaDE-CIQ3

(d) SaDE-CIQ

Figure 4: The quantized images of Baboon with 16 colors obtained by three algorithms.

(a) Lena

(b) MoDE-CIQ

(c) SaDE-CIQ3

(d) SaDE-CIQ

Figure 5: The quantized images of Lena with 16 colors obtained by three algorithms.

(a) Airplane

(b) MoDE-CIQ

(c) SaDE-CIQ3

(d) SaDE-CIQ

Figure 6: The quantized images of Airplane with 16 colors obtained by three algorithms.

According the above results, for the images with contrasting and equally distributed main base colors, the quantization effects of MoDE-CIQ and SaDE-CIQ are similar. But for the images with many shaded regions and extremely unequally distributed base colors, MoDE-CIQ could make the colors more natural and preserve more detail colors. In SaDE-CIQ3, the weighted factors in (10) affect its quantization quality. Thus, we can think MoDE-CIQ is superior to the other two algorithms.

4.2. Experiments for Showing the Advantage of the Multiobjective Model

As the statement on Step 4 of MoDE-CIQ, we can obtain an approximative Pareto solution set. This is an advantage comparing to all single-objective algorithms. The above experiments reserved the approximative Pareto-optimal solutions of all four images. The solution sets corresponding to Peppers, Baboon, Lena, and Airplane, respectively, include 13 solutions (shown in Table 5), 9 solutions (in Table 6), 11 solutions (in Table 7), and 8 solutions (in Table 8). For comparing these optimal solutions, their corresponding MSE values are listed. Figure 7 shows the Pareto front of these Pareto-optimal solutions. These optimization solutions present some quantized images with different effects. Users can select the suitable quantized image according to their requirements for the color gradations and details.

Table 5: Pareto-optimal solutions for Peppers.

Order

MSE

1
24.9238
227.2425
19.5660

2
38.5345
196.8913
25.4623

3
31.0556
208.1886
24.1790

4
34.6405
205.1429
22.2026

5
25.6127
226.7033
19.1029

6
35.2191
197.7366
23.0621

7
31.4675
207.8684
23.3818

8
34.4429
205.8758
22.2859

9
34.8841
204.8451
26.031

10
34.1102
207.7311
23.1158

11
25.9563
217.8530
20.4536

12
28.4238
210.6036
22.2533

13
34.3636
206.7747
21.8853

Table 6: Pareto-optimal solutions for Baboon.

Order

MSE

1
27.3819
212.0693
23.1123

2
27.8841
209.1716
22.9602

3
31.8821
204.127
24.7008

4
29.2681
205.8375
24.4271

5
30.9412
204.9553
24.6433

6
33.8514
202.2041
25.7050

7
30.2455
205.5812
24.5084

8
27.8998
208.0066
227.341

9
27.6801
209.3535
227.341

Table 7: Pareto-optimal solutions for Lena.

Order

MSE

1
24.9313
207.9088
19.8533

2
20.2311
228.5612
14.2847

3
20.6109
228.4630
15.1880

4
26.7185
202.3789
20.9155

5
25.9452
203.7679
20.8445

6
24.5586
209.2558
18.9967

7
23.6997
209.7721
19.8327

8
22.3126
212.7511
20.1288

9
21.1493
216.7855
17.3328

10
24.9279
209.1586
233.6600

11
20.6396
224.5480
233.6600

Table 8: Pareto-optimal solutions for Airplane.

Order

MSE

1
23.9419
220.4130
12.4973

2
21.2536
225.4198
11.4192

3
22.3128
223.4460
11.5732

4
22.0105
225.3840
11.2520

5
22.2876
225.0752
11.4864

6
25.2011
219.2880
13.5785

7
68.6311
212.3398
316.6950

8
22.1871
225.3009
316.6950

Figure 7: Pareto front of MoDE-CIQ.

By the experimental results of the above two parts, MoDE-CIQ is a competitive algorithm for image color quantization.

All the above algorithms were implemented in Visual C++ and the experiments were conducted on a computer with Intel® Xeon® CPU E3-1230 v3 @ 3.30 GHZ and 8 GB RAM.

5. Conclusions

This paper established a multiobjective image color quantization model, in which intracluster distance and intercluster separation are selected as its objective functions. A multiobjective image color quantization algorithm based on self-adaptive hybrid DE (MoDE-CIQ) was proposed to solve this model. MoDE-CIQ emerges the ideas of SaDE-CIQ [19] and a multipopulation DE algorithm VEDE [30], and applies MSE to determine the optimal solution. The multiobjective model and the proposed algorithm present a strategy to obtain a quantized image which holds the smallest color distortion among those images with a balance between the optimal color gradation and the optimal color details. The experimental results indicated that the multiobjective model and MoDE-CIQ are effective and competitive for image color quantization problems.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by Hubei Province Department of Education Major Scientific Research Program of China (D20161306).

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