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

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.

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][27][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 selfadaptive 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
Computational Intelligence and Neuroscience most popular evaluation criterion for color image quantization [29]. Intracluster distance ( max ) (2) and intercluster separation ( min ) (3) come next in importance to MSE. Smaller MSE means smaller color distortion. Smaller max means smoother gradation of similar colors. Larger min 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. 1 and 2 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, max and min , as its subobjective functions. The model can be formulized as follows: Here [0, 255] 3× is decision space. Decision vector is a colormap consisting of randomly selected color triples in the color space [0, 255] 3 . 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 max minimum and min 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.

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, 1 ( ) and 2 ( ) should not become better simultaneously. Namely, when max becomes better (smaller), min should not also become better (larger). In this part, several experiments are conducted to show that the subobjective functions, 1 ( ) and 2 ( ), in the above model are obviously conflicting. Figure 1 shows four common test images (Peppers, Baboon, Lena, and Airplane) with size 512 × 512 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 max and min 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 max and a mixed probability . Here, max = 200. For showing the same relation of MSE, max and min for the different values of , we let take three different values, 0.1, 0.05, and 0.01 in the three algorithms.
For the three algorithms with different , we can get the similar relation of MSE, max and min . So, we only use the part results of SaDE-CIQ1 with = 0.1 as an example to analyze the relation of MSE, max and min . By any image and its quantized image, we can calculate the values of MSE, max and min . Table 1 gives all the objective values max of SaDE-CIQ1 in 10 runs and the corresponding values of MSE and min . 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, max and min . When max becomes better (smaller), min does not become better (larger). When MSE becomes better (smaller), min does not become better (larger). When max becomes better (smaller), MSE also becomes better (smaller). These mean max and min are conflicting, MSE and min are conflicting, and max and MSE are not conflicting. According to the statistical analysis for all test images, there are 15 conflicts between max and min , 16 between MSE and min , and 11 between max and MSE. These statistical data show that any two of MSE, max and min , are in conflict.
In summary, for the conflict of max and min , it is appropriate to select them as the subobjective functions in the above multiobjective image color model. Meanwhile, for the conflicts of MSE with max and min , there does not exist preference when MSE is applied to select the solution in the Pareto set of the above multiobjective model. Computational Intelligence and Neuroscience 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 1 is updated by SaDE-CIQ with 1 ( ) as its objective. The population 2 is updated by SaDE-CIQ with 2 ( ) as its objective. Then, the best individuals of the two populations are exchanged. The update and exchange operations are repeated to achieve  Step 3 (reserve nondominated solutions). All nondominated solutions in are recorded in a set POS. (Note: for an individual max ( = 1, 2, . . . , 2 * NP), if there is no another one max ( ̸ = , = 1, 2, . . . , 2 * NP) such 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.

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

Experiments for
where 1 , 2 , and 3 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.

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: = 16, NP = 100, max = 200, = 5. Mixed probability takes three different values, 0.1, 0.05, and 0.01. 1 , 2 , and 3 take the same values as those in [28]. Table 3 reports the best MSE values and the corresponding objective values max , min in 10 runs. In fact, smaller max is better, larger min is better, and smaller MSE is better. As shown in Table 3 = 0.05, max is best, min is a median, and MSE is similar to the other two values.

Experimental Results. For MoDE-CIQ,
According to the above conclusions, we will take as 0.1 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, max should be as small as possible. So we will take as 0.05 for Airplane in the following comparing experiments.
For comparing MoDE-CIQ, SaDE-CIQ, and SaDE-CIQ3, Table 4 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. 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.

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 Paretooptimal 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.  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 Intel5 Xeon5 CPU E3-1230 v3 @ 3.30 GHZ and 8 GB RAM.

Conclusions
This paper established a multiobjective image color quantization model, in which intracluster distance max and intercluster   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.