Computational Intelligence and Neuroscience

Volume 2016 (2016), Article ID 2450431, 12 pages

http://dx.doi.org/10.1155/2016/2450431

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

^{1}School of Information and Mathematics, Yangtze University, Jingzhou, Hubei 434023, China^{2}School 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.