An Image Enhancement Method Using the Quantum-Behaved Particle Swarm Optimization with an Adaptive Strategy

Abstract

Image enhancement techniques are very important to image processing, which are used to improve image quality or extract the fine details in degraded images. In this paper, two novel objective functions based on the normalized incomplete Beta transform function are proposed to evaluate the effectiveness of grayscale image enhancement and color image enhancement, respectively. Using these objective functions, the parameters of transform functions are estimated by the quantum-behaved particle swarm optimization (QPSO). We also propose an improved QPSO with an adaptive parameter control strategy. The QPSO and the AQPSO algorithms, along with genetic algorithm (GA) and particle swarm optimization (PSO), are tested on several benchmark grayscale and color images. The results show that the QPSO and AQPSO perform better than GA and PSO for the enhancement of these images, and the AQPSO has some advantages over QPSO due to its adaptive parameter control strategy.

1. Introduction

Image enhancement is one of the important image techniques in low-level image processing, the purpose of which is to improve the quality of an image for machine analysis or visual perception of human being [1]. For grayscale image enhancement, the most widelyused method is histogram equalization which is based on the assumption that a uniformly distributed grayscale histogram has the best visual contrast [1, 2]. The degraded images with poor contrast can be enhanced by stretching the dynamic range of the grayscale histogram, namely, by grayscale rescaling [3]. In this method, a linear or nonlinear transform between the degraded image and enhanced image is needed. Generally speaking, various types of nonlinear transforms are needed to enhance various types of degradations. An acceptable approach is to use a set of generalized transforms, which are generally combinations of the basic types of nonlinear transforms. Tubbs used a regularized incomplete Beta function as the generalized transform described by a set of parameters, and consequently the image enhancement problem is reduced to a global optimization problem [4].

Grayscale image enhancement, however, cannot be generalized to color image enhancement directly. Several factors, including selection of the color space, characteristics of the human visual system, and color contrast sensitivity, are not taken into consideration in grayscale image enhancement but have to be all treated in color image enhancement [5–7]. The most popular hardware-oriented model for color images is the RGB (red, green, blue) model that has been widely used in color monitors, color video cameras, and computer multimedia applications. If grayscale image enhancement is applied directly to the three components (R, G, B) of a degraded color image, they may be prone to produce color artifacts which look very strange for human beings [1, 5–8]. Another color space is HIS which stands for three main attributes: hue, intensity, and saturation, generally used to distinguish one color from another [6, 7]. Using the HIS color space in color image processing system can decouple the achromatic and chromatic information. Moreover, this color space is close to the color perceiving properties of the human visual system. Therefore, according to the employed color space, the existing color image enhancement techniques can be classified into three categories: the techniques using the RGB color space [5, 7], the techniques using the LHS color space [2, 9, 10], and other methods [11, 12]. In this work, the color image enhancement approach is based on HIS color space.

The goal of this paper is to propose two novel objective functions for the estimation of the parameters of the regularized incomplete Beta functions, which in turn are used for grayscale and color image enhancement, respectively. The objective functions are optimized by quantum-behaved particle swarm optimization (QPSO).

The QPSO algorithm, as a variant of PSO, was inspired by quantum mechanics and trajectory analysis of the individual particle’s behavior in PSO [13–19]. The particle in QPSO needs no velocity vector but sampling its position directly through a double exponential distribution. Except for the population size, the only algorithmic parameter is the contraction-expansion (CE) coefficient which should be adjusted to balance the global search and the local search of the algorithm. Thus, an efficient control strategy for the CE coefficient can enable the algorithm to perform generally well when QPSO is applied to real-world problems. In this paper, we propose an adaptive control method for the CE coefficient and used the improved QPSO for the enhancement of grayscale and color images, using two proposed new objective functions for estimating the parameters of the Beta functions.

The rest of the paper is organized as follows. Sections 2 and 3 address the concepts and the proposed methods of grayscale and color image enhancement, respectively. Section 4 provides a brief introduction of the QPSO algorithm, followed by the proposal of the improved QPSO with the adaptive control strategy. The experimental results for grayscale and color image enhancement are given in Section 5. The paper is concluded in Section 6.

2. Grayscale Image Enhancement

2.1. Grayscale Rescaling

Generally speaking, the purpose of grayscale image enhancement is to improve the low-level image quality, that is, to highlight some specified features or recover the significant details in degraded images. One of the common characteristics of degraded images is poor contrast. The degraded images with poor contrast can be enhanced by stretching the dynamic range of the grayscale histogram, that is, by grayscale rescaling. The simplest rescaling transform is given by 𝐼^∗_𝑥𝑦=𝑓𝑖_𝑥𝑦, (1) where 𝑖_𝑥𝑦 and 𝐼^∗_𝑥𝑦 are the grayscale values of the pixel at (𝑥,𝑦) within the input image and the output enhanced image, respectively, and 𝑓 is a linear or nonlinear transform. Four typical types of widely used nonlinear transforms for grayscale image enhancement are shown in Figure 1, where vertical axes and abscissa stand for the gray value after and before enhancement processing, respectively. Generally, the input range of a pixel should be first normalized into the range [0,1], and thus (1) is rewritten as 𝑖_𝑥,𝑦=𝑖_min+𝑖_max−𝑖_min×𝑇𝑛𝑖_𝑥,𝑦, (2) where 𝑖_min and 𝑖_max are the minimum and maximum values of 𝑖_𝑥,𝑦, respectively, 𝑛(𝑖_𝑥,𝑦) normalizes 𝑖_𝑥,𝑦 into the range [0,1], and 𝑇(⋅) (0≤𝑇(⋅)≤1) is one kind of nonlinear transforms.

(a)

(b)

(c)

(d)

Figure 1 

Four types of nonlinear transforms for grayscale image enhancement: (a) a transform stretching dark regions; (b) a transform stretching lighter regions; (c) a transform stretching middle and compressing two ends; (d) a transform compressing middle and stretching two ends.

In general, since the types of degradations for an input image are usually unknown and a particular type of nonlinear transforms can only enhance a particular type of degradation, we need various types of nonlinear transforms, with each described by a set of parameters, to enhance various types of degradation. An acceptable approach is to use a set of generalized transforms which combine the four types of nonlinear transforms [20]. In this study, we adopt the method proposed by Tubbs, who used the regularized incomplete Beta functions to simulate the four types of transforms as shown in Figures 1(a)–1(d) [4]. The standard Beta density function and the regularized incomplete Beta function are, respectively, given by 𝑓(𝑥)=𝑥^𝛼−1(1−𝑥)^𝛽−1𝐵,𝑓𝛼,𝛽𝑥,𝛼,𝛽=𝐵⁻¹𝛼,𝛽×^𝑥₀𝑡^𝛼−1(1−𝑡)^𝛽−1𝑑𝑡, (3) where 0≤𝑥≤1, 𝛼,𝛽>0, and 𝐵(𝛼,𝛽) is the Beta function defined by 𝐵𝛼,𝛽=¹₀𝑡^𝛼−1(1−𝑡)^𝛽−1𝑑𝑡. (4) Therefore, the generalized transform can be given by 𝑇𝑖_𝑥𝑦=𝑓𝑖_𝑥,𝑦,𝛼,𝛽=^𝑖𝑥𝑦₀𝑡^𝛼−1(1−𝑡)^𝛽−1𝐵𝛼,𝛽𝑑𝑡, (5) where 𝑖_𝑥,𝑦 denotes the gray value in (𝑥,𝑦), 0≤𝑖_𝑥,𝑦≤1, and 0≤𝑇(𝑖_𝑥,𝑦)≤1.

2.2. Generalized Transformation of Gray Level

Consequently, if the gray level of pixel (𝑥,𝑦) is 𝑓(𝑥,𝑦), which turned into 𝑓^′(𝑥,𝑦) after enhancement, the process of generalized transformation consists of the following steps.

Step 1. Normalize the gray value of each pixel by 𝑢=𝑓𝑥,𝑦𝑥,𝑦−𝑖_min𝑖_max−𝑖_min, (6) where 𝑖_min and 𝑖_max are the minimum and maximum values, respectively, of the input image and 0≤𝑢(𝑥,𝑦)≤1.

Step 2. Transform the input image by the regularized incomplete Beta function 𝑇(𝑖_𝑥𝑦) in (5): 𝑢^′𝑥,𝑦=𝑇𝑢𝑥,𝑦, (7) where 𝑢^′(𝑥,𝑦)∈[0,1] is the output image.

Step 3. Calculate the gray value of the output image: 𝑓^′𝑥,𝑦=𝑖_max−𝑖_min×𝑢^′𝑥,𝑦+𝑖_min. (8)

2.3. Objective Function

In this paper, the image enhancement problem is converted into an optimization so that it is important to determine the objective (fitness) function which is used to evaluate the goodness of solution to the problem. According to the characteristics in grayscale image enhancement, we proposed a fitness function which is formed by combining the AC power measure (𝐹_ac) and the Brenner’s measure (𝐹_br). Both 𝐹_ac and 𝐹_br are used to evaluate the focus accuracy of a grayscale image [21].

For a given grayscale image 𝑀×𝑁 in size, the AC power measure is defined by 𝐹_ac=1𝑛^𝑀𝑁_𝑥=1𝑦=1𝑖²_𝑥𝑦−1𝑛^𝑀𝑁_𝑥=1𝑦=1𝑖_𝑥𝑦², (9) where 𝑖_𝑥𝑦 denotes the gray value in (𝑥,𝑦) and 𝑛(=𝑀×𝑁) is the total number of pixels within the image. It is obvious from the above equation that if the 𝐹_ac value is increased, the average pixel variance of the image is increased and the overall contrast of the image will be improved accordingly.

The Brenner’s measure 𝐹_br is given by 𝐹_br=^𝑀−2𝑁_𝑥=1𝑦=1𝑖_𝑥,𝑦−𝑖_𝑥+2,𝑦². (10)𝐹_br is the total square differences between each pixel and the corresponding pixel located two pixels away. We can found from (10) that an image with good contrast will have a large 𝐹_brvalue.

With the definitions of 𝐹_ac and 𝐹_br, the fitness function for grayscale image enhancement is given by 1Fitness=𝑛^𝑀𝑁_𝑥=1𝑦=1𝑓^′2−1𝑥,𝑦𝑛^𝑀𝑁_𝑥=1𝑦=1𝑓^′𝑥,𝑦²+1𝑛^𝑀−2𝑁_𝑥=1𝑦=1𝑓^′𝑥,𝑦−𝑓^′𝑥+2,𝑦², (11) where 𝑓^′(𝑥,𝑦) is the pixel value of (𝑥,𝑦) after generalized transformation. Therefore, the goal of grayscale image enhancement is to find a proper parameter set (𝛼,𝛽) such that 𝑓^′(𝑥,𝑦) obtained after generalized transformation minimizes the above fitness function.

3. Color Image Enhancement

As mentioned in Section 2, for grayscale image enhancement, the general method is histogram equalization or its variations. However, grayscale image enhancement cannot be generalized to color image enhancement directly. Several factors, such as selection of the color space, characteristics of the human visual system, and color contrast sensitivity, need not to be taken into consideration in grayscale image enhancement, but should all be treated in color image enhancement [5–7].

The most popular hardware-oriented model is the RGB (red, green, blue) model used in color monitors, color video cameras, and computer multimedia applications. Histogram equalization and its variants are quite useful in emphasizing important details in grayscale images; nevertheless, if applied individually to the three components (R, G, B) of a degraded color image, they would destroy the natural color balance in the scene. Image enhancement approaches applied directly to the three components (R, G, B) of a degraded color image may easily produce color artifacts which may make the enhanced image look very strange for human beings frequently. Thus, it is inappropriate for the human visual system to apply color image enhancement directly in the RGB color space [1, 5–8].

The characteristics (three main attributes) generally used to distinguish one color from another are hue, intensity, and saturation (HIS). Using the HIS color space in color image processing system can decouple the achromatic and chromatic information. Besides, this color space is close to the color perceiving properties of the human visual system. Therefore, the proposed color image enhancement approach in this paper is applied in the HIS color space.

Here it is assumed that each input degraded color image is originally represented in the RGB color space and is converted into the HIS color space for enhancement. When an input degraded color image in the HIS color space is obtained, it is necessary to normalize the input range of a pixel into [0,1] and then determine a set of generalized transforms for the degraded color image.

The regularized incomplete Beta function in (5) 𝑇(𝑖_𝑥𝑦) is used as the generalized transform to transform the intensity of the pixel (𝑥,𝑦), that is, 𝑖^′𝑥,𝑦=𝑇𝑖𝑥,𝑦, (12) where 𝑖^′(𝑥,𝑦)∈[0,1]. Then the intensity 𝑓^′(𝑥,𝑦) of the output image can be given by 𝑓^′𝑥,𝑦=𝑖_max−𝑖_min×𝑖^′𝑥,𝑦+𝑖_min. (13) Because the information in the hue component 𝐻 has no clear application to image enhancement [11] based on the experiments obtained in this study, we do not attempt any form of hue processing. That is, in the proposed approach the hue component 𝐻 in the HIS color space is left unchanged. The last characteristic, saturation component 𝑆 should be homogenized first in this approach.

After the pretreatment mentioned above, we can come into the main processing of color image enhancement. In the proposed enhancement for color image, the fitness function is formed by three performance parameters, namely, the AC power measure (𝐹_ac), the Brenner’s measure (𝐹_br), and the information-noise change measure (𝐼𝑁_𝑐).

Similar to (9) for a grayscale image, the AC power of a given color image 𝑀×𝑁 in size is defined by 𝐹_ac=1𝑛^𝑀𝑁_𝑥=1𝑦=1𝑖²_𝑥𝑦−1𝑛^𝑀𝑁_𝑥=1𝑦=1𝑖_𝑥𝑦², (14) where 𝑖_𝑥𝑦 is the pixel value at (𝑥,𝑦) and 𝑛(=𝑀×𝑁) is the total number of pixels within the image. The Brenner’s measure 𝐹_br is given by 𝐹_br=^𝑀−2𝑁_𝑥=1𝑦=1𝑖_𝑥,𝑦−𝑖_𝑥+2,𝑦², (15) which shares the same meaning as that in (10).𝐼𝑁_𝑐 is given by 𝐼𝑁_𝑐=_{𝑛(ℎ)>𝑡}1, (16) where 𝑛(ℎ) is the number of pixels in the input histogram with level ℎ and 𝑡 is a threshold. Here 𝐼𝑁_𝑐 is an auxiliary measure which can prevent the effect of overthresholding (resulting in some important details to be dropped), or overenhancing of noise.

In this work, the fitness function𝑓(⋅) for color image enhancement is given by Fitness=𝐼𝑁_𝑐×𝐹_ac+𝐹_br=∑_{𝑛(ℎ)>𝑡}1×1𝑛^𝑀𝑁∑_𝑥=1∑_𝑦=1𝑓^′2−1𝑥,𝑦𝑛^𝑀𝑁∑_𝑥=1∑_𝑦=1𝑓^′𝑥,𝑦²+∑^𝑀−2𝑁_𝑥=1∑_𝑦=1𝑓^′𝑥,𝑦−𝑓^′𝑥+2,𝑦², (17) from which we can see that the larger the fitness function value is, the better the enhanced color image will be.

4. QPSO and QPSO with Adaptive Strategy

4.1. The QPSO Algorithm

The particle swarm optimization (PSO) method is a member of a wider class of Swarm Intelligence methods used for solving GO problems, which was originally proposed by Kennedy and Eberhart as a simulation of social behavior of bird flock and was initially introduced in [22, 23]. Since the origin of PSO, there has been a lot of work done in the improvement and applications of the algorithm [24–33]. In PSO, the particles represent the candidate solutions to the problem, fly through a multidimensional search space, and evaluate their position to a goal at every iteration as well as sharing memories of “best” position to find out the optima or suboptima finally. Since its origin, the PSO algorithm has already been used in a large variety of application areas due to its easy implementation and low requirement in memory and CPU speed. However, as proved by Van Den Bergh [34], PSO is not a global convergence-guaranteed algorithm because the particles are restricted to a finite search space at each iteration, which weakens the global search ability of the algorithm and makes the PSO algorithm easily trap into the local optima. In order to overcome this shortage, Sun et al. proposed a novel variant of PSO, named quantum-behaved particle swarm optimization (QPSO), in which each individual particle has quantum behavior [13, 14]. As has already been proved in theory and several applications, QPSO has a better performance compared with PSO both in precision and efficiency since the sampling space of each particle at each iteration can cover the whole feasible space of the problem at hand [35–44]. In this paper, we introduce an adaptive mechanism into the QPSO and therefore propose an adaptive QPSO (AQPSO).

In the original PSO with 𝑀 individuals, each individual is treated as a volumeless particle in the N-dimensional space, with the position vector and velocity vector of particle 𝑖 at 𝑛th iteration represented as 𝑋_𝑖,𝑛=(𝑋¹_𝑖,𝑛,𝑋²_𝑖,𝑛,…,𝑋^𝑁_𝑖,𝑛) and 𝑉_𝑖,𝑛=(𝑉¹_𝑖,𝑛,𝑉²_𝑖,𝑛,…,𝑉^𝑁_𝑖,𝑛). Vector 𝑃_𝑖,𝑛=(𝑃¹_𝑖,𝑛,𝑃²_𝑖,𝑛,…,𝑃^𝑁_𝑖,𝑛) is the best previous position (the position giving the best objective function value since initialization of the population) of particle 𝑖and is called personal best (pbest) position, and vector 𝐺_𝑛=(𝐺¹_𝑛,𝐺²_𝑛,…,𝐺^𝑁_𝑛) is the position of the best particle among all the particles in the population found so far and is known as global best (gbest) position. Without loss of generality, we consider the following minimization problems: Minimize𝑓(𝑋),s.t.𝑋∈𝑆⊆𝑅^𝑛, (18) where 𝑓(𝑋) is an objective function and 𝑆 is the feasible space. Accordingly, 𝑃_𝑖,𝑛 can be updated by 𝑃_𝑖,𝑛=𝑋_𝑖,𝑛if𝑓𝑋_𝑖,𝑛<𝑓𝑃_{𝑖,𝑛−1}𝑃_{𝑖,𝑛−1}if𝑓𝑋_𝑖,𝑛≥𝑓𝑃_{𝑖,𝑛−1}. (19) Since 𝐺_𝑛 is the personal best position giving the best fitness value among all particles, it can be found by 𝐺_𝑛=𝑃_𝑔,𝑛, (20) where 𝑔=argmin_{1≤𝑖≤𝑀}[𝑓(𝑃_𝑖,𝑛)]. With the above definitions, each particle in the original PSO updates its velocity and position by the following equations: 𝑉^𝑗_𝑖,𝑛+1=𝑉^𝑗_𝑖,𝑛+𝑐₁𝑟^𝑗_𝑖,𝑛𝑋^𝑗_𝑖,𝑛−𝑃^𝑗_𝑖,𝑛+𝑐₂𝑅^𝑗_𝑖,𝑛𝑋^𝑗_𝑖,𝑛−𝐺^𝑗_𝑛,𝑋^𝑗_𝑖,𝑛+1=𝑋^𝑗_𝑖,𝑛+𝑉^𝑗_𝑖,𝑛+1, (21)(22) for 𝑖=1,2,…,𝑀;𝑗=1,2,…,𝑁, where 𝑐₁ and 𝑐₂ are called acceleration coefficients. The parameters 𝑟^𝑗_𝑖,𝑛 and 𝑅^𝑗_𝑖,𝑛 are two different random numbers distributed uniformly on (0,1), that is, 𝑟^𝑗_𝑖,𝑛,𝑅^𝑗_𝑖,𝑛∼𝑈(0,1). Generally, the value of 𝑉^𝑗_𝑖,𝑛 is restricted in the interval [−𝑉_max,𝑉_max].

In [45], Clerc and Kennedy undertook the first formal analysis of the particle’s trajectory and the stability properties of the PSO algorithm. They simplified the particle swarm to a second-order linear dynamical system whose stability depends on the system poles or the eigenvalues of the state matrix. Their analyses essentially revealed the fact that the particle swarm may converge if each particle converges toward its local attractor, 𝑝_𝑖,𝑛=(𝑝¹_𝑖,𝑛,𝑝²_𝑖,𝑛,…,𝑝^𝑁_𝑖,𝑛) defined at the coordinates 𝑝^𝑗_𝑖,𝑛=𝑐₁𝑟^𝑗_𝑖,𝑛𝑃^𝑗_𝑖,𝑛+𝑐₂𝑅^𝑗_𝑖,𝑛𝐺^𝑗_𝑛𝑐₁𝑟^𝑗_𝑖,𝑛+𝑐₂𝑅^𝑗_𝑖,𝑛,1≤𝑗≤𝑁, (23) or 𝑝^𝑗_𝑖,𝑛=𝜑^𝑗_𝑖,𝑛𝑃^𝑗_𝑖,𝑛+1−𝜑^𝑗_𝑖,𝑛𝐺^𝑗_𝑛, (24) where 𝜑^𝑗_𝑖,𝑛=𝑐₁𝑟^𝑗_𝑖,𝑛/(𝑐₁𝑟^𝑗_𝑖,𝑛+𝑐₂𝑅^𝑗_𝑖,𝑛) with regard to the random numbers 𝑟^𝑗_𝑖,𝑛 and 𝑅^𝑗_𝑖,𝑛 in (21) and (23). The acceleration coefficients 𝑐₁ and 𝑐₂ in the original PSO are generally set to be equal, that is, 𝑐₁=𝑐₂ so that 𝜑^𝑗_𝑖,𝑛 is a sequence of uniformly distributed random number over (0,1). As a result, (24) can be restated as 𝑝^𝑗_𝑖,𝑛=𝜑^𝑗_𝑖,𝑛𝑃^𝑗_𝑖,𝑛+1−𝜑^𝑗_𝑖,𝑛𝐺^𝑗_𝑛,𝜑^𝑗_𝑖,𝑛∼𝑈(0,1). (25) The above equation indicates that 𝑝_𝑖,𝑛 is a stochastic point that lies in a hyperrectangle, with 𝑃_𝑖,𝑛 and 𝐺_𝑛 being two ends of its diagonal, and that moves following 𝑃_𝑖,𝑛 and 𝐺_𝑛. In the process of convergence, the particle moves around and careens toward point 𝑝_𝑖 with its kinetic energy (velocity) declining to zero, like a returning satellite orbiting the earth. As such, the particle in PSO can be considered as the one flying in an attraction potential field centered at point 𝑝_𝑖,𝑛 in Newtonian space. It has to be in bound state for the sake of avoiding explosion and guaranteeing convergence. If these conditions are generalized to the case that the particle in PSO moves in quantum space, it is also indispensable that the particle moves in a quantum potential field to ensure the bound state. From the perspective of quantum mechanics, the bound state in quantum space, however, is entirely different from that in Newtonian space, which leads to the proposal of the QPSO algorithm.

In QPSO, each single particle is assumed to be a spinless one with quantum behavior. Thus state of the particle is characterized by wavefunction 𝜓, where |𝜓|² is the probability density function of its position. At the 𝑛th iteration, particle 𝑖 moves in 𝑁-dimensional space with a 𝛿potential well centered at 𝑝^𝑗_𝑖,𝑛 on 𝑗th dimension for 1≤𝑗≤𝑁. Let 𝑌^𝑗_𝑖,𝑛+1=|𝑋^𝑗_𝑖,𝑛+1−𝑝^𝑗_𝑖,𝑛|; we can get the following normalized wavefunction at iteration 𝑛+1 satisfying the bound condition that 𝜓(𝑌^𝑗_𝑖,𝑛+1)→0 as 𝑌^𝑗_𝑖,𝑛+1→∞: 𝜓𝑌^𝑗_𝑖,𝑛+1=1𝐿^𝑗_𝑖,𝑛−𝑌exp^{𝑗𝑖,𝑛+1}𝐿^𝑗_𝑖,𝑛, (26) where 𝐿^𝑗_𝑖,𝑛 is the characteristic length of the wavefunction. By the definition of wavefunction, the probability density function is given by 𝑄𝑌^𝑗_𝑖,𝑛+1=𝜓𝑌^𝑗_𝑖,𝑛²=1𝐿^𝑗_𝑖,𝑛exp−2𝑌^𝑗_𝑖,𝑛+1𝐿^𝑗_𝑖,𝑛, (27) and thus the probability distribution function is 𝐹𝑌^𝑗_𝑖,𝑛+1=1−exp−2𝑌^𝑗_𝑖,𝑛+1𝐿^𝑗_𝑖,𝑛. (28) Using Monte Carlo method, we can obtain the 𝑗th component of position particle 𝑖 at the (𝑛+1)th iteration by 𝑋^𝑗_𝑖,𝑛+1=𝑝^𝑗_𝑖,𝑛±𝐿^𝑗_𝑖,𝑛21ln𝑢^𝑗_𝑖,𝑛+1,𝑢^𝑗_𝑖,𝑛+1∼𝑈(0,1), (29) where 𝑢^𝑗_𝑖,𝑛 is a sequence of random numbers uniformly distributed in (0,1). As proposed in [14], the value of 𝐿^𝑗_𝑖,𝑛 is given by 𝐿^𝑗_𝑖,𝑛=2𝛼𝑋^𝑗_𝑖,𝑛−𝐶^𝑗_𝑛, (30) where 𝐶_𝑛=𝑃_𝑛=(𝐶¹_𝑛,𝐶²_𝑛,…,𝐶^𝑁_𝑛) is called mean best (mbest) position defined by the mean of the pbest positions of all particles and thus 𝐶^𝑗_𝑛=𝑃^𝑗_𝑛=(1/𝑀)∑^𝑀_𝑖=1𝑃^𝑗_𝑖,𝑛(1≤𝑗≤𝑁). Therefore the position of the particle updates as follows: 𝑋^𝑗_𝑖,𝑛+1=𝑝^𝑗_𝑖,𝑛±𝛼𝑋^𝑗_𝑖,𝑛−𝐶^𝑗_𝑛1⋅ln𝑢^𝑗_𝑖,𝑛+1. (31) The parameter 𝛼 in (30) or (31) is known as the contraction-expansion (CE) coefficient, which can be tuned to control the convergence speed of the algorithms. The PSO with (31) is known as quantum-behaved particle swarm optimization (QPSO), the procedure of which is outlined in Algorithm 1, where rand𝑖(⋅), 𝑖=1,2,3, are separately generated random numbers uniformly distributed in (0,1).

Begin  Initialize the population;  While (termination condition = false)  Do   Compute mean best position 𝐶;   Properly select the value of 𝛼;   for (𝑖=1 to 𝑀)    Evaluate the objective function value 𝑓 (𝑋𝑖);    Update 𝑃𝑖 and 𝐺;     For 𝑗=1 to 𝑁      𝜑𝑗𝑖,𝑛= rand1(·);      𝑃𝑗𝑖,𝑛=𝜑𝑗𝑖,𝑛𝑃𝑗𝑖,𝑛+1−𝜑𝑗𝑖,𝑛𝐺𝑗𝑛;      𝑢𝑗𝑖,𝑛= rand2(·);      if (rand3(·) < 0.5)       𝑋𝑗𝑖,𝑛+1=𝑝𝑗𝑖,𝑛+𝛼𝑋𝑗𝑖,𝑛−𝐶𝑗𝑛ln1/𝑢𝑗𝑖,𝑛;      else       𝑋𝑗𝑖,𝑛+1=𝑝𝑗𝑖,𝑛−𝛼𝑋𝑗𝑖,𝑛−𝐶𝑗𝑛ln1/𝑢𝑗𝑖,𝑛;      end if     end for    end for   end do end

Algorithm 1 

Procedure of the QPSO.

4.2. QPSO with Adaptive Strategy

It has been proved that the CE coefficient 𝛼 must be set as 𝛼<1.781 to guarantee the boundedness of the particles so that the whole particle swarm may converge [15]. Whenever 𝛼 is set to be larger than 𝛼>1.781, the particles will explode. As a result, it is necessary to set 𝛼 smaller than 1.781 for the purpose of convergence when the algorithm is being applied to real-world problems. In [15], two methods of controlling 𝛼 were analyzed comprehensively. When fixed-valued 𝛼 is used, it is suggested that fixing the value of 𝛼 as 0.75 during the search process can generate good algorithmic performance in general. On the other hand, it has been verified that decreasing 𝛼 linearly from 𝛼₀ to 𝛼₁ (𝛼₀>𝛼₁) can yield generally good algorithmic performance.

In this paper, we propose an adaptive control method for the CE coefficient by using the following error function: Δ𝐹_𝑖,𝑛=𝑓𝑃_𝑖,𝑛−𝑓𝐺_𝑛min𝑓𝑃_𝑖,𝑛,𝑓𝐺_𝑛, (32) where 𝑓(𝑃_𝑖,𝑛) is the fitness values of the pbest position of particle 𝑖 and 𝑓(𝐺_𝑛) is the fitness value of the gbest position at the 𝑛th iteration, respectively. min(𝑥₁,𝑥₂) returns the minimum of 𝑥₁ and 𝑥₂. Equation (32) measures the proximity of a particle to the gbest position. It is obvious that the smaller the value of Δ𝐹_𝑖,𝑛, is the closer the particle’s pbest position to the gbest position, and accordingly the narrower the search range of the particle, leading the particle to loss of global search ability. If all the particles’ pbest positions cluster to the gbest position, that is, the average value of Δ𝐹_𝑖,𝑛 over all the particles is small, the whole particle swarm may encounter premature convergence, since the search scope of the particle is determined by the multiplication of the CE coefficient and the distance from its current position to the mean best position. If the particles cluster together around the gbest position, a larger CE coefficient α can prevent them from further clustering. Therefore, we propose an adaptive tuning method for 𝛼 to help enhance the global search ability of the QPSO algorithm.

In the proposed adaptive QPSO (AQPSO) algorithm, each particle has its own CE coefficient during each iteration, instead of sharing the CE coefficient with the other particles. Generally, the value of 𝛼 for the particle which is far away from the gbest position has a larger value, while that for the particle close to the gbest position has a smaller one that can make the particle fly toward the gbest position more rapidly. Here, the proximity of the particle to the gbest position is defined by the logarithmic value of Δ𝐹_𝑖,𝑛, namely, 𝑧_𝑖,𝑛=lnΔ𝐹_𝑖,𝑛=ln𝑓𝑃_𝑖,𝑛−𝑓𝐺_𝑛−lnmin𝑓𝑃_𝑖,𝑛,𝑓𝐺_𝑛. (33) The CE coefficient for each particle is set according to the value of 𝑧_𝑖,𝑛 using the following piecewise function: 𝛼_𝑖,𝑛𝑧_𝑖,𝑛=0.6,𝑧_𝑖,𝑛>0,0.7,−2<𝑧_𝑖,𝑛≤0,0.6+0.1×𝑘,−𝑘−1<𝑧_𝑖,𝑛(),()≤−𝑘,𝑘=2,3,41.0+0.2×𝑘−4,−𝑘−1<𝑧_𝑖,𝑛≤𝑘,(𝑘=5,6,7),1.8,𝑧_𝑖,𝑛≤−8. (34) The graph of the adaptive CE coefficient is shown in Figure 1. When 𝑧_𝑖,𝑛≤−8, the value of 𝛼_𝑖,𝑛 is larger than 1.781, which means that the particle is in explosion and flies away rapidly from the gbest position. This piecewise function for the value of the CE coefficient means that if the distance between the particle and the gbest position is large, the CE coefficient should be set small to make the particle converge fast; if the distance is small, the CE coefficient can be large so that the particle can converge slowly to perform global search. This adaptive control method for the CE coefficient helps to maintain the diversity of the swarm at a certain level and ensures the global search ability of the algorithm during the whole search process (see Figure 2).

Figure 2 

The adaptive CE coefficient for each particle.

5. Experimental Results

5.1. Results for Grayscale Image Enhancement

The experiments on enhancement of the degraded grayscale image “frog” were performed by using GA, PSO, QPSO, and AQPSO, respectively. The “frog” is 256×256 in size. Figures 3(a) and 3(b) show its image and grayscale histogram, respectively, from which we can see that the contrast is poor and the details in skin and background are not clear. The most popular method for grayscale image enhancement is known to be histogram equalization, by which the image “frog” enhanced is shown in Figure 3(c). The corresponding grayscale histogram in Figure 3(d) indicates that the frequency of each gray level is the same, the entropy of the image now is the biggest, and the information combined is the largest. Nevertheless, it was pointed in [16] that the same frequency for each gray level is not bounded to result in the best visual effects, since the robustness of a grayscale image is directly related to the distribution of its gray level.

(a) Original image “frog”

(b) The rayscale histogram of “frog”

(c) The “frog” after histogram equalization

(d) The grayscale histogram of the “frog” in (c)

Figure 3 

The image “frog” and its histogram equalization image, as well as their grayscale histograms.

In the experiments for GA, the population size was set to be 60, the crossover and mutation probabilities were 0.6 and 0.01, respectively, and 20 binary bits composed a chromosome. For PSO, the inertia weight decreased linearly from 0.9 to 0.4 during the search process, and the two acceleration coefficients were fixed at 2.0. For QPSO, the CE coefficient decreased linearly from 1.0 to 0.5 in the course of search. The population sizes for all PSO variants, including PSO, QPSO, and AQPSO, were set to be 30. The maximum number of iterations for a single run of each algorithm was 500. Every algorithm was performed for 30 trial runs, and mean best fitness and standard deviation were recorded for performance comparison.

The estimated parameters 𝛼 and 𝛽 corresponding to the best fitness value obtained over 30 trial runs are listed in Table 1, which shows that the best fitness values obtained by QPSO and AQPSO are better than those by PSO and GA. Between QPSO and AQPSO, the latter appears to estimate the parameters more accurately than the former. In order to evaluate the average performance of each competitor algorithms, we recorded in Table 2 the mean best fitness values and standard deviations over 30 trial runs of GA, PSO, QPSO, and AQPSO. It can be observed that AQPSO obtained the mean best fitness value over 30 runs and is the best among all the tested algorithms and that the second best performing algorithms is QPSO. Table 3 lists the 𝑡 test results between the AQPSO and the other algorithms, showing that the outperformance of the AQPSO over its competitors is significant. Figure 4 shows the enhanced images by different approaches based on the proposed objective function in (11) for grayscale image, as well as their grayscale histograms and transformation lines. The enhanced image by each algorithm was yielded through the transforms with the parameters 𝛼 and 𝛽 estimated by the algorithm. Obviously, the image texture that was enhanced by any of GA, PSO, QPSO, or AQPSO is nicer than that in Figure 3(c). However, the best fitness obtained by AQPSO is larger than that by any other competitors, implying that AQPSO had a better performance than the others.

Table 1 

The estimated parameters corresponding to the best fitness values obtained by GA, PSO, QPSO, and AQPSO.

Table 2 

Mean best fitness and standard deviation over 30 trial runs of GA, PSO, QPSO, and AQPSO.

Table 3 

Unpaired t-test between the performances of the algorithms.

Figure 4 

The comparison of enhancement using GA, PSO, QPSO, and AQPSO for “frog.”

5.2. Results for Color Image Enhancement

Two color images, “Lenna” and “Waterlilies,” are used as benchmarks to evaluate the performance of the QPSO and AQPSO algorithms in color image enhancement. The two images are originally represented in the RGB color space, 256∗256 in size. Two simple approaches for color image enhancement that equalize the histograms in R, G, B and histograms in H, I, S, respectively, were also employed for performance comparison. In the experiments for GA, population size was set to be 60, crossover operator and mutation operator were 0.6 and 0.01, respectively, 20 binary bits composes a chromosome, and moreover, the maximum of iterations was 500. In the experiments for PSO, QPSO, and AQPSO, each algorithm used 30 particles and lasted for a maximum of iterations of 500. Each of the tested population-based optimization algorithms performed 30 trial runs on each color image, with the mean fitness values and the standard deviation being recorded in Table 3.

It is shown in Table 4 that the mean fitness values obtained by AQPSO are slightly larger than those by PSO and GA, even than that by QPSO, which means that the AQPSO-based color image enhancement approach had the best performance among all the competitors. The significance of the AQPSO’s outperformance is verified by 𝑡 test as shown in Table 5. Tables 6 and 7 list the estimated parameters 𝛼 and 𝛽 corresponding to the best fitness value obtained over 30 trial runs. It is shown that the best fitness values obtained by QPSO and AQPSO are better than those by PSO and GA, implying that QPSO and AQPSO were able to estimate the parameters more accurately, and that AQPSO had better performance than QPSO as can be observed. Figures 5 and 6 show the enhanced images for “Lenna” and “Waterlilies” using different approaches based on the proposed objective function in (17) for color image. The enhanced image by each algorithm was yielded through the transforms with the parameters 𝛼 and 𝛽 listed in Tables 5 and 6. As can be observed in Figures 5 and 6, the image texture that was enhanced by any of GA, PSO, QPSO, or AQPSO is nicer than those by equalization of histogram in R, G, B or in H, I, S.

Table 4 

Experimental results of each algorithm on “Lenna” and “Waterlilies”.

Table 5 

Unpaired t test between the performances of the algorithms for “Lenna” and “Waterlilies”.

Table 6 

The estimated parameters for Lenna corresponding to the best fitness values obtained by GA, PSO, QPSO, and AQPSO.

Table 7 

The estimated parameters for Waterlilies corresponding to the best fitness values obtained by GA, PSO, QPSO, and AQPSO.

(a) Degraded Lenna

(b) Equalization of histogram in R, G, B

(c) Equalization of histogram in H, I, S

(d) Enhanced by GA

(e) Enhanced by PSO

(f) Enhanced by QPSO

(g) Enhanced by AQPSO

Figure 5 

Comparison of enhancements using different approaches for “Lenna”.

(a) Degraded Waterlilies

(b) Equalization of histogram in R, G, B

(c) Equalizing histogram in H, I, S

(d) Enhanced by GA

(e) Enhanced by PSO

(f) Enhanced by QPSO

(g) Enhanced by AQPSO

Figure 6 

Comparison of enhancements using different approaches for “Waterlilies”.

6. Conclusion

This paper proposed a novel objective function to evaluate the effectiveness of grayscale image enhancement based on the normalized incomplete Beta function and modified this objective for color image enhancement in HIS space. Furthermore, the AQPSO algorithm which employs an adaptive control method for the CE coefficient was proposed to optimize the objective function for obtaining a proper set of parameters of the Beta function and achieving enhancement of a grayscale or color image.

AQPSO, along with QPSO, PSO, and GA, was tested on some well-known benchmark images for performance comparison on image enhancement. To do so, two groups of experiments were performed, one for the enhancement of a grayscale image and the other for the enhancement of a color image. The experimental results showed that QPSO and AQPSO were able to estimate the parameters more accurately and thus had better performance for both grayscale and color image enhancement than their competitor. Moreover, AQPSO generated better enhanced images than the original QPSO, as can be observed from the results.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 61103051), by the National Natural Science Foundation of China (Grants no. 61170119 and 61105128), by Natural Science Foundation of Jiangsu Province, China (Grant no. BK2010143), by the Ministry of Education Research in the humanities and social sciences planning fund (Grant no. 12YJAZH120), and by the ZheJiang Provincial Natural Science Foundation of China (Grants no. Y1101237, LY12F02012, and Y107759).