Fuzzy entropy based optimal thresholding using bat algorithm

Image segmentation is a very significant process in image analysis. Much effort based on thresholding has been made on this field as it is simple and intuitive, commonly used thresholding approaches are to optimize a criterion such as between-class variance or entropy for seeking appropriate threshold values. However, a mass of computational cost is needed and efficiency is broken down as an exhaustive search is utilized for finding the optimal thresholds, which results in application of evolutionary algorithm and swarm intelligence to obtain the optimal thresholds. This paper considers image thresholding as a constrained optimization problem and optimal thresholds for 1-level or multi-level thresholding in an image are acquired by maximizing the fuzzy entropy via a newly proposed bat algorithm. The optimal thresholding is achieved through the convergence of bat algorithm. The proposed method has been tested on some natural and infrared images. The results are compared with the fuzzy entropy based methods that are optimized by artificial bee colony algorithm (ABC), genetic algorithm (GA), particle swarm optimization (PSO) and ant colony optimization (ACO); moreover, they are also compared with thresholding methods based on criteria of between-class variance and Kapur's entropy optimized by bat algorithm. It is demonstrated that the proposed method is robust, adaptive, encouraging on the score of CPU time and exhibits the better performance than other methods involved in the paper in terms of objective function values.     

Human mental search-based multilevel thresholding for image segmentation

Multilevel thresholding is one of the principal methods of image segmentation. These methods enjoy image histogram for segmentation. The quality of segmentation depends on the value of the selected thresholds. Since an exhaustive search is made for finding the optimum value of the objective function, the conventional methods of multilevel thresholding are time-consuming computationally, especially when the number of thresholds increases. Use of evolutionary algorithms has attracted a lot of attention under such circumstances. Human mental search algorithm is a population-based evolutionary algorithm inspired by the manner of human mental search in online auctions. This algorithm has three interesting operators: (1) clustering for finding the promising areas, (2) mental search for exploring the surrounding of every solution using Levy distribution, and (3) moving the solutions toward the promising area. In the present study, multilevel thresholding is proposed for image segmentation using human mental search algorithm. Kapur (entropy) and Otsu (between-class variance) criteria were used for this purpose. The advantages of the proposed method are described using twelve images and in comparison with other existing approaches, including genetic algorithm, particle swarm optimization, differential evolution, firefly algorithm, bat algorithm, gravitational search algorithm, and teaching-learning-based optimization. The obtained results indicated that the proposed method is highly efficient in multilevel image thresholding in terms of objective function value, peak signal to noise, structural similarity index, feature similarity index, and the curse of dimensionality. In addition, two nonparametric statistical tests verified the efficiency of the proposed algorithm, statistically.     

Modified bacterial foraging algorithm based multilevel thresholding for image segmentation

Multilevel thresholding is one of the most popular image segmentation techniques. In order to determine the thresholds, most methods use the histogram of the image. This paper proposes multilevel thresholding for histogram-based image segmentation using modified bacterial foraging (MBF) algorithm. To improve the global searching ability and convergence speed of the bacterial foraging algorithm, the best bacteria among all the chemotactic steps are passed to the subsequent generations. The optimal thresholds are found by maximizing Kapur's (entropy criterion) and Otsu's (between-class variance) thresholding functions using MBF algorithm. The superiority of the proposed algorithm is demonstrated by considering fourteen benchmark images and compared with other existing approaches namely bacterial foraging (BF) algorithm, particle swarm optimization algorithm (PSO) and genetic algorithm (GA). The findings affirmed the robustness, fast convergence and proficiency of the proposed MBF over other existing techniques. Experimental results show that the Otsu based optimization method converges quickly as compared with Kapur's method.     

An efficient method for multilevel color image thresholding using cuckoo search algorithm based on minimum cross entropy

Among various thresholding methods, minimum cross entropy is implemented for its effectiveness and simplicity. Although it is efficient and gives excellent result in case of bi-level thresholding, but its evaluation becomes computationally costly when extended to perform multilevel thresholding owing to the exhaustive search performed for the optimum threshold values. Therefore, in this paper, an efficient multilevel thresholding technique based on cuckoo search algorithm is adopted to render multilevel minimum cross entropy more practical and reduce the complexity. Experiments have been conducted over different color images including natural and satellite images exhibiting low resolution, complex backgrounds and poor illumination. The feasibility and efficiency of proposed approach is investigated through an extensive comparison with multilevel minimum cross entropy based methods that are optimized using artificial bee colony, bacterial foraging optimization, differential evolution, and wind driven optimization. In addition, the proposed approach is compared with thresholding techniques depending on between-class variance (Otsu) method and Tsalli’s entropy function. Experimental results based on qualitative results and different fidelity parameters depicts that the proposed approach selects optimum threshold values more efficiently and accurately as compared to other compared techniques and produces high quality of the segmented images.     

Multi-objective and multi-level image thresholding based on dominance and diversity criteria

The Kapur and Otsu methods are widely used image thresholding approaches and they are very efficient in bi-level thresholding applications. Evolutionary algorithms have been developed to extend the Kapur and Otsu methods to the multi-level thresholding case. However, there remains an unsolved argument that neither Kapur nor Otsu objective can optimally fit diverse content contained in different kinds of images. This paper proposes a multi-objective model which seeks to find the Pareto-optimal set with respect to Kapur and Otsu objectives. Based on dominance and diversity criteria, we developed a hybrid multi-objective particle swarm optimization (MOPSO) method by incorporating several intelligent search strategies. The ensemble strategy is also applied to automatically select the best search strategy to perform at various algorithm stages according to its historic performances. The experimental result shows that the solutions to our multi-objective model consistently produce equal or better segmentation results than those by the optimal solutions to the original Kapur and Otsu models, and that the proposed hybrid algorithm with and without the ensemble strategy produces a better approximation to the ideal Pareto front than those obtained by two other MOPSO variants and the MOEA/D. In comparison with the most recent multilevel thresholding methods, our approach also consistently obtains better performance in the segmentation result for several benchmark images.     

A Context Sensitive Multilevel Thresholding Using Swarm Based Algorithms

In this paper, a comprehensive energy function is used to formulate the three most popular objective functions: Kapur's, Otsu and Tsalli's functions for performing effective multilevel color image thresholding. These new energy based objective criterions are further combined with the proficient search capability of swarm based algorithms to improve the efficiency and robustness. The proposed multilevel thresholding approach accurately determines the optimal threshold values by using generated energy curve, and acutely distinguishes different objects within the multi-channel complex images. The performance evaluation indices and experiments on different test images illustrate that Kapur's entropy aided with differential evolution and bacterial foraging optimization algorithm generates the most accurate and visually pleasing segmented images.      

Optimal segmentation of brain MRI based on adaptive bacterial foraging algorithm

Segmentation of brain magnetic resonance images (MRIs) can be used to identify various neural disorders. The MRI segmentation facilitates in extracting different brain tissues such as white matter, gray matter and cerebrospinal fluids. Segmentation of these tissues helps in determining the volume of the tissues in three-dimensional brain MRI, which yields in analyzing many neural disorders such as epilepsy and Alzheimer disease. In this article, multilevel thresholding based on adaptive bacterial foraging (ABF) algorithm is presented for brain MRI segmentation. The proposed ABF algorithm employs an adaptive step size to improve both exploration and exploitation capability of the BF algorithm. Maximization of the measure of separability on the basis of the entropy (Kapur) method and the between-class variance (Otsu) method, which are the two popular thresholding techniques, are employed to evaluate the performance of the proposed method. Application results to axial, T2-weighted brain MRI slices are provided to show the performance of the proposed segmentation approach. These results are compared with bacterial foraging (BF) algorithm, particle swarm optimization (PSO) algorithm and genetic algorithm (GA) in terms of solution quality, robustness and computational efficiency.     

A fast technique for image segmentation based on two Meta-heuristic algorithms

Image segmentation is a primary task in image processing which is widely used in object detection and recognition. Multilevel thresholding is one of the prominent technique in the field of image segmentation. However, the computational cost of multilevel thresholding increases exponentially as the number of threshold value increases, which leads to use of meta-heuristic optimization to find the optimal number of threshold. To overcome this problem, this paper investigates the ability of two nature-inspired algorithms namely: antlion optimisation (ALO) and multiverse optimization (MVO). ALO is a population-based method and mimics the hunting behaviour of antlions in nature. Whereas, MVO is based on the multiverse theory which depicts that there is over one universe exist. These two metaheuristic algorithms are used to find the optimal threshold values using Kapur’s entropy and Otsu’s between class variance function. They examine the outcomes of the proposed algorithm with other evolutionary algorithms based on cost value, stability analysis, feature similarity index (FSIM), structural similarity index (SSIM), peak signal to noise ratio (PSNR), computational time. We also provide Wilcoxon test which justify the response of these parameters. The experimental results showed that the proposed algorithm gives better results than other existing methods. It is noticed that MVO is faster than other algorithms. The proposed method is also tested on medical images to detect the tumor from MRI T1-weighted contrast-enhanced brain images.

     

A new active contour remote sensing river image segmentation algorithm inspired from the cross entropy

The CV (Chan–Vese) model is a piecewise constant approximation of the Mumford and Shah model. It assumes that the original image can be segmented into two regions such that each region can be represented as constant grayscale value. In fact, the objective functional of the CV model actually finds a segmentation of the image such that the within-class variance is minimized. This is equivalent to the Otsu image thresholding algorithm which also aims to minimize the within-class variance. Similarly to the Otsu image thresholding algorithm, cross entropy is another widely used image thresholding algorithm and it finds a segmentation such that the cross entropy of the segmented image and the original image is minimized. Inspired from the cross entropy, a new active contour image segmentation algorithm is proposed. The region term in the new objective functional is the integral of the logarithm of the ratio between the grayscale of the original image and the mean value computed from the segmented image weighted by the grayscale of the original image. The new objective functional can be solved by the level set evolution method. A distance regularized term is added to the level set evolution equation so the level set need not be reinitialized periodically. A fast global minimization algorithm of the objective functional is also proposed which incorporates the edge term originated from the geodesic active contour model. Experimental results show that, the algorithm proposed can segment images more accurately than the CV model and the implementation speed of the fast global minimization algorithm is fast.     

基于NW小世界邻居的粒子群多阈值分割算法

针对多阈值分割问题,提出了一种新的多阈值分割算法.此算法采用相对类内方差代替传统Otsu算法中的绝对类内方差,改善了传统Otsu对小对象分割不理想的弱点;采用NW小世界模型作为粒子群优化的社会认知拓扑结构,具有较好的全局寻优能力和较快的收敛速度.实验结果显示此算法具有好的性能.     

一种基于遗传算法的最小交叉熵阈值选择方法

最小交叉熵阈值法(MCET) 在二级阈值中是有效的, 但在多极阈值的穷尽搜索中却要付出昂贵的时间代价. 鉴于此, 提出一种基于遗传算法(GA) 的MCET选择方法: 在执行图像分割(IS) 任务之前, 先将IS 转化为在一定约束 条件下待优化的问题; 在寻找待优化问题最优解的计算过程中引入一种回归设计技巧以存储中间结果; 使用这种回 归设计技巧, 在一组标准测试图像上利用GA搜索待优化问题的最优解. 实验结果表明, 利用所提出的方法获得的多 个阈值非常接近于穷尽搜索获得的结果.

     

An evolutionary gray gradient algorithm for multilevel thresholding of brain MR images using soft computing techniques

The conventional two dimensional (2-D) histogram based Otsu’s method gives unreliable results while considering multilevel thresholding of brain magnetic resonance (MR) images, because the edges of the brain regions are not preserved due to the local averaging process involved. Moreover, some of the useful pixels present inside the off-diagonal regions are ignored in the calculation. This article presents an evolutionary gray gradient algorithm (EGGA) for optimal multilevel thresholding of brain MR images. In this paper, more edge information is preserved by computing 2-D histogram based gray gradient. The key to our success is the use of the gray gradient information between the pixel values and the pixel average values to minimize the information loss. In addition, the speed improvement is achieved. Theoretical formulations are derived for computing the maximum between class variance from the 2-D histogram of the brain image. A first-hand fitness function is suggested for the EGGA. A novel adaptive swallow swarm optimization (ASSO) algorithm is introduced to optimize the fitness function. The performance of ASSO is validated using twenty three standard Benchmark test functions. The performance of ASSO is better than swallow swarm optimization (SSO). The optimum threshold value is obtained by maximizing the between class variance using ASSO. Our method is tested using the standard axial T₂ − weighted brain MRI database of Harvard medical education using 100 slices. Performance of our method is compared to the Otsu’s method based on the one dimensional (1-D) and the 2-D histogram. The results are also compared among four different soft computing techniques. It is observed that results obtained using our method is better than the other methods, both qualitatively and quantitatively. Benefits of our method are – (i) the EGGA exhibits better objective function values; (ii) the EGGA provides us significantly improved results; and (iii) more computational speed is achieved.     

高斯尺度空间下估计背景的自适应阈值分割算法

为有效分割非均匀光照图像,提出一种在高斯尺度空间下估计背景的自适应阈值分割算法. 首先,利用二维高斯函数对待处理图像进行卷积操作来构建一个高斯尺度空间,在此空间下进行背景估计,并采用背景差法来消除非均匀光照干扰,从而提取出目标图像;然后,采用 矫正进行增强处理以突出较暗目标信息;最后,经强调谷底的最大类间方差法进行全局分割得到最终结果. 为验证算法的有效性,对非均匀光照条件下文本图像以及非文本图像进行了测试,并与基于偏移场的模糊C均值方法、灰度波动变换自适应阈值分割算法和自适应最小误差阈值分割算法,在错误分割率和运行时间上进行了对比. 实验结果表明,对比以上三种方法,该算法的分割结果更为理想.     

二维Otsu阈值法的快速算法

最大类间方差法(Otsu)是图像分割的经典算法,在其基础之上发展起来的二维Otsu阈值分割法由于计算复杂而制约了其应用。针对这一缺点,提出一种改进的二维Otsu阈值法的快速算法。首先将原始二维直方图划分成M×M个区域,将每个区域视为1个点,构造新的二维直方图,在其上利用二维Otsu以及快速递推算法,得到分割阈值所处的区域编号;既而对所确定的区域再次使用二维Otsu算法得到原始图像的分割阈值。实验结果证明,改进算法有效地提高了计算速度,降低了算法的空间复杂度,且分割效果与原始算法基本一致。     

A Local Contrast Fusion Based 3D Otsu Algorithm for Multilevel Image Segmentation

To overcome the shortcomings of 1D and 2D Otsu’s thresholding techniques, the 3D Otsu method has been developed. Among all Otsu’s methods, 3D Otsu technique provides the best threshold values for the multi-level thresholding processes. In this paper, to improve the quality of segmented images, a simple and effective multilevel thresholding method is introduced. The proposed approach focuses on preserving edge detail by computing the 3D Otsu along the fusion phenomena. The advantages of the presented scheme include higher quality outcomes, better preservation of tiny details and boundaries and reduced execution time with rising threshold levels. The fusion approach depends upon the differences between pixel intensity values within a small local space of an image; it aims to improve localized information after the thresholding process. The fusion of images based on local contrast can improve image segmentation performance by minimizing the loss of local contrast, loss of details and gray-level distributions. Results show that the proposed method yields more promising segmentation results when compared to conventional 1D Otsu, 2D Otsu and 3D Otsu methods, as evident from the objective and subjective evaluations.      

Robust and automated unimodal histogram thresholding and potential applications

In this paper, three new histogram-based algorithms are presented to segment images expressing unimodal intensity histograms. These algorithms are applied to laser scanning confocal microscope images (known to often exhibit unimodal histograms) to identify fluorescent signals, and other applications are also shown. The first algorithm facilitates linear diffusion to investigate dynamic histogram features in scale-space. The second algorithm is based on a histogram comparison between a reference area and the whole image at reduced scale. The third algorithm uses the maximisation of a between-class variance criterion applied to image histograms. Results obtained from automatic thresholding of confocal microscopy images show good agreement between the algorithms. Further applications to segment other images are also shown.     

A novel hybrid algorithm of gravitational search algorithm with genetic algorithm for multi-level thresholding

The multi-level thresholding is a popular method for image segmentation. However, the method is computationally expensive and suffers from premature convergence when level increases. To solve the two problems, this paper presents an advanced version of gravitational search algorithm (GSA), namely hybrid algorithm of GSA with genetic algorithm (GA) (GSA-GA) for multi-level thresholding. In GSA-GA, when premature convergence occurred, the roulette selection and discrete mutation operators of GA are introduced to diversify the population and escape from premature convergence. The introduction of these operators therefore promotes GSA-GA to perform faster and more accurate multi-level image thresholding. In this paper, two common criteria (1) entropy and (2) between-class variance were utilized as fitness functions. Experiments have been performed on six test images using various numbers of thresholds. The experimental results were compared with standard GSA and three state-of-art GSA variants. Comparison results showed that the GSA-GA produced superior or comparative segmentation accuracy in both entropy and between-class variance criteria. Moreover, the statistical significance test demonstrated that GSA-GA significantly reduce the computational complexity for all of the tested images.     

Multilevel thresholding selection based on the artificial bee colony algorithm for image segmentation

Multilevel thresholding is an important technique for image processing and pattern recognition. The maximum entropy thresholding (MET) has been widely applied in the literature. In this paper, a new multilevel MET algorithm based on the technology of the artificial bee colony (ABC) algorithm is proposed: the maximum entropy based artificial bee colony thresholding (MEABCT) method. Four different methods are compared to this proposed method: the particle swarm optimization (PSO), the hybrid cooperative-comprehensive learning based PSO algorithm (HCOCLPSO), the Fast Otsu’s method and the honey bee mating optimization (HBMO). The experimental results demonstrate that the proposed MEABCT algorithm can search for multiple thresholds which are very close to the optimal ones examined by the exhaustive search method. Compared to the other four thresholding methods, the segmentation results of using the MEABCT algorithm is the most, however, the computation time by using the MEABCT algorithm is shorter than that of the other four methods.     

Whale Optimization Algorithm and Moth-Flame Optimization for multilevel thresholding image segmentation

Determining the optimal thresholding for image segmentation has got more attention in recent years since it has many applications. There are several methods used to find the optimal thresholding values such as Otsu and Kapur based methods. These methods are suitable for bi-level thresholding case and they can be easily extended to the multilevel case, however, the process of determining the optimal thresholds in the case of multilevel thresholding is time-consuming. To avoid this problem, this paper examines the ability of two nature inspired algorithms namely: Whale Optimization Algorithm (WOA) and Moth-Flame Optimization (MFO) to determine the optimal multilevel thresholding for image segmentation. The MFO algorithm is inspired from the natural behavior of moths which have a special navigation style at night since they fly using the moonlight, whereas, the WOA algorithm emulates the natural cooperative behaviors of whales. The candidate solutions in the adapted algorithms were created using the image histogram, and then they were updated based on the characteristics of each algorithm. The solutions are assessed using the Otsu’s fitness function during the optimization operation. The performance of the proposed algorithms has been evaluated using several of benchmark images and has been compared with five different swarm algorithms. The results have been analyzed based on the best fitness values, PSNR, and SSIM measures, as well as time complexity and the ANOVA test. The experimental results showed that the proposed methods outperformed the other swarm algorithms; in addition, the MFO showed better results than WOA, as well as provided a good balance between exploration and exploitation in all images at small and high threshold numbers.     

Automatic multi-thresholds selection for image segmentation based on evolutionary approach

Automatic thresholding has been widely used in machine vision for automatic image segmentation. Otsu’s method selects an optimum threshold by maximizing the between-class variance in a grayscale image. However, the method becomes time-consuming when extended to multi-level threshold problems, because excessive iterations are required in order to compute the cumulative probability and the mean of class. In this paper, we focus on the issue of automatic selection for multi-level thresholding, and we greatly improve the efficiency of Otsu’s method for image segmentation based on evolutionary approaches. We have investigated and evaluated the performance of the Otsu and Valleyemphasis thresholding methods. Based on our evaluation results, we have developed many different algorithms for automatic threshold selection based on the evolutionary method using the Modified Adaptive Genetic Algorithm and the Hill Climbing Algorithm. The experimental results show that the evolutionary approach achieves a satisfactory segmentation effect and that the processing time can be greatly reduced when the number of thresholds increases.