若干仿生算法的理论及其在函数优化和图像多阈值分割中的应用
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摘要
最优化方法作为一个重要的科学分支，一直受到人们的广泛重视，它对多个学科产生了重大影响，并在诸多工程领域得到迅速推广和应用，己成为不同领域中以及人们的日常工作与生活中不可或缺的工具。很多优化问题己被证明是NP完全问题，至今没有有效的多项式时间解法，用传统的最优化方法求解，需要的计算时间与问题的规模成指数关系。因此，人们转而求其次，发展了很多仿生算法，希望在有限的时间内求得问题的次优解或近优解，如遗传算法、分布估计算法、粒子群算法、禁忌搜索以及其混合优化策略等。本文的研究紧紧围绕几种仿生算法的理论及其在函数优化、图像阈值分割上面的应用展开，具体地说，本文的研究内容及主要解决的问题如下所述。
1)首先介绍了仿生算法之遗传算法的基础理论研究概况和遗传算法的各种改进策略及其已取得的成果。然后在此基础上，我们深入分析了一种伪遗传算法所采用的二进制编码互补双亲策略初始化种群的方法；证明了以此种方式初始化种群能使变异算子极限搜索概率提高1/|HL|2；并且也分析了它的优良模式的生存能力和全局收敛性。接下来，基于上述的分析对伪遗传算法进行了二个方面的改进，改进后的算法称为GACPS算法。经过对一类自变量非对称的测试函数的仿真实验证实，改进后的算法在求解精确度、稳定性和收敛速度等方面都有很明显的提高。此外，我们也指出了GACPS算法研究的意义和下一步研究的内容。
2)在上文关于遗传算法理论研究的基础上，结合近年来仿生算法应用于图像阈值分割的概况，我们把遗传算法应用到图像分割领域，提出一种自动多阈值图像分割算法AMT-BSGA。首先将一幅图像看成是由像素值组成的总体，运用分块采样得到若干子样本；其次在每一个子样本中运用遗传算法来使样本的均值与方差比极大化；再基于获得的样本信息对阈值数目和阈值进行自动预测；最后利用一种确定性的算法对阈值数目和阈值做进一步的优化。该算法无需事先考虑图像的纹理和分割数等先验信息，具有较高的易用性；其计算复杂性对图像阈值个数敏感性较低；无需进行灰度直方图分析。在Berkeley图像分割数据集上的大量仿真实验结果表明，本章所提算法能获得较准确、快速和稳定的图像分割。此外，我们也指出了AMT-BSGA的进一步研究的思路和方向。
3)在上文对遗传算法理论及其在函数优化、图像多阈值分割中应用的深入研究基础上，结合遗传算法的发展过程，我们研究了分布估计算法。首先介绍了分布估计算法基本原理和流程，分布估计算法基础理论研究和分布估计算法的种类及其改进。接下来我们基于贝叶斯定理和最优解的概率分布提出了一种新的分布估计算法即贝叶斯预测型进化算法(BFEA)。它通过预测最优解所在的子空间来导引算法的搜索,即按照一定的规则在含有最优解概率较大的子空间产生较多的个体,在最优解概率较小的子空间产生较少的个体。它可方便的引入专家知识、能高效利用所有先前代蕴含的信息且能以很快的速度收敛到最优解子空间。文中从理论上分析了它的收敛性、收敛速度和逆收敛算子。我们发现它是有效解决遗传算法中的连锁和欺骗问题的一种新方法。文中的理论分析与对比实验证实了贝叶斯预测型进化算法求解较精确、稳定和快速。此外，我们也指出了贝叶斯预测型进化算法的下一步研究的思路和内容。
4)受到上文把贝叶斯定理和遗传算法相结合的启发，我们尝试把贝叶斯定理和粒子群算法相结合。在研究了粒子群算法的流程、特点和应用情况以及粒子群算法的理论和改进方法的基础上，接下来，我们结合贝叶斯定理和灰度图像阈值分割的特点提出了一个高效并且更简单的改进粒子群算法，称为贝叶斯粒子群算法(Bayesian ParticleSwarm Optimization algorithm, BPSO)。在贝叶斯粒子群算法中，我们设计了一种方法去自动地和分别地指派粒子群算法中“社会影响”(social influence)部分的收缩系数(constriction coefficient)的大小，以至于各个粒子们可以根据历史信息以及与当前最优粒子之间距离的大小而拥有不同程度的探索和开发能力。另外，我们根据图像多阈值分割时阈值从小到大排列的特点设计了一种种群初始化策略，这种策略使算法搜索效率更高。相对于现有的三种先进的算法来说，大量的在Berkely图像数据集上的仿真实验显示了贝叶斯粒子群算法可以得到高效的、稳定的和比较平滑的分割结果。此外，我们也指出了贝叶斯粒子群算法进一步研究的方向和内容。
在本文的结论部分，我们对全文主要研究内容进行总结，讨论了现有工作中的不足之处，并且指出了在本文研究基础之上的后续研究内容和思想方法。
Optimization which has greatly influenced many disciplines and obtains rapid promotionand applications in so many engineering domains has received widespread attention as animportant scientific branch. Optimization has become an indispensable tool for many differentareas. Many optimization problems have been proved to be NP-Complete problems (NPC).So far there is no effective method of Polynomial time for NPC, and the computing time of atraditional optimization method grows exponentially with the scale of the problem. Thus,instead, researchers hoping to achieve optimal or near-optimal solutions in limited time havedeveloped a lot of Bionical algorithms such as, Genetic Algorithms, Estimation OfDistribution Algorithms, Particle Swarm Optimization, Tabu Search, hybrid optimizationstrategies and so on. This thesis focuses on Bionical Algorithms and Their Application onFunction Optimization and Multilevel Thresholding. The main works and innovative pointsare as follows:
1) Traditional Genetic Algorithms initialize their individuals stochastically and generatetoo many eccentric and homogeneous individuals. This will cause premature convergence andslow down the speed. In this thesis this important problem is studied via thecomplementary-parent strategy of initializing population in PGA. We analyze it and concludethat the limiting probability of the traditional mutation operator based on this strategy is1/|HL|2higher than on the traditional one. This strategy is more efficient by applying toCoarse-grained parallel genetic algorithm which develops the parallelism among populations.For PGA, we also discuss the reproductive capacity of excellent schemata based on theschema theorem and demonstrate the global convergence using homogeneous finite Markovchain. And then, we present an improved algorithm named GACPS. With some of typicalunsymmetrical functions tested, simulation show the quality of GACPS is much higher thanPGA on precision, stability and convergence rate.
2) Multilevel thresholding is an important technique for image compression, imageanalysis and pattern recognition. However, it is still a hard problem to determine the numberof thresholds automatically. So, a new multilevel thresholding method called as AutomaticMultilevel Thresholding Algorithm For Image Segmentation based on Block Sampling and Genetic Algorithm (AMT-BSGA), which can automatically determine the appropriate numberof thresholds and the proper threshold values, is proposed on the basis of Block Sampling andGenetic Algorithm. In AMT-BSGA, an image is treated as a population with the gray valuesof pixels as the individuals. First, an image is evenly divided into several blocks, and a sampleis drawn from each block. Then, genetic algorithm based optimizing is applied to each sampleso as to maximize the ratio of mean and variance for the sample. Based on the optimizedsamples, the number of thresholds and threshold values are preliminarily determined. Finally,a deterministic method is implemented to further optimize the number of thresholds andthreshold values. AMT-BSGA can work without knowing in advance other auxiliaryinformation, such as contextual or textual properties, and the number of thresholds. It has alower computing complexity which is almost independent from the number of thresholds andcan avoid the burden of analyzing histograms. AMT-BSGA can produce effective, efficientand smoother results, which has been verified by extensive simulations on Berkeley datasets.
3) Based on Bayesian theorem and the probability distribution of promising solutionsand integrating with the basic principle of evolutionary computation, we proposes a novelalgorithm named Bayesian Forecasting Evolutionary Algorithm (BFEA). Our proposedmethod guides the exploration of the search space according to the prediction probability ofevery subspace including the optimal solutions. Moreover, prior information about theproblem is incorporated into the algorithm easily and much more information in the generatedpopulations is used. More importantly, BFEA is a new method to solve linkage problem anddeceptive problem effectively. This algorithm can converge faster to the subspaces withoptimal solution and its convergence, convergence rate and counteraction operator are alsoanalyzed theoretically. Theoretical analyses and tests have demonstrated the proposed methodhas a simpler algorithm model, lower computing complexity and higher computationaccuracy.
4) A simpler and efficient PSO algorithm based on Bayesian theorem and the charactersof intensity images is proposed, called as Bayesian Particle Swarm Optimization algorithm(BPSO). In BPSO, a new method is designed to assign the constriction coefficient of the“social influence” term for each particle automatically and separately based on Bayesiantheorem, so that they can have different levels of exploration and exploitation capabilities. And a new population initialization strategy is adopted to make the search more efficient,according to the characters of multilevel thresholding in an image arranged from a low graylevel to a high one. The experimental results indicate that BPSO can produce effective,efficient and smoother segmentation results in comparison with three existing methods onBerkely datasets.
1)首先介绍了仿生算法之遗传算法的基础理论研究概况和遗传算法的各种改进策略及其已取得的成果。然后在此基础上，我们深入分析了一种伪遗传算法所采用的二进制编码互补双亲策略初始化种群的方法；证明了以此种方式初始化种群能使变异算子极限搜索概率提高1/|HL|2；并且也分析了它的优良模式的生存能力和全局收敛性。接下来，基于上述的分析对伪遗传算法进行了二个方面的改进，改进后的算法称为GACPS算法。经过对一类自变量非对称的测试函数的仿真实验证实，改进后的算法在求解精确度、稳定性和收敛速度等方面都有很明显的提高。此外，我们也指出了GACPS算法研究的意义和下一步研究的内容。
2)在上文关于遗传算法理论研究的基础上，结合近年来仿生算法应用于图像阈值分割的概况，我们把遗传算法应用到图像分割领域，提出一种自动多阈值图像分割算法AMT-BSGA。首先将一幅图像看成是由像素值组成的总体，运用分块采样得到若干子样本；其次在每一个子样本中运用遗传算法来使样本的均值与方差比极大化；再基于获得的样本信息对阈值数目和阈值进行自动预测；最后利用一种确定性的算法对阈值数目和阈值做进一步的优化。该算法无需事先考虑图像的纹理和分割数等先验信息，具有较高的易用性；其计算复杂性对图像阈值个数敏感性较低；无需进行灰度直方图分析。在Berkeley图像分割数据集上的大量仿真实验结果表明，本章所提算法能获得较准确、快速和稳定的图像分割。此外，我们也指出了AMT-BSGA的进一步研究的思路和方向。
3)在上文对遗传算法理论及其在函数优化、图像多阈值分割中应用的深入研究基础上，结合遗传算法的发展过程，我们研究了分布估计算法。首先介绍了分布估计算法基本原理和流程，分布估计算法基础理论研究和分布估计算法的种类及其改进。接下来我们基于贝叶斯定理和最优解的概率分布提出了一种新的分布估计算法即贝叶斯预测型进化算法(BFEA)。它通过预测最优解所在的子空间来导引算法的搜索,即按照一定的规则在含有最优解概率较大的子空间产生较多的个体,在最优解概率较小的子空间产生较少的个体。它可方便的引入专家知识、能高效利用所有先前代蕴含的信息且能以很快的速度收敛到最优解子空间。文中从理论上分析了它的收敛性、收敛速度和逆收敛算子。我们发现它是有效解决遗传算法中的连锁和欺骗问题的一种新方法。文中的理论分析与对比实验证实了贝叶斯预测型进化算法求解较精确、稳定和快速。此外，我们也指出了贝叶斯预测型进化算法的下一步研究的思路和内容。
4)受到上文把贝叶斯定理和遗传算法相结合的启发，我们尝试把贝叶斯定理和粒子群算法相结合。在研究了粒子群算法的流程、特点和应用情况以及粒子群算法的理论和改进方法的基础上，接下来，我们结合贝叶斯定理和灰度图像阈值分割的特点提出了一个高效并且更简单的改进粒子群算法，称为贝叶斯粒子群算法(Bayesian ParticleSwarm Optimization algorithm, BPSO)。在贝叶斯粒子群算法中，我们设计了一种方法去自动地和分别地指派粒子群算法中“社会影响”(social influence)部分的收缩系数(constriction coefficient)的大小，以至于各个粒子们可以根据历史信息以及与当前最优粒子之间距离的大小而拥有不同程度的探索和开发能力。另外，我们根据图像多阈值分割时阈值从小到大排列的特点设计了一种种群初始化策略，这种策略使算法搜索效率更高。相对于现有的三种先进的算法来说，大量的在Berkely图像数据集上的仿真实验显示了贝叶斯粒子群算法可以得到高效的、稳定的和比较平滑的分割结果。此外，我们也指出了贝叶斯粒子群算法进一步研究的方向和内容。
在本文的结论部分，我们对全文主要研究内容进行总结，讨论了现有工作中的不足之处，并且指出了在本文研究基础之上的后续研究内容和思想方法。
Optimization which has greatly influenced many disciplines and obtains rapid promotionand applications in so many engineering domains has received widespread attention as animportant scientific branch. Optimization has become an indispensable tool for many differentareas. Many optimization problems have been proved to be NP-Complete problems (NPC).So far there is no effective method of Polynomial time for NPC, and the computing time of atraditional optimization method grows exponentially with the scale of the problem. Thus,instead, researchers hoping to achieve optimal or near-optimal solutions in limited time havedeveloped a lot of Bionical algorithms such as, Genetic Algorithms, Estimation OfDistribution Algorithms, Particle Swarm Optimization, Tabu Search, hybrid optimizationstrategies and so on. This thesis focuses on Bionical Algorithms and Their Application onFunction Optimization and Multilevel Thresholding. The main works and innovative pointsare as follows:
1) Traditional Genetic Algorithms initialize their individuals stochastically and generatetoo many eccentric and homogeneous individuals. This will cause premature convergence andslow down the speed. In this thesis this important problem is studied via thecomplementary-parent strategy of initializing population in PGA. We analyze it and concludethat the limiting probability of the traditional mutation operator based on this strategy is1/|HL|2higher than on the traditional one. This strategy is more efficient by applying toCoarse-grained parallel genetic algorithm which develops the parallelism among populations.For PGA, we also discuss the reproductive capacity of excellent schemata based on theschema theorem and demonstrate the global convergence using homogeneous finite Markovchain. And then, we present an improved algorithm named GACPS. With some of typicalunsymmetrical functions tested, simulation show the quality of GACPS is much higher thanPGA on precision, stability and convergence rate.
2) Multilevel thresholding is an important technique for image compression, imageanalysis and pattern recognition. However, it is still a hard problem to determine the numberof thresholds automatically. So, a new multilevel thresholding method called as AutomaticMultilevel Thresholding Algorithm For Image Segmentation based on Block Sampling and Genetic Algorithm (AMT-BSGA), which can automatically determine the appropriate numberof thresholds and the proper threshold values, is proposed on the basis of Block Sampling andGenetic Algorithm. In AMT-BSGA, an image is treated as a population with the gray valuesof pixels as the individuals. First, an image is evenly divided into several blocks, and a sampleis drawn from each block. Then, genetic algorithm based optimizing is applied to each sampleso as to maximize the ratio of mean and variance for the sample. Based on the optimizedsamples, the number of thresholds and threshold values are preliminarily determined. Finally,a deterministic method is implemented to further optimize the number of thresholds andthreshold values. AMT-BSGA can work without knowing in advance other auxiliaryinformation, such as contextual or textual properties, and the number of thresholds. It has alower computing complexity which is almost independent from the number of thresholds andcan avoid the burden of analyzing histograms. AMT-BSGA can produce effective, efficientand smoother results, which has been verified by extensive simulations on Berkeley datasets.
3) Based on Bayesian theorem and the probability distribution of promising solutionsand integrating with the basic principle of evolutionary computation, we proposes a novelalgorithm named Bayesian Forecasting Evolutionary Algorithm (BFEA). Our proposedmethod guides the exploration of the search space according to the prediction probability ofevery subspace including the optimal solutions. Moreover, prior information about theproblem is incorporated into the algorithm easily and much more information in the generatedpopulations is used. More importantly, BFEA is a new method to solve linkage problem anddeceptive problem effectively. This algorithm can converge faster to the subspaces withoptimal solution and its convergence, convergence rate and counteraction operator are alsoanalyzed theoretically. Theoretical analyses and tests have demonstrated the proposed methodhas a simpler algorithm model, lower computing complexity and higher computationaccuracy.
4) A simpler and efficient PSO algorithm based on Bayesian theorem and the charactersof intensity images is proposed, called as Bayesian Particle Swarm Optimization algorithm(BPSO). In BPSO, a new method is designed to assign the constriction coefficient of the“social influence” term for each particle automatically and separately based on Bayesiantheorem, so that they can have different levels of exploration and exploitation capabilities. And a new population initialization strategy is adopted to make the search more efficient,according to the characters of multilevel thresholding in an image arranged from a low graylevel to a high one. The experimental results indicate that BPSO can produce effective,efficient and smoother segmentation results in comparison with three existing methods onBerkely datasets.
引文
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