基于马尔可夫随机场和模糊聚类的图像分割算法研究
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摘要
目前,基于马尔可夫的图像分割和基于模糊的图像分割已经成为图像处理中的重要领域。我们的研究主要集中在像素表现出的属性如何因区域的不同而不同。基于给定图像中像素之间的相似性和相异性,我们研究、开发并实现了一种新的图像分割方法。在本论文中,我们将计算机视觉领域中的分割定义为将一幅数字图像分离为多个像素集的过程,也称为超像素过程。我们认为分割的作用是：简化或改变图像的表现形式,使得图像更有意义而且更容易分析。图像分割通常用于定位图像中的物体、边界直线以及曲线。更精确地说,图像分割是给图像中的每个像素分配一个标签,使得具有相同标签的像素拥有某种共同的视觉特性。
现已有许多通用的算法和技术用于图像分割。由于图像分割问题没有统一的解决方法,因此为了更加有效地解决某特定领域中的图像分割问题,这些算法和技术通常要与相关领域的知识结合使用。
本文的中心内容是开发一个完全无监督的算法来实现图像分割。已有的文献达不到这样的目标,它们提供的许多算法只能解决这个极具挑战性的问题的一些子问题。
无监督分割是在事先不知道区域的数量的情况下识别并定位给定图像的组成区域的过程。我们还认为,这个问题可以在贝叶斯框架内进行阐述,并且通过使用一个假设模型能够将无监督分割转化为一个优化问题。
在整个图像分割领域,普遍采用的是一个分层的图像模型,其基本成分是各种形式的马尔可夫随机场。高斯马尔可夫随机场模型用于对图像区域的纹理内容进行建模,帕兹模型为图像分割提供了一个正则化函数。
这些高度复杂的模型的优化问题是几十年来一直极具挑战性的研究课题。本论文的主要贡献在于将单一优化过程这一新技术用于无监督分割。我们希望这些算法能促进未来分层图像模型的研究,特别是有助于发现能够更进一步拟合真实世界数据的深层次模型。
本文首先对围绕马尔可夫随机场模型及其优化的大量文献进行了阐述和分析,其中涉及到选择合适的特征来辨别被观察图像的纹理内容。在阐述和分析了大量文献的基础上,我们提出了新的算法,实现了源于这两个领域的概念之间的融合。
以往应用在统计力学上的算法是这项工作的重要组成部分。多样化的马尔可夫链蒙特卡罗算法被普遍运用,特别是可逆跳跃采样算法具有十分重要的意义。多个此类算法结合在一起构成了单一优化框架,该框架是本文提出的几个最成功算法的核心。
对于彩色图像分割,我们提出了一个新的方法,该方法将区域定义为属于同一类的像素的连通集合,并且同时考虑了像素的连通性和色度属性,以便建立非等概率的类。
我们将一个像素集合的颜色连通度定义为属于同一颜色区间的像素集合的连通性测量。我们假设图像中的某个区域的像素可以对应的被关联到一个像素类,这个像素类对应的像素集合的颜色连通度应该表现出很高的数值,辨别出这些集合是问题所在。
本文定义了一个关于分割马尔可夫的初始数据结构,即颜色连通度,它将有组织、有层次地计算图像中包含的所有可能的像素集的颜色连通度。
我们把每幅图像分解为R、G、B三个颜色通道,并对每个通道分别进行分析。为了从树边际分类过程得到的结果中收集属于同一区域的像素,我们用区域邻接图对图像进行建模,并进一步对它进行模糊分析。
最后,我们的实验通过用相似度图表示图像分区结果,证明了本文方法的有效性。
本文的目的是研究将先验信息应用于图像处理和计算图像纹理分割的效果。特别是,我们探讨了利用马尔可夫随机场模型对图像纹理分割中衰减系数分布的先验信息进行建模的适用性。这涉及到选择不同的模型并用它们对样本图片进行拟合。第二个目标是用这些模型来帮助解决一些图像处理问题并帮助确定它们的使用相比没有考虑先验信息的方法是否能够改善结果。
一幅图像可能具有很多特征,包括高层特征和底层特征。基于高层特征(例如线、边、区域、形状和运动等)的分割方法对于些简单图像的小范围应用来说效果很好。另外,从精度角度来看,基于高层特征的方法得到的分割结果比较粗糙,因为这些方法考虑的是区域同质性以及分割后的线、边和区域的连通性。在另一方面,底层特征(例如亮度、颜色、纹理和坐标等)能够有效地用于获得更高精度的分割结果。这些基于底层特征的方法能够应用于像素级并得到精度更高的像素聚类,进而得到比较好的分割精度。
基于这种背景,我们的工作主要基于某些领域彼此有联系的大量已有文献。我们的目标是通过使用分割技术对原始图像进行分割来改善或者增强这些不仅亮度暗而且对比度低的彩色图像。现有的文献还无法达到这个目标,它们提供的许多方法都只能解决这个高挑战性问题的子问题。
图像处理中的许多问题可以看成以寻找最小化代价函数的参数估计为目的的优化问题。我们考虑使用马尔可夫随机场来解决我们的优化问题。另一个方法是使用贝叶斯最大化。贝叶斯最大化后验概率估计的目标是找到后验概率的最大值。后验概率是似然分布和先验分布的结合。其中,似然分布与解空间的度量数据有关,先验分布则包含的是可能解的先验信息。如第三、四、五章所述,这个分布已经被模型化为模糊分布或者马尔可夫随机场。
为了达到更好的目标,本文主要关注图像纹理分割领域,将像素定位到我们想要提取信息的更佳位置。
我们认为,从定义上讲,图像纹理分割允许我们知道轮廓的边界和图像中区域的内容,就如图1.1所述的那样。
图像分割已被发现是非常有用的,它具有挑战性的研究领域是了解和确定更多的信息来分析数据。这些数据可以来自气象图片、林业图片、海上浮油图像、脑部医疗图像以及安全部门或临近国家在机场共享的面部图像数据。
对于基于底层特征的图像分析,把待观测图像分割成未知数量的、彼此不同的、且在一定程度上同质的区域仍然是一个基本的问题。有很多关于这些算法的直接应用,例如超声图像的分割、通过合成孔径雷达图像辨别农作物、核磁共振图像和X射线图像的分割、遥感图像的分割以及用于防治质量检查的图像分割等。另外,图像分割也通常被看作是为复杂视觉系统的高层处理提供入口的早期关键处理过程。许多不同的方法已应用于图像分割,然而,既完全无监督又具有鲁棒性的方法还没有被实现。因此,现在非常需要新方法根据提取信息的上下文环境来提高图像分割的效果。此外,新方法应该帮助理解坐标平面或空间中像素的能量分布函数。
图像处理就其本身而言并不是一个新的问题,但它多年来一直不断发展。当图像很清晰或者可读性高的时候,更容易实现对图像较好的解释和分析。出于这个原因,我们需要改善和增强某些可见性差或者物体外观不清晰的图像。正如前面所提到的,图像处理对于全球安全受到威胁的世界来说是很有用的。随着全球动态和安全问题的不断变化,许多国家需要了解和知道如何处理来自我们日常生活中不同方面的图像。值得注意的应用包括：
第一,通过卫星图像的灾难(例如海啸和风暴)信息预测
为了预测灾祸和自然灾难,我们需要分析卫星图像,为了分析卫星图像,我们需要将图像细分成小图像,因为图像越小,细节就越多,解释也越准确。好的、精确的图像分析能够有效地帮助人们避免自然灾难,例如洪水、地震等等。
第二,安全系统
图像处理在安全方面有着巨大的使用价值,特别是现在世界各地恐怖主义盛行。一个好的、精确的图像分析能够帮助检测、定位和逮捕犯罪分子。此外,国家之间可以共享通缉犯的照片并且可以建立固定的图像数据库帮助跟踪流窜在不同国家和大洲的所有罪犯。
第三,与护照相符的指纹数据库
我们使用指纹可以检索到一个国家或区域以及临近其他国家的所有国民的任何信息,只要这个国家或地区建立了独一无二的数据库,就像使用独一无二的护照一样。
在本文中我们不仅提供了多种用于彩色图像分区和分割的方法,还提供了相关方法来增强彩色图像,这对分析和理解一大批彩色图像时很必要的。我们还提供了一个基于马尔可夫随机场模型和模糊图像分割的纹理分析和分类方法。有时候图像不是很清晰,所以我们需要额外的分析来增强和调整这些图像。
为了能够在马尔可夫随机场模型中检测到纹理,我们通过在分割区域上施加合理的物理约束(比如区域的连通性)来对图像(纹理)进行分割。这些技术已经被证明能够改善分割结果。在这些技术中,参数概率模型虽然没有充分的物理证明,但是仍经常用于对被观测图像数据进行建模,因为该模型易于计算。我们希望通过提供一个更好的模型能够改善已有的基于马尔可夫随机场的图像分割技术。
多年以来,马尔可夫模型应用越来越广泛,已经成为分类、降噪以及模式识别领域中功能强大的统计工具。多组分图像的分割过程中需要一个不一般的模型化步骤,以便给不同的像素集合分配标签,其中像素集合由具有相似行为的像素组成。
图像分割面临的诸多挑战促使研究人员研发出模糊分割算法,该算法将图像区域看作模糊子集(模糊对象),图像中的每个像素可以被划分到多个潜在的类中,并且不同对象的亮度之间的界限可以明确界定。由于固有物体材料的各向异性以及成像设备本身的人为因素影响,图像可能会变模糊(例如物体模糊不清、噪声以及背景亮度不均匀等)。本文基于模糊集理论可以有效地对图像中像素的模糊现象进行建模。
通过查阅文献可知,已经有一些图像分割方法是基于模糊的概念提出的。在这些方法中,模糊连通性和模糊聚类是最常用的两种技术。另外,基于模糊规则的方法、模糊阈值、模糊马尔可夫随机场以及模糊区域增长等技术在基于区域的模糊分割中也均被提及。
图像分割领域中的马尔可夫随机场理论是物理学中用于分析空间或上下文相关性的概率论的一个分支。马尔可夫随机场理论的基础来源于磁性材料的统计物理学,但是它也经常被用在图像处理技术中,因为这种方法定义了一个能够描述相邻像素之间连贯性的模型。
我们利用MRF在非监督情况下分割灰度级图像。MRF是依据局部空间相互作用对图像像素的联合概率分布进行建模的强有力工具,通常也称为条件迭代模式(ICM)。
我们主要考虑在没有任何关于模型参数的先验信息的情况下使用MRF对灰度级图像进行分割。很多方法预先假设分类的数量是已知的。在这种情况下,我们用期望最大化(EM)算法来对参数进行估计并用最小信息长度(MML)算法来估计类的数量。
第二章详细阐述了马尔可夫随机场理论在图像分割、图像分区和图像复原等领域的开发和应用。我们对原始图像进行分割和分析,并根据不同的上下文解释提取的信息。
我们还考虑了一个关于节点和标签的示例。我们强调,这些概念需要了解马尔可夫随机场以及怎么将它们用于对图像进行建模。对于一幅图像,定义一个概率密度分布,要求必须给每个可能的标签结构或图像中的亮度分配一个概率。
然后,我们通过直方图曲线分析了图像的亮度变化。图像中每个不同的坐标位置对应一个像素,是构成图像的最小单元。不同类型的图像,其每个像素点(x,y)所输出的值的性质不同。大多数图像是对应一个特定的物理量度量结果,如亮度、温度、距离或能量。这个度量结果可以采用任何的数值形式。
灰度级图像度量的是光的强度。每个像素是一个标量,该标量与亮度成比例。最小的亮度称为黑色,最大的亮度称为白色。图2.3给出了一个典型的例子。彩色图像度量的是光的强度和色度。
灰度直方图描述了灰度图像中每个像素值出现的相对频率。图2.3(b)描述了直方图均衡化,图2.3(c)给出了图2.3(a)中图像的亮度直方图。灰度直方图为图像中的亮度提供了一个简易的概述,但却不能传达任何有关像素之间空间关系的信息。在此示例中,图像不包含许多亮度非常低或非常高的像素。灰度直方图中的峰值有可能对应图像中的对象,但是如果不从视觉上对图像进行检查很难确定它是不是对的。
在本文中,我们分析并强调了大部分在统计力学领域发展起来的马尔可夫随机场理论。
关于吉布斯分布,我们强调马尔可夫随机场和吉布斯分布是等价的。一个马尔可夫随机场的吉布斯分布只是那个马尔可夫随机场的联合概率。P(f)是一个在点阵S上的吉布斯分布。则P(f)由2.2.2节中的方程(2.4)给出其中
Z是一个被称为分区函数的归一化常量。计算分区函数涉及到把所有可能的阵列归一化,可能的阵列的数量为Mm,M是每个节点的标签数目,m是点阵含有的节点数,Z有时被称为系统的自由能量。
公式2.4中的能量函数方程U(f)是点阵S上所有集团的集团势函数Vc(f)之和,如图公式2.6所示。高能量的阵列发生的概率较小。
对于所有可能的标签阵列,能量U(f)和集团势函数Vc(f)都应该是正的,才能使方程2.4进行正常的归一化。这种正性约束可以通过将集团势函数减去L域中势函数的最小值来强制执行,如公式2.7所示。这样做是为了保证除了在公式3.19事先统一定义的之外所有的集团势函数对所有可能的标签都是正数。
一个集团的阶由集团的节点数量给出。一阶集团势函数是一个节点的标签函数,二阶集团势函数是两个节点的标签函数并且是能够传达上下文信息或者对相邻节点的相关性进行建模的最低阶集团势函数。
如此,我们通过在2.5.1节设立的实验假设来完成本章节内容。
本节还分析了一个基于多个马尔可夫信息理论的彩色图像分区算法。第一次分割我们尝试使用粗糙的阈值分割技术,而第二次分割通过考虑图像中的每个像素来减少分类错误。
阈值和数据融合技术可以帮助我们根据不同的区域来对像素进行重新划分。结果表明,在每个迭代步骤中最相关的颜色空间选择可以提供最相关的分割结果。
第三章中,我们研究了基于模糊的图像分割和基于马尔可夫随机场的图像分割方法。为了保持一致性,我们首先定义分割图像中所涉及到实体的用语。
-区域：一个区域是一组连通的像素集合,其面积大于零,在一个分割后的图像中具有相同的标签。它在之后的无监督中也会提及到。
-边缘：边缘是位于两个相邻区域的交界处的连通的像素集合。
对于大多数现实世界中的图像,其中的亮度、颜色、纹理、物体边界以及形状等会存在不规则的起伏变化。导致这些起伏的原因是多样的、复杂的,它们往往是由于不均匀的照明、物体表面方向和纹理的不规则起伏变化、复杂的场景几何以及噪声等因素造成的。因此,这样的图像处理可以转化为统计推论问题,这就需要对图像像素定义一个统计模型。
我们也认为,常染色体模型或高斯模型可以通过它的均值和方差参数项定义一个连续的标签集合L。它的最大优点是,归一化常数可以在解析解的形式下求得。这有助于提高该模型的计算效率。方程(3.1)给出了一个节点标签关于邻近节点标签的条件概率密度函数。
非连续自适应模型的设计允许边缘的形成,同时仍然保持边缘周围的平滑性。边缘可以被视为近似平滑区域之间的边界。图像中位于边缘上的点被归类为离群点,不对他们进行平滑处理。随后我们给出了四个用于非连续自适应模型的势函数示例。
我们还着重强调了3.6.1小节中的Gibbs采样算法和Metropolis采样算法,来更新点阵S中节点i处的当前标签,并计算当前标签的概率与更新后标签的概率之间的比值。更新后的标签会按照算法3中的概率值被接受。
最后,在3.7.6节中,我们用图像仿真进行了假设检验,结果是令人满意的。
在第四章中我们重点介绍了一些纹理分割方法。纹理分割方法最初源自German,主要针对亮度相似的区域间的边界问题。我们强调了区域合并结果的判别标准：首先,通过类似于分割步骤中的测试方法,合并后的区域必须是均匀的；其次,伪似然度比率必须比预定义的阈值小。图像中所有的区域会迭代进行合并,直到没有能进一步合并的可能。共生矩阵中的元素表示那些在图像网格的某个特殊分割中两个灰度值出现的相对频率。
正如第一章中所介绍的,我们还考虑了基于模型的统计。如果通过模型得出的图像有问题,那么构成模型参数的最明显的那些特征向量将将对自己进行估计,因为从定义上讲,结合我们的先验知识,它们将会形成更加无损的特征集合。
在整篇文章中,分层的高斯马尔可夫随机场(GMRF)通常被选为区域模型。在选定了一组特征集后,为了获得所需的分割结果,某种形式的分类过程仍然是必需的。正如在引言中所介绍的,它通常包括两个步骤。这些算法将进行更详细地调试。
为了实现粗分割,首先要把图像分为若干个非重叠的窗口,然后分别计算每个窗口中的特征向量。如上所述,每个特征向量构成GMRF模型参数的估计值,具体地像PHI、均值和方差等。最小乘参数估计值虽然影响估计模型的稳定性,但是仍会被使用,因为它允许以最小的计算负担获得估计值。
对于局部空间统计,我们认为来自局部空间灰度级统计的纹理特征是这个纹理分割领域使用最广泛的,例如：使用合成孔径雷达图像来辨别农作物；超声图像的纹理辨别；以及更多的应用于遥感领域的技术。
另外,根据对人类早期视觉系统的理解,多通道滤波方法已经应用于纹理分割。它通过一系列的心理物理实验演变而来。视觉处理过程被简单地假设为空间域的特征检测,早期视觉系统的初始模型,就是在这种假设基础上提出的。最后,我们以图4.6(a)熊猫原始图像为例,对其进行了分割并分析了图像直方图曲线的方差,还得出了高斯模型曲线以及一些图像仿真结果。
第五章中,我们通过将给定图像分离为均匀的、有组织的区域来阐述马尔可夫随机场的优点。如果观测图像少定义为一个M×N的长方形矩阵Ω,那么标签x也将定义为同等大小的矩阵。为了对原始图像进行建模,需要考虑分层模型中节点标签之间的交互。不同之处在于选择最大似然(ML)还是MAP标准进行模型参数估计
为了提取信息,我们以图5.1的图像为例来对这个图像中的现象进行分析和建模。航空公司的飞机在起飞或着陆的过程中有时会由于鸟、蛇、狗等动物而遭受安全问题,而且这些动物可能会给航空公司造成非常昂贵的损失。在这一章中,我们尝试改进一个模型算法来定位这些鸟类所在的区域半径,因为几乎所有的飞机着陆事故都是由于这些鸟类导致的。
通过这个例子,我们看到,图像的直方图变换是一个数据立方体,其由任意给定位置的像素点的固定邻域的像素灰度直方图组成。这样的转换在图像处理应用中非常有用,例如分类与分割,尤其是在处理纹理特征时可以通过它们的像素亮度和颜色的分布来区分。
我们知道高斯马尔可夫随机场(GMRF)是广泛应用于各种自然纹理和人造纹理的建模过程。大量的分类和重建实验证明了它的有效性。
我们也在5.5.6节中提供了仿真实验,设计了一个用于图像测试的算法。我们将这个算法用于处理测试图像集,展示了基于模糊隐马尔可夫链(HMC)的分割过程。在尝试了多张图像后,我们设定迭代次数能够得到成功的子类数量。
我们在第六章强调了用马尔科夫链蒙特卡洛算法从后验分布进行采样的无监督图像分割方法。我们注意到无监督分割算法比那些基于特征的算法有优势,因为其不需要任何窗口来估计区域模型参数,即对于图形元素的最小尺寸及形状都没有限制。对于一个特殊的模型,我们认为贝叶斯参数估计能够找出最适合某些特定观测数据的参数值。
然后我们用仿真算法对石油泄漏以及动植物系统图像进行图像分割和分区测试。
在石油泄漏的假设检验示例中,我们使用图像分割算法来研究石油泄漏对生态系统环境的影响。我们的海洋生态环境经常可能出现两栖类和爬行类物种的安全性问题以及水产养殖业的安全性问题。我们的想法是识别出由各个领域的人员造成的可能影响这些物种正常生活的任何障碍。这里我们强调了这一点,并通过分割图像以及提取有用的信息来准确地解释这些问题。
我们知道石油泄漏是指由于人类的活动导致液体石油烃释放到环境中,特别是海洋区域,这是一种污染。这个词通常专指海上漏油,即石油被释放到海洋和沿海水域,但石油泄漏也可能发生在陆地上。石油泄漏的原因可能是来自于油轮、海上平台、钻探设备和油井,也可能是精炼石油产品(如汽油,柴油)以及它们的副产品的泄漏,还可能是大型船舶使用的锅炉燃油或任何形式的废油。石油进入海洋环境的另一个重要的途径是通过自然渗透。
在本节,我们还通过图像6.1(从a到f)来展示基于不同区域贝叶斯估计的图像分区算法。
我们认为,对于一个给定的特殊模型,贝叶斯参数估计的作用是找出该模型关于某些观测数据的最佳拟合参数值。为了达到这个目标,必须制定一个关于选择标准的初始假设。如果我们的主要目的是选择一个参数估计方法来最小化平均误分概率,通过使用0-1效用函数,贝叶斯决策过程就变成选择参数来最大化后验概率,也就是最大化后验(MAP)概率估计。
最后,我们介绍了关于噪声图像和纹理图像的无监督分割算法。使用的模型都已经被定义,特别是各向同性的分层马尔可夫随机场以及高斯马尔可夫随机场区域模型。模拟退火算法被用于优化模型参数、类的数量以及像素的标签数量,这需要马尔可夫链蒙特卡洛(MCMC)采样算法能够从该模型的后验分布中对这些参数进行采样。同时使用吉布斯采样算法、Metropolis-Hastings算法和可逆跳MCMC算法对合并后的模型空间进行快速而完整的搜索。
实验结果展示了合成图像通过快速分割完成了对原始底层图像的准确估计。真实世界中的图像也能够成功地完成分割。实验结果同样表现出意想不到的收敛性,这进一步证明了在用模拟退火实现图像分割时甚至在区域数量已知时使用可逆跳跃MCMC算法的合理性。
Currently, Markov Random Field and Fuzzy Image Segmentation algorithms have become important fields in image processing research. In image processing system, segmentation process is one of the most important steps. More precisely, image segmentation is defined as the process of assigning a label to every pixel in an image such that pixels with the same label share certain visual characteristics. Several general-purpose algorithms and techniques have been developed for image segmentation. In order to effectively solve image segmentation problem for a specific problem domain, these techniques often have to be combined with knowledge domains, as there is no general solution to image segmentation problems.
Our research focused on how pixels display differently with different regions. We investigated a new approach to image segmentation, developed and implemented algorithms based on similarities or dissimilarities of the pixels in a given image.
The central theme of this dissertation is the development of a fully Markov unsupervised algorithm to achieve image segmentation. Existing literature falls short of such a goal, only providing many algorithms capable of solving subsets of this highly challenging problem. Unsupervised segmentation is the process of identifying and locating the constituent regions of an observed image, while having no prior knowledge of the number of regions. In this regard, in our work we consider the use of Bayesian framework to model the image, so that the unsupervised segmentation process can be considered as a problem of single optimization. However, we have discovered that throughout the literature, the commonly adopted optimization model is a hierarchical image model whose underlying components are various forms of Markov Random Fields. Gaussian Markov Random Field models are used to model the textural content of the image regions, while a Potts model provides a regularization function for the image segmentation as could be seen in our work.
The optimization process of such highly complicated models is an area that has been challenging researchers for several decades. So the main contribution of this thesis is to develop new techniques for Markov unsupervised segmentation to be carried out using a single optimization process. It is hoped that these algorithms will facilitate and enhance the future study of hierarchical image models and in particular enable the discovery of further models capable of more precise fitting of real world data.
Markov Random Field segmentation models and their optimizations deal with the selection of features to identify the textural contents of an observed image. In the light of the above, new algorithms have been proposed for the attainment of a balance between the two concepts: unsupervised segmentation and optimization process. Furthermore, Statistical Mechanics forms an important part of this work. The prevalent use of Markov Chain Monte Carlo algorithms is encountered and in particular, the use of the Reversible Jump Sampling algorithm being of great significance is encountered too. The combination of several of these algorithms enabled us to achieve the single optimization framework, which makes us obtained the most successful novel algorithms used in this dissertation.
However, for the color images segmentation, we have proposed a method which considers that the regions are defined as connected sets of pixels belonging to the same class. Our approach considers simultaneously both the connectivity and colorimetric properties of the pixels in order to build classes which can be non-equiprobable.
We defined the color connectivity degree of a set of pixels as a connectivity measurement of a set of pixels whose colors belong to a color interval. We suppose that pixels of each region in the image can be associated to a class of pixels, and that a class is a set of pixels whose color connectivity degree presents a high value. The problem here is identifying these sets.
In this context, we have defined an original data structure of Markovianity, the color connectivity degree, which counts in an organized and hierarchical way the color connectivity degrees of all possible sets of pixels that an image can contain.
For each one of these images, we decompose the image at three composite (RGB) and propose an analysis method for each. In order to gather pixels belonging to the same region from the results obtained by the Tree Marginal Classification processes, we model the image by a Region Adjacency Graph and propose a fuzzy analysis for it.
Finally, through our various experiments with the partitioning of image by similarity graphs, the efficiency of this method has been effectively proven.
现已有许多通用的算法和技术用于图像分割。由于图像分割问题没有统一的解决方法,因此为了更加有效地解决某特定领域中的图像分割问题,这些算法和技术通常要与相关领域的知识结合使用。
本文的中心内容是开发一个完全无监督的算法来实现图像分割。已有的文献达不到这样的目标,它们提供的许多算法只能解决这个极具挑战性的问题的一些子问题。
无监督分割是在事先不知道区域的数量的情况下识别并定位给定图像的组成区域的过程。我们还认为,这个问题可以在贝叶斯框架内进行阐述,并且通过使用一个假设模型能够将无监督分割转化为一个优化问题。
在整个图像分割领域,普遍采用的是一个分层的图像模型,其基本成分是各种形式的马尔可夫随机场。高斯马尔可夫随机场模型用于对图像区域的纹理内容进行建模,帕兹模型为图像分割提供了一个正则化函数。
这些高度复杂的模型的优化问题是几十年来一直极具挑战性的研究课题。本论文的主要贡献在于将单一优化过程这一新技术用于无监督分割。我们希望这些算法能促进未来分层图像模型的研究,特别是有助于发现能够更进一步拟合真实世界数据的深层次模型。
本文首先对围绕马尔可夫随机场模型及其优化的大量文献进行了阐述和分析,其中涉及到选择合适的特征来辨别被观察图像的纹理内容。在阐述和分析了大量文献的基础上,我们提出了新的算法,实现了源于这两个领域的概念之间的融合。
以往应用在统计力学上的算法是这项工作的重要组成部分。多样化的马尔可夫链蒙特卡罗算法被普遍运用,特别是可逆跳跃采样算法具有十分重要的意义。多个此类算法结合在一起构成了单一优化框架,该框架是本文提出的几个最成功算法的核心。
对于彩色图像分割,我们提出了一个新的方法,该方法将区域定义为属于同一类的像素的连通集合,并且同时考虑了像素的连通性和色度属性,以便建立非等概率的类。
我们将一个像素集合的颜色连通度定义为属于同一颜色区间的像素集合的连通性测量。我们假设图像中的某个区域的像素可以对应的被关联到一个像素类,这个像素类对应的像素集合的颜色连通度应该表现出很高的数值,辨别出这些集合是问题所在。
本文定义了一个关于分割马尔可夫的初始数据结构,即颜色连通度,它将有组织、有层次地计算图像中包含的所有可能的像素集的颜色连通度。
我们把每幅图像分解为R、G、B三个颜色通道,并对每个通道分别进行分析。为了从树边际分类过程得到的结果中收集属于同一区域的像素,我们用区域邻接图对图像进行建模,并进一步对它进行模糊分析。
最后,我们的实验通过用相似度图表示图像分区结果,证明了本文方法的有效性。
本文的目的是研究将先验信息应用于图像处理和计算图像纹理分割的效果。特别是,我们探讨了利用马尔可夫随机场模型对图像纹理分割中衰减系数分布的先验信息进行建模的适用性。这涉及到选择不同的模型并用它们对样本图片进行拟合。第二个目标是用这些模型来帮助解决一些图像处理问题并帮助确定它们的使用相比没有考虑先验信息的方法是否能够改善结果。
一幅图像可能具有很多特征,包括高层特征和底层特征。基于高层特征(例如线、边、区域、形状和运动等)的分割方法对于些简单图像的小范围应用来说效果很好。另外,从精度角度来看,基于高层特征的方法得到的分割结果比较粗糙,因为这些方法考虑的是区域同质性以及分割后的线、边和区域的连通性。在另一方面,底层特征(例如亮度、颜色、纹理和坐标等)能够有效地用于获得更高精度的分割结果。这些基于底层特征的方法能够应用于像素级并得到精度更高的像素聚类,进而得到比较好的分割精度。
基于这种背景,我们的工作主要基于某些领域彼此有联系的大量已有文献。我们的目标是通过使用分割技术对原始图像进行分割来改善或者增强这些不仅亮度暗而且对比度低的彩色图像。现有的文献还无法达到这个目标,它们提供的许多方法都只能解决这个高挑战性问题的子问题。
图像处理中的许多问题可以看成以寻找最小化代价函数的参数估计为目的的优化问题。我们考虑使用马尔可夫随机场来解决我们的优化问题。另一个方法是使用贝叶斯最大化。贝叶斯最大化后验概率估计的目标是找到后验概率的最大值。后验概率是似然分布和先验分布的结合。其中,似然分布与解空间的度量数据有关,先验分布则包含的是可能解的先验信息。如第三、四、五章所述,这个分布已经被模型化为模糊分布或者马尔可夫随机场。
为了达到更好的目标,本文主要关注图像纹理分割领域,将像素定位到我们想要提取信息的更佳位置。
我们认为,从定义上讲,图像纹理分割允许我们知道轮廓的边界和图像中区域的内容,就如图1.1所述的那样。
图像分割已被发现是非常有用的,它具有挑战性的研究领域是了解和确定更多的信息来分析数据。这些数据可以来自气象图片、林业图片、海上浮油图像、脑部医疗图像以及安全部门或临近国家在机场共享的面部图像数据。
对于基于底层特征的图像分析,把待观测图像分割成未知数量的、彼此不同的、且在一定程度上同质的区域仍然是一个基本的问题。有很多关于这些算法的直接应用,例如超声图像的分割、通过合成孔径雷达图像辨别农作物、核磁共振图像和X射线图像的分割、遥感图像的分割以及用于防治质量检查的图像分割等。另外,图像分割也通常被看作是为复杂视觉系统的高层处理提供入口的早期关键处理过程。许多不同的方法已应用于图像分割,然而,既完全无监督又具有鲁棒性的方法还没有被实现。因此,现在非常需要新方法根据提取信息的上下文环境来提高图像分割的效果。此外,新方法应该帮助理解坐标平面或空间中像素的能量分布函数。
图像处理就其本身而言并不是一个新的问题,但它多年来一直不断发展。当图像很清晰或者可读性高的时候,更容易实现对图像较好的解释和分析。出于这个原因,我们需要改善和增强某些可见性差或者物体外观不清晰的图像。正如前面所提到的,图像处理对于全球安全受到威胁的世界来说是很有用的。随着全球动态和安全问题的不断变化,许多国家需要了解和知道如何处理来自我们日常生活中不同方面的图像。值得注意的应用包括：
第一,通过卫星图像的灾难(例如海啸和风暴)信息预测
为了预测灾祸和自然灾难,我们需要分析卫星图像,为了分析卫星图像,我们需要将图像细分成小图像,因为图像越小,细节就越多,解释也越准确。好的、精确的图像分析能够有效地帮助人们避免自然灾难,例如洪水、地震等等。
第二,安全系统
图像处理在安全方面有着巨大的使用价值,特别是现在世界各地恐怖主义盛行。一个好的、精确的图像分析能够帮助检测、定位和逮捕犯罪分子。此外,国家之间可以共享通缉犯的照片并且可以建立固定的图像数据库帮助跟踪流窜在不同国家和大洲的所有罪犯。
第三,与护照相符的指纹数据库
我们使用指纹可以检索到一个国家或区域以及临近其他国家的所有国民的任何信息,只要这个国家或地区建立了独一无二的数据库,就像使用独一无二的护照一样。
在本文中我们不仅提供了多种用于彩色图像分区和分割的方法,还提供了相关方法来增强彩色图像,这对分析和理解一大批彩色图像时很必要的。我们还提供了一个基于马尔可夫随机场模型和模糊图像分割的纹理分析和分类方法。有时候图像不是很清晰,所以我们需要额外的分析来增强和调整这些图像。
为了能够在马尔可夫随机场模型中检测到纹理,我们通过在分割区域上施加合理的物理约束(比如区域的连通性)来对图像(纹理)进行分割。这些技术已经被证明能够改善分割结果。在这些技术中,参数概率模型虽然没有充分的物理证明,但是仍经常用于对被观测图像数据进行建模,因为该模型易于计算。我们希望通过提供一个更好的模型能够改善已有的基于马尔可夫随机场的图像分割技术。
多年以来,马尔可夫模型应用越来越广泛,已经成为分类、降噪以及模式识别领域中功能强大的统计工具。多组分图像的分割过程中需要一个不一般的模型化步骤,以便给不同的像素集合分配标签,其中像素集合由具有相似行为的像素组成。
图像分割面临的诸多挑战促使研究人员研发出模糊分割算法,该算法将图像区域看作模糊子集(模糊对象),图像中的每个像素可以被划分到多个潜在的类中,并且不同对象的亮度之间的界限可以明确界定。由于固有物体材料的各向异性以及成像设备本身的人为因素影响,图像可能会变模糊(例如物体模糊不清、噪声以及背景亮度不均匀等)。本文基于模糊集理论可以有效地对图像中像素的模糊现象进行建模。
通过查阅文献可知,已经有一些图像分割方法是基于模糊的概念提出的。在这些方法中,模糊连通性和模糊聚类是最常用的两种技术。另外,基于模糊规则的方法、模糊阈值、模糊马尔可夫随机场以及模糊区域增长等技术在基于区域的模糊分割中也均被提及。
图像分割领域中的马尔可夫随机场理论是物理学中用于分析空间或上下文相关性的概率论的一个分支。马尔可夫随机场理论的基础来源于磁性材料的统计物理学,但是它也经常被用在图像处理技术中,因为这种方法定义了一个能够描述相邻像素之间连贯性的模型。
我们利用MRF在非监督情况下分割灰度级图像。MRF是依据局部空间相互作用对图像像素的联合概率分布进行建模的强有力工具,通常也称为条件迭代模式(ICM)。
我们主要考虑在没有任何关于模型参数的先验信息的情况下使用MRF对灰度级图像进行分割。很多方法预先假设分类的数量是已知的。在这种情况下,我们用期望最大化(EM)算法来对参数进行估计并用最小信息长度(MML)算法来估计类的数量。
第二章详细阐述了马尔可夫随机场理论在图像分割、图像分区和图像复原等领域的开发和应用。我们对原始图像进行分割和分析,并根据不同的上下文解释提取的信息。
我们还考虑了一个关于节点和标签的示例。我们强调,这些概念需要了解马尔可夫随机场以及怎么将它们用于对图像进行建模。对于一幅图像,定义一个概率密度分布,要求必须给每个可能的标签结构或图像中的亮度分配一个概率。
然后,我们通过直方图曲线分析了图像的亮度变化。图像中每个不同的坐标位置对应一个像素,是构成图像的最小单元。不同类型的图像,其每个像素点(x,y)所输出的值的性质不同。大多数图像是对应一个特定的物理量度量结果,如亮度、温度、距离或能量。这个度量结果可以采用任何的数值形式。
灰度级图像度量的是光的强度。每个像素是一个标量,该标量与亮度成比例。最小的亮度称为黑色,最大的亮度称为白色。图2.3给出了一个典型的例子。彩色图像度量的是光的强度和色度。
灰度直方图描述了灰度图像中每个像素值出现的相对频率。图2.3(b)描述了直方图均衡化,图2.3(c)给出了图2.3(a)中图像的亮度直方图。灰度直方图为图像中的亮度提供了一个简易的概述,但却不能传达任何有关像素之间空间关系的信息。在此示例中,图像不包含许多亮度非常低或非常高的像素。灰度直方图中的峰值有可能对应图像中的对象,但是如果不从视觉上对图像进行检查很难确定它是不是对的。
在本文中,我们分析并强调了大部分在统计力学领域发展起来的马尔可夫随机场理论。
关于吉布斯分布,我们强调马尔可夫随机场和吉布斯分布是等价的。一个马尔可夫随机场的吉布斯分布只是那个马尔可夫随机场的联合概率。P(f)是一个在点阵S上的吉布斯分布。则P(f)由2.2.2节中的方程(2.4)给出其中
Z是一个被称为分区函数的归一化常量。计算分区函数涉及到把所有可能的阵列归一化,可能的阵列的数量为Mm,M是每个节点的标签数目,m是点阵含有的节点数,Z有时被称为系统的自由能量。
公式2.4中的能量函数方程U(f)是点阵S上所有集团的集团势函数Vc(f)之和,如图公式2.6所示。高能量的阵列发生的概率较小。
对于所有可能的标签阵列,能量U(f)和集团势函数Vc(f)都应该是正的,才能使方程2.4进行正常的归一化。这种正性约束可以通过将集团势函数减去L域中势函数的最小值来强制执行,如公式2.7所示。这样做是为了保证除了在公式3.19事先统一定义的之外所有的集团势函数对所有可能的标签都是正数。
一个集团的阶由集团的节点数量给出。一阶集团势函数是一个节点的标签函数,二阶集团势函数是两个节点的标签函数并且是能够传达上下文信息或者对相邻节点的相关性进行建模的最低阶集团势函数。
如此,我们通过在2.5.1节设立的实验假设来完成本章节内容。
本节还分析了一个基于多个马尔可夫信息理论的彩色图像分区算法。第一次分割我们尝试使用粗糙的阈值分割技术,而第二次分割通过考虑图像中的每个像素来减少分类错误。
阈值和数据融合技术可以帮助我们根据不同的区域来对像素进行重新划分。结果表明,在每个迭代步骤中最相关的颜色空间选择可以提供最相关的分割结果。
第三章中,我们研究了基于模糊的图像分割和基于马尔可夫随机场的图像分割方法。为了保持一致性,我们首先定义分割图像中所涉及到实体的用语。
-区域：一个区域是一组连通的像素集合,其面积大于零,在一个分割后的图像中具有相同的标签。它在之后的无监督中也会提及到。
-边缘：边缘是位于两个相邻区域的交界处的连通的像素集合。
对于大多数现实世界中的图像,其中的亮度、颜色、纹理、物体边界以及形状等会存在不规则的起伏变化。导致这些起伏的原因是多样的、复杂的,它们往往是由于不均匀的照明、物体表面方向和纹理的不规则起伏变化、复杂的场景几何以及噪声等因素造成的。因此,这样的图像处理可以转化为统计推论问题,这就需要对图像像素定义一个统计模型。
我们也认为,常染色体模型或高斯模型可以通过它的均值和方差参数项定义一个连续的标签集合L。它的最大优点是,归一化常数可以在解析解的形式下求得。这有助于提高该模型的计算效率。方程(3.1)给出了一个节点标签关于邻近节点标签的条件概率密度函数。
非连续自适应模型的设计允许边缘的形成,同时仍然保持边缘周围的平滑性。边缘可以被视为近似平滑区域之间的边界。图像中位于边缘上的点被归类为离群点,不对他们进行平滑处理。随后我们给出了四个用于非连续自适应模型的势函数示例。
我们还着重强调了3.6.1小节中的Gibbs采样算法和Metropolis采样算法,来更新点阵S中节点i处的当前标签,并计算当前标签的概率与更新后标签的概率之间的比值。更新后的标签会按照算法3中的概率值被接受。
最后,在3.7.6节中,我们用图像仿真进行了假设检验,结果是令人满意的。
在第四章中我们重点介绍了一些纹理分割方法。纹理分割方法最初源自German,主要针对亮度相似的区域间的边界问题。我们强调了区域合并结果的判别标准：首先,通过类似于分割步骤中的测试方法,合并后的区域必须是均匀的；其次,伪似然度比率必须比预定义的阈值小。图像中所有的区域会迭代进行合并,直到没有能进一步合并的可能。共生矩阵中的元素表示那些在图像网格的某个特殊分割中两个灰度值出现的相对频率。
正如第一章中所介绍的,我们还考虑了基于模型的统计。如果通过模型得出的图像有问题,那么构成模型参数的最明显的那些特征向量将将对自己进行估计,因为从定义上讲,结合我们的先验知识,它们将会形成更加无损的特征集合。
在整篇文章中,分层的高斯马尔可夫随机场(GMRF)通常被选为区域模型。在选定了一组特征集后,为了获得所需的分割结果,某种形式的分类过程仍然是必需的。正如在引言中所介绍的,它通常包括两个步骤。这些算法将进行更详细地调试。
为了实现粗分割,首先要把图像分为若干个非重叠的窗口,然后分别计算每个窗口中的特征向量。如上所述,每个特征向量构成GMRF模型参数的估计值,具体地像PHI、均值和方差等。最小乘参数估计值虽然影响估计模型的稳定性,但是仍会被使用,因为它允许以最小的计算负担获得估计值。
对于局部空间统计,我们认为来自局部空间灰度级统计的纹理特征是这个纹理分割领域使用最广泛的,例如：使用合成孔径雷达图像来辨别农作物；超声图像的纹理辨别；以及更多的应用于遥感领域的技术。
另外,根据对人类早期视觉系统的理解,多通道滤波方法已经应用于纹理分割。它通过一系列的心理物理实验演变而来。视觉处理过程被简单地假设为空间域的特征检测,早期视觉系统的初始模型,就是在这种假设基础上提出的。最后,我们以图4.6(a)熊猫原始图像为例,对其进行了分割并分析了图像直方图曲线的方差,还得出了高斯模型曲线以及一些图像仿真结果。
第五章中,我们通过将给定图像分离为均匀的、有组织的区域来阐述马尔可夫随机场的优点。如果观测图像少定义为一个M×N的长方形矩阵Ω,那么标签x也将定义为同等大小的矩阵。为了对原始图像进行建模,需要考虑分层模型中节点标签之间的交互。不同之处在于选择最大似然(ML)还是MAP标准进行模型参数估计
为了提取信息,我们以图5.1的图像为例来对这个图像中的现象进行分析和建模。航空公司的飞机在起飞或着陆的过程中有时会由于鸟、蛇、狗等动物而遭受安全问题,而且这些动物可能会给航空公司造成非常昂贵的损失。在这一章中,我们尝试改进一个模型算法来定位这些鸟类所在的区域半径,因为几乎所有的飞机着陆事故都是由于这些鸟类导致的。
通过这个例子,我们看到,图像的直方图变换是一个数据立方体,其由任意给定位置的像素点的固定邻域的像素灰度直方图组成。这样的转换在图像处理应用中非常有用,例如分类与分割,尤其是在处理纹理特征时可以通过它们的像素亮度和颜色的分布来区分。
我们知道高斯马尔可夫随机场(GMRF)是广泛应用于各种自然纹理和人造纹理的建模过程。大量的分类和重建实验证明了它的有效性。
我们也在5.5.6节中提供了仿真实验,设计了一个用于图像测试的算法。我们将这个算法用于处理测试图像集,展示了基于模糊隐马尔可夫链(HMC)的分割过程。在尝试了多张图像后,我们设定迭代次数能够得到成功的子类数量。
我们在第六章强调了用马尔科夫链蒙特卡洛算法从后验分布进行采样的无监督图像分割方法。我们注意到无监督分割算法比那些基于特征的算法有优势,因为其不需要任何窗口来估计区域模型参数,即对于图形元素的最小尺寸及形状都没有限制。对于一个特殊的模型,我们认为贝叶斯参数估计能够找出最适合某些特定观测数据的参数值。
然后我们用仿真算法对石油泄漏以及动植物系统图像进行图像分割和分区测试。
在石油泄漏的假设检验示例中,我们使用图像分割算法来研究石油泄漏对生态系统环境的影响。我们的海洋生态环境经常可能出现两栖类和爬行类物种的安全性问题以及水产养殖业的安全性问题。我们的想法是识别出由各个领域的人员造成的可能影响这些物种正常生活的任何障碍。这里我们强调了这一点,并通过分割图像以及提取有用的信息来准确地解释这些问题。
我们知道石油泄漏是指由于人类的活动导致液体石油烃释放到环境中,特别是海洋区域,这是一种污染。这个词通常专指海上漏油,即石油被释放到海洋和沿海水域,但石油泄漏也可能发生在陆地上。石油泄漏的原因可能是来自于油轮、海上平台、钻探设备和油井,也可能是精炼石油产品(如汽油,柴油)以及它们的副产品的泄漏,还可能是大型船舶使用的锅炉燃油或任何形式的废油。石油进入海洋环境的另一个重要的途径是通过自然渗透。
在本节,我们还通过图像6.1(从a到f)来展示基于不同区域贝叶斯估计的图像分区算法。
我们认为,对于一个给定的特殊模型,贝叶斯参数估计的作用是找出该模型关于某些观测数据的最佳拟合参数值。为了达到这个目标,必须制定一个关于选择标准的初始假设。如果我们的主要目的是选择一个参数估计方法来最小化平均误分概率,通过使用0-1效用函数,贝叶斯决策过程就变成选择参数来最大化后验概率,也就是最大化后验(MAP)概率估计。
最后,我们介绍了关于噪声图像和纹理图像的无监督分割算法。使用的模型都已经被定义,特别是各向同性的分层马尔可夫随机场以及高斯马尔可夫随机场区域模型。模拟退火算法被用于优化模型参数、类的数量以及像素的标签数量,这需要马尔可夫链蒙特卡洛(MCMC)采样算法能够从该模型的后验分布中对这些参数进行采样。同时使用吉布斯采样算法、Metropolis-Hastings算法和可逆跳MCMC算法对合并后的模型空间进行快速而完整的搜索。
实验结果展示了合成图像通过快速分割完成了对原始底层图像的准确估计。真实世界中的图像也能够成功地完成分割。实验结果同样表现出意想不到的收敛性,这进一步证明了在用模拟退火实现图像分割时甚至在区域数量已知时使用可逆跳跃MCMC算法的合理性。
Currently, Markov Random Field and Fuzzy Image Segmentation algorithms have become important fields in image processing research. In image processing system, segmentation process is one of the most important steps. More precisely, image segmentation is defined as the process of assigning a label to every pixel in an image such that pixels with the same label share certain visual characteristics. Several general-purpose algorithms and techniques have been developed for image segmentation. In order to effectively solve image segmentation problem for a specific problem domain, these techniques often have to be combined with knowledge domains, as there is no general solution to image segmentation problems.
Our research focused on how pixels display differently with different regions. We investigated a new approach to image segmentation, developed and implemented algorithms based on similarities or dissimilarities of the pixels in a given image.
The central theme of this dissertation is the development of a fully Markov unsupervised algorithm to achieve image segmentation. Existing literature falls short of such a goal, only providing many algorithms capable of solving subsets of this highly challenging problem. Unsupervised segmentation is the process of identifying and locating the constituent regions of an observed image, while having no prior knowledge of the number of regions. In this regard, in our work we consider the use of Bayesian framework to model the image, so that the unsupervised segmentation process can be considered as a problem of single optimization. However, we have discovered that throughout the literature, the commonly adopted optimization model is a hierarchical image model whose underlying components are various forms of Markov Random Fields. Gaussian Markov Random Field models are used to model the textural content of the image regions, while a Potts model provides a regularization function for the image segmentation as could be seen in our work.
The optimization process of such highly complicated models is an area that has been challenging researchers for several decades. So the main contribution of this thesis is to develop new techniques for Markov unsupervised segmentation to be carried out using a single optimization process. It is hoped that these algorithms will facilitate and enhance the future study of hierarchical image models and in particular enable the discovery of further models capable of more precise fitting of real world data.
Markov Random Field segmentation models and their optimizations deal with the selection of features to identify the textural contents of an observed image. In the light of the above, new algorithms have been proposed for the attainment of a balance between the two concepts: unsupervised segmentation and optimization process. Furthermore, Statistical Mechanics forms an important part of this work. The prevalent use of Markov Chain Monte Carlo algorithms is encountered and in particular, the use of the Reversible Jump Sampling algorithm being of great significance is encountered too. The combination of several of these algorithms enabled us to achieve the single optimization framework, which makes us obtained the most successful novel algorithms used in this dissertation.
However, for the color images segmentation, we have proposed a method which considers that the regions are defined as connected sets of pixels belonging to the same class. Our approach considers simultaneously both the connectivity and colorimetric properties of the pixels in order to build classes which can be non-equiprobable.
We defined the color connectivity degree of a set of pixels as a connectivity measurement of a set of pixels whose colors belong to a color interval. We suppose that pixels of each region in the image can be associated to a class of pixels, and that a class is a set of pixels whose color connectivity degree presents a high value. The problem here is identifying these sets.
In this context, we have defined an original data structure of Markovianity, the color connectivity degree, which counts in an organized and hierarchical way the color connectivity degrees of all possible sets of pixels that an image can contain.
For each one of these images, we decompose the image at three composite (RGB) and propose an analysis method for each. In order to gather pixels belonging to the same region from the results obtained by the Tree Marginal Classification processes, we model the image by a Region Adjacency Graph and propose a fuzzy analysis for it.
Finally, through our various experiments with the partitioning of image by similarity graphs, the efficiency of this method has been effectively proven.
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