区域多尺度马尔可夫随机场的遥感影像分类

代沁伶; 罗斌; 郑晨; 王雷光

doi:10.11834/jrs.20208287

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区域多尺度马尔可夫随机场的遥感影像分类
Regional multiscale Markov random field for remote sensing image classification
2020年24卷第3期页码：245-253
纸质出版日期： 2020-03-07 ，
DOI： 10.11834/jrs.20208287

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引用
代沁伶,罗斌,郑晨,王雷光.2020.区域多尺度马尔可夫随机场的遥感影像分类.遥感学报,24(3): 245-253

Dai Q L, Luo B, Zheng C and Wang L G. 2020. Regional multiscale Markov random field for remote sensing image classification. Journal of Remote Sensing（Chinese）， 24(3): 245-253
代沁伶,罗斌,郑晨,王雷光.2020.区域多尺度马尔可夫随机场的遥感影像分类.遥感学报,24(3): 245-253 DOI： 10.11834/jrs.20208287.

Dai Q L, Luo B, Zheng C and Wang L G. 2020. Regional multiscale Markov random field for remote sensing image classification. Journal of Remote Sensing（Chinese）， 24(3): 245-253 DOI： 10.11834/jrs.20208287.

区域多尺度马尔可夫随机场的遥感影像分类 Regional multiscale Markov random field for remote sensing image classification 1 2 first-author 代沁伶，1982年生,女，博士研究生, 研究方向为数字图像处理及遥感应用。E-mail: daiqinling@126.com 代沁伶，1982年生,女，博士研究生, 研究方向为数字图像处理及遥感应用。E-mail: daiqinling@126.com daiqinling@126.com 3 corresp 罗斌，1980年生,男，教授，研究方向为遥感影像分析。E-mail:luob@whu.edu.cn 罗斌，1980年生,男，教授，研究方向为遥感影像分析。E-mail:luob@whu.edu.cn luob@whu.edu.cn 4 5 6 武汉大学印刷与包装系, 武汉 430079 武汉大学印刷与包装系, 武汉 430079 School of Printing and Packaging, Wuhan University, Wuhan 430079，China School of Printing and Packaging, Wuhan University, Wuhan 430079，China 西南林业大学设计学院, 昆明 650224 西南林业大学设计学院, 昆明 650224 School of Design, Southwest Forestry University, Kunming 650224, China School of Design, Southwest Forestry University, Kunming 650224, China 武汉大学测绘遥感信息工程国家重点实验室, 武汉 430079 武汉大学测绘遥感信息工程国家重点实验室, 武汉 430079 State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan;University, Wuhan 430079, China State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan;University, Wuhan 430079, China 河南大学数学与统计学院, 开封 475000 河南大学数学与统计学院, 开封 475000 School of Mathematics and Statistics, Henan University, Kaifeng 475000, China School of Mathematics and Statistics, Henan University, Kaifeng 475000, China 西南林业大学大数据与人工智能研究院, 昆明 650224 西南林业大学大数据与人工智能研究院, 昆明 650224 Institutes of Big Data and Artificial Intelligence, Southwest Forestry University, Kunming 650224, China Institutes of Big Data and Artificial Intelligence, Southwest Forestry University, Kunming 650224, China 西南林业大学林业生态大数据国家林业局重点实验室, 昆明 650224 西南林业大学林业生态大数据国家林业局重点实验室, 昆明 650224 Key Laboratory of State Forestry Administration on Forestry and Ecological Big Data, Southwest Forestry University, Kunming 650224, China Key Laboratory of State Forestry Administration on Forestry and Ecological Big Data, Southwest Forestry University, Kunming 650224, China 多尺度分析技术广泛应用于高分辨率遥感影像的特征提取和建模。分解层数受制于影像的大小，下采样小波变换实现的影像多尺度表达难以描述大范围的空间模式，导致分类结果出现“胡椒盐”现象；面向对象的影像分析技术虽避免了“胡椒盐”现象，但由于仅利用了单尺度的的特征，也难以描述影像多层次的空间模式，导致分类精度较低。为改善分类结果中的“胡椒盐”现象和提高分类精度，提出了一种结合区域多尺度遥感影像分割和马尔可夫随机场的分类方法。首先，获得原始影像过分割区域，依据区域内亮度均值以及区域间的共享边界长度信息，提取影像低频和高频特征，采用该低频特征波段代替原始影像，重复分割与特征波段提取过程，形成影像的区域多尺度表达。然后，以原始图像为初始尺度，以分割区域为处理单元，以更细尺度分类结果为标记场先验，以当前高频特征建立特征场，逐层分类、投影，获得最终尺度分类结果。合成纹理影像和多光谱遥感影像的实验表明：相比于小波域多尺度建模方法和单尺度区域建模方法，本文提出的方法可以有效提高分类精度，并避免“胡椒盐”现象的产生。 Multiscale analysis technique can describe an image from different resolutions and has been widely used for extracting features and modelling of remotely sensed images. The subsampled wavelet transforms are commonly employed for establishing the multiscale representation of an image. However, the wavelet-based features cannot describe patterns with long spatial spans and often result in noisy classification results. By contrast, the object-based image analysis can create classification maps composed of compact land objects. However, features extracted from a single scale still cannot provide discriminating information for the land cover classification. To improve the classification accuracy and alleviate noisy thematic maps, a regional multiscale classification method is proposed.The proposed method consists of two main blocks, including establishing a regional multiscale image representation and classifying on the basis of Markov Random Field (MRF). In the first block, the mean shift segmentation method is employed to create initial over-segmented regions. Thereafter, a rule combining the grey values in the regions and the shared boundary lengths among regions is designed to extract the low-frequency part of an image. The current high-frequency part is obtained by subtracting the original image and the current low-frequency portion. A regional multiscale representation can be iteratively established by replacing the original image with the low-frequency part and repeating the segmenting and decomposing process. In the second block, the classification result obtained from the original image is considered the prior of the label field in the first decomposed level, and the high-frequency part in the first level models the feature field of MRF. The classification result in the first level is obtained by solving an objective function consisting of feature and label energies. By iteratively projecting the current classification result to the next level and modelling the feature field with the high-frequency part, the final classification map can be acquired in the coarsest scale.To examine the effectiveness of the proposed method, three groups (six images in total), such as Prague textures, synthetic remote sensing textures, and QuickBird multispectral images, are adopted. The proposed regional multiscale MRF (RMRF) is compared with the Iterated Conditional Model (ICM), Graph Cut (GC), and Wavelet-based MRF classification methods (WMRF). Visual inspection and quantitative measurement are employed for the comparison. As GC fails to integrate the spatial information among neighbouring pixels, the classification results are dominated by the pixel spectral values. Textures or land cover patterns with complex spectral heterogeneity cannot be properly captured. WMRF provides a pixel-level multiscale representation, which is also insufficient for describing texture patterns across a large spatial span. ICM, GC and WMRF may create noisy thematic maps with different extensions when the spectral heterogeneity of a given land cover pattern is extreme. On the contrary, RMRF has a flexible framework to extract and model spatial information. RMF can capture the essential features of land objects with large spectral heterogeneity, resulting in maps with less noise.This study proposes an RMRF for classification. Experiments demonstrate that the established multiscale representation can efficiently describe texture patterns with wide spatial spans, such as land objects with complex structures. By combining the multiscale representation with MRF model, RMRF can achieve a high-precision semantic segmentation of ground objects. The adaptive estimation of the decomposition layer number and parameters of the algorithm are the on-going works. 高分辨率遥感影像分类区域分割马尔可夫随机场多尺度模型 high resolution remote sensing image classifications regional segmentation Markov random field multiscale model 1　引　言 1 遥感影像地物分类是赋予影像每个像素地物类别的过程，一直是遥感影像分析中的重点和热点问题( 宫鹏等，2006 宫鹏等，2006 )。而随着高空间分辨率遥感影像的涌现，影像中地物信息更加丰富，地物光谱类内异质性和类间同质性变化的趋势增强，像素类属的不确定性增加，地物空间尺度和语义尺度变异加剧，导致传统基于光谱、逐像素的分类方法面临极大挑战，分类结果往往出现明显的误分和“胡椒盐”现象( 宫鹏， 2009 宫鹏， 2009 ； Blaschke， 2010 Blaschke， 2010 )。为应对这些挑战，需要充分利用高分辨率影像中的空间信息，需要在不同或多个空间尺度空间上表示地物对象。整合空间信息，提取多尺度的特征，以及采用高性能的分类器或建模方法，成为高分辨率影像解译常用的技术方法( Noda 等，2002 Noda 等，2002 ； Blaschke，2010 Blaschke，2010 ； Moser 等， 2013 Moser 等， 2013 )。不同窗口大小提取多尺度共生纹理( Huang等，2007 Huang等，2007 )，多尺度结构元素形态滤波提取形态特征( Benediktsson等， 2003 Benediktsson等， 2003 )，多尺度Gabor滤波( 张刚和马宗民， 2010 张刚和马宗民， 2010 )，多尺度引导滤波( 王雷光等， 2018 王雷光等， 2018 )，以及多尺度小波分解(Malfait 和Roose，1997； Nod 等， 2002)，均可以实现影像的空间特征多尺度表达。多尺度特征与高性能分类器或特征建模方法，如支持向量机(Chang 和Lin， 2011)、随机森林分类器( Gislason 等， 2006 Gislason 等， 2006 )、马尔可夫随机场( Li， 2009) Li， 2009) ）的结合，极大的提高了分类的精度。高分辨率影像解译的另一类方法是基于深度学习的方法。这类方法不显式的提取特征，而是通过对大量样本的训练实现由影像到地物类别“端对端”的映射( Long 等， 2015 Long 等， 2015 ； Marmanis 等，2018 Marmanis 等，2018 )，目前受到了极大的关注。但是，由于海量数据的标记和训练过程均需要极高的时间代价，极大制约了这类方法的应用。相较而言，基于多尺度特征提取和建模的影像分类思路，对样本的依赖性较小，具有较强的适用性。因规则的金字塔结构和完备的理论，下采样小波变换常被用来实现影像的多尺度表达，通过多尺度小波系数的堆叠获取特征向量(Malfait 和Roose， 1997)。基于小波变换的分类方法可以在低分辨率尺度上保持地物对象的完整，并在高分辨率尺度上保持地物边界的分割精度。但是，该类方法本质上仍是逐像素分类的方法，一般而言，低分辨率尺度上的一个像素对应相邻高分辨率尺度上一个2×2的局部窗。而小波分解的层数受制于影像的尺寸大小（一般不超过3层），特征向量只能描述遥感影像小范围的（不超过8×8的局部窗宽范围）纹理特征。在高空间分辨率的遥感影像中，同物异谱和异物同谱现象加剧，所以像素级的小范围多尺度特征难以有效地描述各类地物的整体纹理模式，这造成了同质地物内部具有较大光谱差异的分类结果中“胡椒盐”现象。以过分割区域为处理单元，描述和建模影像的形状、拓扑、语义、空间上下文等特征，面向对象的遥感影像分析方法能从根本上避免“胡椒盐”现象的产生，近年来也广受关注( Blaschke， 2010 Blaschke， 2010 ； Moser 等，2013 Moser 等，2013 )。例如， Li 等(2010) Li 等(2010) 建模区域邻接图，提取区域的形状、光谱和纹理等特征，通过区域合并实现分类。 Yu和Clausi (2008) Yu和Clausi (2008) 使用马尔可夫随机场MRF(Markov Random Field)描述区域的边界信息，通过寻找目标函数的最优解合并区域，实现分类。 Zhang和Ji (2010) Zhang和Ji (2010) 提出一种多粒度的特征，该特征综合影像中的点、线和区域信息，然后使用条件随机场和贝叶斯网络分别描述特征中的无向关系和有向关系，并通过概率推断得到最终的分类结果。 Zheng和Wang (2015) Zheng和Wang (2015) 提出一种对象级MRF模型，在建立邻接区域节点关系时，考虑了区域的边界和大小信息，实现地物对象间关系的交互, 通过贝叶斯准则推理得到最终的分类结果。然而，这类方法也往往只利用了单尺度的特征，存在难以描述影像中复杂、大范围的空间模式的问题。受上述文献的启发，考虑到区域级建模方法中使用多尺度信息可以克服小波域多尺度技术的不足，本文提出一种影像区域级多尺度表示方法，并与马尔可夫随机场相结合，实现了一种基于区域多尺度建模的遥感影像分类方法 RMRF(Regional Multiscale Markov Random Field)。 2　区域多尺度表达 1 多尺度表达是多尺度区域分割技术的核心步骤。通过多尺度表达，一幅影像可以分解为不同尺度的高频和低频系数矩阵。对任一观测影像为 <math id="M1"><mi mathvariant="bold-italic">Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166810&type= 2.28600001 2.62466669 ，首先通过区域分割算法形成初始过分割区域，然后定义区域的邻接关系和区域特征，最后通过对低频分量的层次分割和分解获得区域多尺度表达。具体而言：（1）使用均值漂移区域分割算法(Comaniciu 和Meer， 2002)获得影像 <math id="M2"><mi mathvariant="bold-italic">Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166821&type= 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166870&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166865&type= 3.04799986 3.72533321 相邻的区域集合。（3）计算邻域系统 <math id="M8"><mi mathvariant="bold-italic">N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166877&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166874&type= 2.62466669 2.62466669 的邻接相关度 <math id="M9"><mi mathvariant="bold-italic">C</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>n</mi><mo>;</mo><mi>j</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166910&type= 2.62466669 3.72533321 定义为区域 <math id="M13"><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166922&type= 3.04799986 3.72533321 与邻接区域 <math id="M14"><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166932&type= 2.87866688 3.72533321 公共边界的长度，记为区域标记为 <math id="M15"><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166941&type= 3.04799986 3.72533321 且8邻域内区域标记为 <math id="M16"><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166955&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166950&type= 2.87866688 3.72533321 的像素对出现的次数。（4）计算区域集合 <math id="M17"><mi mathvariant="bold-italic">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166963&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166958&type= 2.37066674 2.62466669 的特征系统 <math id="M18"><mi mathvariant="bold-italic">M</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166968&type= 34.54399872 4.48733330 ，其中特征 <math id="M19"><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166978&type= 3.72533321 4.06400013 表示区域 <math id="M20"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166988&type= 3.21733332 4.06400013 的光谱均值。（5）计算当前尺度的低频分量 <math id="M21"><mi mathvariant="bold-italic">L</mi><mo>=</mo><munderover><mo>∪</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167001&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21166998&type= 14.22399998 7.70466709 ，其中 <math id="M22"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167005&type= 3.47133350 4.06400013 表示区域 <math id="M23"><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167013&type= 3.04799986 3.72533321 的低频信息，其特征 <math id="M24"><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167020&type= 3.97933340 4.06400013 定义为 <math display="block" id="M25"><msubsup><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msub><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow></msubsup><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">m</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mrow><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow></msubsup><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167030&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167027&type= 34.71333313 23.11400032 （6）计算当前尺度的高频分量 <math id="M26"><mi mathvariant="bold-italic">H</mi><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="bold-italic">Y</mi><mo>-</mo><mi mathvariant="bold-italic">L</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167039&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167034&type= 18.54199982 3.72533321 。（7）以低频分量 <math id="M27"><mi mathvariant="bold-italic">L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167041&type= 2.03200006 2.62466669 代替原始影像 <math id="M28"><mi mathvariant="bold-italic">Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167053&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167050&type= 2.28600001 2.62466669 ，重复步骤（1）—（6），获取下一尺度的低频和高频分量。上述多尺度分解过程中，采用均值漂移算法作为区域分割方法，主要考虑到其支持多波段影像的处理，形成的区域同质性较好、对象边界定位较准确，其可用性已在众多面向对象的遥感影像分析应用中获得了证实( Wang 等，2015 Wang 等，2015 ；Zheng 和 Wang, 2015 Wang, 2015 ; Zheng 等, 2017 Zheng 等, 2017 )。低频分量由上一尺度影像中区域及其相邻区域光谱均值特征的加权平均计算，归一化的边界长度量化了各相邻区域间的影响。因此，低频分量保持了影像的主要结构信息，而由影像与低频分量差值计算的高频部分则描述了影像的细节信息。图1 图1 给出了一幅纹理影像进行3次分解建立的影像区域多尺度表达和分类过程示例。如图1 图1 右侧区域多尺度表达模块所示，第1次分解的输入为原始影像，第2和第3次分解的输入影像分别是上一尺度分解获得的低频分量 L 1 和 L 2 。由图1 图1 可见，右上角的横竖纹理经若干次分解后，逐渐在低频图像中被描述成同质纹理区域，而高频细节信息则保留在高频分量中。提出的多尺度分解方法可以有效的将原始影像分解成不同尺度的高频细节和低频。随着分解层数的增多，基本区域单元会越来越大，相应的高、低频部分就可以描述影像中空间跨度更大的纹理模式。相比下采样小波变换，本方法通过非下采样方式建立多尺度表达，可以实现任意多尺度的高低频信息分解。分解的层数可根据影像自身的纹理特点确定。一般来说，影像纹理模式空间跨度越大，分解的层数相应就越多。 3　基于区域多尺度的马尔可夫随机场建模 1 3.1　马尔可夫随机场 2 马尔可夫随机场MRF(Markov Random Field)模型是一种概率图模型( Li， 2009 Li， 2009 )。假设观测影像 <math id="M29"><mi mathvariant="bold-italic">Y</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><msub><mrow><mo stretchy="false">}</mo></mrow><mrow><mi mathvariant="bold-italic">i</mi><mo>∈</mo><mi mathvariant="bold-italic">S</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167070&type= 17.52599907 4.48733330 的位置集合为 <math id="M30"><mi mathvariant="bold-italic">S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167081&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167077&type= 2.03200006 2.62466669 ，其中 <math id="M31"><msub><mrow><mi>Y</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167088&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167084&type= 2.87866688 3.72533321 表示位置 <math id="M32"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167092&type= 1.10066664 2.62466669 处影像的特征。定义标记场 <math id="M33"><mi mathvariant="bold-italic">X</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><msub><mrow><mo stretchy="false">}</mo></mrow><mrow><mi mathvariant="bold-italic">i</mi><mo>∈</mo><mi mathvariant="bold-italic">S</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167102&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167099&type= 18.20333290 4.48733330 ，其中位置 <math id="M34"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167107&type= 1.10066664 2.62466669 处的标记是 <math id="M35"><msub><mrow><mi>X</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">Λ</mi><mo>=</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>K</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167116&type= 32.25799942 4.57200003 ， K 是影像分类的类别数。 <math id="M36"><mi mathvariant="bold-italic">x</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><msub><mrow><mo stretchy="false">}</mo></mrow><mrow><mi mathvariant="bold-italic">i</mi><mo>∈</mo><mi mathvariant="bold-italic">S</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167124&type= 16.76399994 4.48733330 是标记场 <math id="M37"><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167130&type= 2.45533323 2.62466669 的一个实现，代表了一种影像的分类结果。 MRF模型将影像分类问题转化为在给定观测影像 <math id="M38"><mi mathvariant="bold-italic">Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167135&type= 2.28600001 2.62466669 下标记场 <math id="M39"><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167142&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167140&type= 2.45533323 2.62466669 最优现实 <math id="M40"><mover accent="true"><mrow><mi mathvariant="bold-italic">x</mi></mrow><mo stretchy="true">̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167145&type= 1.77800000 2.62466669 的估计，根据Bayesian公式，即求解目标函数 <math display="block" id="M41"><mover accent="true"><mrow><mi mathvariant="bold-italic">x</mi></mrow><mo stretchy="true">̂</mo></mover><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="bold-italic">x</mi></mrow></munder><mo stretchy="false">{</mo><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">X</mi><mo>=</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>=</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167151&type= 54.35600281 4.40266657 式中， <math id="M42"><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>=</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167157&type= 14.22399998 3.47133350 代表标记场的联合概率分布； <math id="M43"><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">X</mi><mo>=</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167169&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167166&type= 16.93333435 3.47133350 代表似然函数，表示在实现 <math id="M44"><mi mathvariant="bold-italic">x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167171&type= 1.77800000 2.62466669 下观测影像 <math id="M45"><mi mathvariant="bold-italic">Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167177&type= 2.28600001 2.62466669 的联合概率分布。根据Hammersley-Clifford定理( Li， 2009 Li， 2009 )，联合概率分布 <math id="M46"><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>=</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167184&type= 14.22399998 3.47133350 服从Gibbs分布 <math display="block" id="M47"><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>=</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>Z</mi></mrow></mfrac><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mo>-</mo><mi mathvariant="bold-italic">U</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167191&type= 40.72466660 7.62000036 式中， <math id="M48"><mi>Z</mi><mo>=</mo><munder><mo movablelimits="false">∑</mo><mrow><mi mathvariant="bold-italic">x</mi><mo>∈</mo><mi mathvariant="bold-italic">X</mi></mrow></munder><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mo>-</mo><mi mathvariant="bold-italic">U</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167198&type= 30.22600174 6.51933336 是归一化常数， <math id="M49"><mi mathvariant="bold-italic">U</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167205&type= 7.87400007 3.47133350 是能量函数。假设 <math id="M50"><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>=</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167212&type= 14.22399998 3.47133350 各样本之间相互独立和标记场具有马氏性，有 <math display="block" id="M51"><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>=</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="false">∏</mo><mrow><mi mathvariant="bold-italic">i</mi><mo>∈</mo><mi mathvariant="bold-italic">S</mi></mrow></munder><mrow><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="bold-italic">t</mi></mrow></msub><mo>,</mo><mi>t</mi><mo>∈</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167216&type= 44.02666855 6.51933336 式中， <math id="M52"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167222&type= 3.47133350 3.72533321 表示区域 i 的邻域集合。同时，假设 <math id="M53"><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">X</mi><mo>=</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167232&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167229&type= 16.93333435 3.47133350 各点之间相互独立，则 <math display="block" id="M54"><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">X</mi><mo>=</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="false">∏</mo><mrow><mi mathvariant="bold-italic">i</mi><mo>∈</mo><mi mathvariant="bold-italic">S</mi></mrow></munder><mrow><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167235&type= 36.99933624 6.51933336 将和代入，则 <math id="M55"><mover accent="true"><mrow><mi mathvariant="bold-italic">x</mi></mrow><mo stretchy="true">̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167242&type= 1.77800000 2.62466669 可以通过逐点求最优 <math id="M56"><msub><mrow><mover accent="true"><mrow><mi>x</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167251&type= 2.37066674 3.72533321 得到 <math display="block" id="M57"><mover accent="true"><mrow><mi>x</mi></mrow><mo stretchy="true">̂</mo></mover><mo>=</mo><munder><mrow><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mo stretchy="false">{</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">Λ</mi><mo>,</mo><mi>i</mi><mo>∈</mo><mi>S</mi><mo stretchy="false">}</mo></mrow></munder><munder><mo movablelimits="false">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></munder><mrow><mo stretchy="false">{</mo><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>⋅</mo><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mi>t</mi><mo>∈</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167261&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167257&type= 61.04467010 6.77333355 3.2　本文方法 2 本节将区域多尺度表达与上述MRF模型相结合，提出一种多尺度的区域级分类方法。首先，构建影像区域多尺度表达；然后，在观测影像上使用MRF模型得到初始结果；再按照从观测影像到最高尺度的顺序逐层投影各层结果，得到最终分类结果。具体方法：（1）观测影像的多尺度表达。使用第2节提出的方法将观测影像进行 J 层分解，各尺度高频部分分别记为 <math id="M58"><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>i</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167270&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167265&type= 3.55599999 2.96333337 ( <math id="M59"><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167275&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167272&type= 19.04999924 3.04799986 )。（2）观测影像的初始尺度分类。定义MRF模型中标记场的概率 <math id="M60"><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="bold-italic">t</mi></mrow></msub><mo>,</mo><mi>t</mi><mo>∈</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167282&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167279&type= 21.67466736 4.57200003 服从多层逻辑模型( Li， 2009 Li， 2009 )：即 <math id="M61"><mi>U</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>j</mi><mo>∈</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></munder><mrow><mi>V</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167286&type= 29.54866600 7.45066643 ，其中 <math display="block" id="M62"><mi>V</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mi>β</mi></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mtd></mtr><mtr><mtd><mi>β</mi></mtd><mtd><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≠</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167297&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167293&type= 34.12066650 9.65199947 式中， <math id="M63"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167304&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167300&type= 1.86266661 3.38666677 是势函数参数。MRF模型中似然函数 <math id="M64"><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167306&type= 17.44133377 4.57200003 假设服从正态分布 <math display="block" id="M65"><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">)</mo></mrow><mrow><mi>D</mi><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold">Σ</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup></mrow></mfenced></mrow></mfrac><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold">Σ</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167317&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167313&type= 67.14067078 16.00200081 式中， <math id="M66"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167324&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167321&type= 2.53999996 2.62466669 是特征向量 <math id="M67"><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167332&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167328&type= 2.53999996 4.57200003 的维数， <math id="M68"><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167335&type= 3.04799986 4.74133348 和 <math id="M69"><msubsup><mrow><mi mathvariant="bold">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167342&type= 3.72533321 4.48733330 是第 <math id="M70"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167349&type= 1.69333339 2.62466669 类的特征均值向量和协方差矩阵， <math id="M71"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">Λ</mi><mo>=</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>K</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167356&type= 30.56466866 3.47133350 。然后，将和带入目标函数，根据迭代条件模式算法ICM（Iterated Conditional Model）( Besag， 1986 Besag， 1986 )，即可得到观测影像的初始分类结果 <math id="M72"><msup><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167364&type= 3.55599999 2.96333337 。（3）以观测影像为起点，将第 <math id="M73"><mi>i</mi><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167371&type= 6.77333355 2.62466669 层( <math id="M74"><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167381&type= 19.04999924 3.04799986 )分类结果 <math id="M75"><msup><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mi>i</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167388&type= 6.43466663 2.96333337 投影至第 <math id="M76"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167393&type= 1.10066664 2.62466669 层作为标记场的先验信息，以第 <math id="M77"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167406&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167403&type= 1.10066664 2.62466669 层的高频部分 <math id="M78"><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>i</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167412&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167409&type= 3.55599999 2.96333337 作为特征，利用ICM方法求解MRF模型得到该层的分类结果 <math id="M79"><msup><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mi>i</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167416&type= 3.13266659 2.96333337 ， <math id="M80"><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167422&type= 19.04999924 3.04799986 。 <math id="M81"><msup><mrow><mi mathvariant="bold-italic">G</mi></mrow><mrow><mi>J</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167429&type= 3.47133350 2.96333337 为最终的分类结果。其中，初始的观测影像层以非监督的K均值聚类结果作为各类别的初始先验概率。本方法通过影像的区域多尺度表示得到了影像各尺度用于建模的高低频特征,而各尺度的特征场以当前尺度的高频特征作为 <math id="M82"><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167450&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167447&type= 2.62466669 4.57200003 ，标记场则考虑了对应特征间的空间关系。如图1 图1 所示，通过从观测影像到最高层尺度的顺序投影，由每个尺度上的特征信息得到当前尺度的分类结果；然后将该结果作为先验信息，以先验概率的形式传递给下一层的标记场，而下一层的标记场在先验信息的基础上，通过和特征场的迭代循环，将当前尺度的特征信息与先验信息进行了融合。所以，最终尺度的分类结果综合了全部尺度上的特征信息。 4　实验与分析 1 为验证算法的有效性，研究考虑4种对比方法，即单尺度ICM算法( Besag， 1986 Besag， 1986 )，图割算法 GC(Graph Cut)( Boykov等， 2001 Boykov等， 2001 )，小波域多尺度MRF模型 WMRF(Wavelet based MRF )( Noda 等， 2002 Noda 等， 2002 )，以及改进的对象级MRF模型 OMRF(Object based MRF)(Zheng 和 Wang，2015 Wang，2015 )。其中，GC用来对比本文方法与经典分类算法的差异；单尺度ICM算法用来说明MRF模型在单尺度上的表现；WMRF用来说明MRF模型在小波域多尺度上的表现；OMRF用于对比分析现有基于区域对象建模的MRF分类方法的性能差异。研究分别采用彩色纹理影像、合成遥感影像和QuickBird多光谱遥感影像进行实验。由于遥感影像中包含丰富的纹理信息，第1组彩色纹理影像用来检测方法对多光谱纹理的描述能力；然后，第2组合成遥感影像检测方法对不同地物类别的分类效果；最后，QuickBird多光谱遥感影像检测方法在实际遥感影像中的应用。 4.1　合成彩色纹理影像的实验 2 首先对Prague纹理库(Mikeš 等， 2015)的彩色纹理影像进行分类。图2 图2 (a)是两幅256 <math id="M83"><mo>×</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167454&type= 2.03200006 2.62466669 256的原始纹理影像。实验中，RMRF进行2层多尺度分解。从结果可以看出，GC方法难以考虑影像纹理的空间结构，结果易受影像光谱值影响，导致图2 图2 (c)上图左上角的误分。单尺度ICM方法因不能获得影像多个尺度上的特征，所以难以描述影像中的纹理信息，导致图2 图2 (d)下图右侧和中间的误分和斑点噪声。WMRF模型因为在多个尺度上对影像进行分析，所以提高了分类的精度，图2 图2 (e)上图左上角纹理得以正确识别。但是，WMRF是像素级的多尺度，只能描述局部的纹理，图2 图2 (e)上图线状黑点和图2 图2 (e)下图黑色鱼鳞线，WMRF都出现了部分误分和视觉噪声。RMRF和OMRF方法均以区域为基本分析单位，由于均利用了区域的大小、邻接关系等信息，获得纹理对象的完整性较好，获得了优于前面3组实验的分类结果。尤其是RMRF方法，以区域为基本单位实现多尺度分解，可以在不同尺度上描述空间跨度较大的纹理模式和较高层的影像信息。 4.2　合成多光谱遥感影像实验 2 选取2幅合成多光谱遥感影像进一步验证算法的性能。图3 图3 (a)是两幅256 <math id="M84"><mo>×</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167472&type= 2.03200006 2.62466669 256的合成遥感影像，包含了城镇、农田、林地、水域等多个地物类别。在这2组实验中，本文RMRF方法分别进行3层和4层区域多尺度分解。从分类结果中可以看出：由于受左下角的建筑阴影(上图)和林地与水体光谱值接近(下图)的影响，GC结果不理想。单尺度ICM和WMRF在影像左上的农田和左下的城镇(上图)，以及影像上方的林地等都存在比较明显的误分(下图)。也就是说，这几种算法都不能处理地物类内光谱差异比较大的情况。而区域多尺度的RMRF能描述较大范围的纹理信息，进而较好的分割出不同地物，对于纹理对象的边界检测效果要优于其他方法。OMRF在地物类内光谱差异较大时，如建筑物混杂植被的区域中（下图），也会出现误分。 4.3　QuickBird遥感影像实验 2 采用两幅QuickBird多光谱真彩色遥感影像验证算法在实际中的应用。图4 图4 (a)是原始实验遥感影像，分别取至武汉郊区和巴格达地区内河港口区域。武汉数据的大小为256 <math id="M85"><mo>×</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167492&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167489&type= 2.03200006 2.62466669 256，包含农田、城镇和林地3类地物；巴格达数据大小为448 <math id="M86"><mo>×</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=21167494&type= 2.03200006 2.62466669 500，主要包含水体和城市两类。实验中，两组数据分别进行了8层和3层区域多尺度分解。从图4 图4 可以看出：武汉影像中同物异谱、异物同谱现象明显，比如城镇中的屋顶和阴影；巴格达影像中右上角水面上的船舶和水面的光谱差异比较大、右下角林地与水面的光谱差异较小。这导致GC、ICM和WMRF的分类结果中出现了明显的误分，且结果细碎。OMRF实现的单一尺度的对象级建模，建模结果受制于区域分割对象的大小，将部分农田划分成了林地。与这4种方法相比，RMRF通过区域多尺度的表达，描述了影像中更复杂的信息，所以结果区域性完整，不同地物边界定位也更精确，分类精度更高。 4.4　定量分析 2 根据混淆矩阵计算得到Kappa系数和整体精度OA(Overall Accuracy )( Cohen， 1960 Cohen， 1960 )来定量分析上面6组实验。Kappa系数和OA取值区间均为[0,1]，两个指标越大，相应实验的分类精度就越高。表1 表1 给出5种实验方法在这6组实验中的Kappa系数和OA。从表1 表1 可以看出，基于区域对象建模的分类方法OMRF和RMRF的精度要普遍优于其余像素级分类方法；本文方法RMRF在6组实验中都取得了最优或次优的结果。注：黑体和下划线分别代表最优及次优精度。 2 2 GC (Boykov等，2001) 2 ICM (Besag，1986) 2 WMRF (Noda等，2002) 2 OMRF (Zheng和Wang，2015) 2 RMRF Kappa OA Kappa OA Kappa OA Kappa OA Kappa OA 纹理1 0.8261 0.8711 0.9499 0.9656 0.9548 0.9692 0.9921 0.9947 0.9914 0.9943 纹理2 0.8724 0.9024 0.9618 0.9710 0.9789 0.9841 0.9834 0.9876 0.9901 0.9926 合成1 0.8546 0.8843 0.9646 0.9730 0.9762 0.9819 0.9934 0.9950 0.9869 0.9901 合成2 0.4291 0.4834 0.8686 0.9052 0.8749 0.9101 0.9445 0.9619 0.9585 0.9719 遥感1 0.4691 0.5236 0.6392 0.7012 0.6934 0.7596 0.7837 0.8399 0.8694 0.9110 遥感2 0.9143 0.9555 0.9176 0.9574 0.9132 0.9550 0.9669 0.9835 0.9462 0.9728 本文所有实验均在CPU 2.5 GHz, 内存16 GB， Windows 10操作系统，Matlab R2015b环境下进行。其中,本文方法中使用的均值漂移算法及GC算法均由C++/Matlab混合编程实现。表2 表2 统计了各数据和方法处理的耗时。虽然该耗时会随着软硬件环境的变化而变化，但并不影响方法间的对比分析。可以看出，单尺度方法（GC和ICM）处理速度要显著快于多尺度方法，反映出多尺度特征建模具有较高计算复杂度。而相同算法在处理尺寸相同的不同影像时，处理时间也出现较大差异，如GC方法对于纹理1和纹理2的处理时间相差了约2倍。其耗时差异的决定因素是算法迭代次数的影响。而WMRF、OMRF及RMRF方法的处理时间，除了与单尺度处理时的迭代次数相关，还与多尺度分解的层数相关。WMRF、OMRF及RMRF等多尺度方法，虽然单一尺度的迭代次数一般要少于ICM方法，耗时更短，但是在构建影像多尺度表达会耗费较多的时间。本文RMRF方法中，多尺度表达的构建耗时（表2 表2 中括号内时间）均超过全部处理时间的80%。注：括号内数据为RMRF方法构建影像多尺度表达耗费的时间。 GC (Boykov等，2001) ICM (Besag，1986) WMRF (Noda等，2002) OMRF (Zheng和Wang，2015) RMRF 纹理1 0.1174 0.1249 0.9287 1.8701 2.0555（1.7786）纹理2 0.3325 0.3470 1.8550 2.0113 2.5890（2.1023）合成1 0.2103 0.2005 0.5065 4.6955 3.6366（3.1463）合成2 0.1746 0.1788 1.6494 2.1174 3.6167（3.1208）遥感1 0.1729 0.1769 1.3598 6.9833 8.9599（8.2574）遥感2 0.5346 0.5555 4.1951 11.1382 15.0682（13.5388） s 5　结　论 1 针对下采样小波变换实现的影像多尺度表达难以描述大范围的空间模式，导致分类结果出现“胡椒盐”现象。传统面向对象的影像分析技术仅利用了单尺度的的特征，难以描述影像中多层次的空间模式，导致分类精度较低。本文提出将多尺度分析技术从像素级拓展至区域级的方法，基于多尺度MRF建模的方法，实现了遥感影像分类。实验证明：以区域为基本单位构建的区域多尺度表达可以描述空间跨度更大的纹理模式和复杂的影像信息，克服了小波域多尺度表达只能描述影像局部纹理的不足，同时避免了单尺度区域建模方法对于分割尺度较敏感的问题。提出的RMRF模型将区域级多尺度表达与MRF模型相结合，实现了地物对象的高精度分类。算法自身的分解层数和参数的自适应估计，以及提高区域多尺度表达的相关代码的运行效率，是进一步研究的重点。参考文献(References) Benediktsson J A , Pesaresi M and Amason K . 2003 . 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