顾及时变特性的时序极化SAR图像自适应超像素生成方法

顾及时变特性的时序极化SAR图像自适应超像素生成方法 Adaptive Superpixel Generation for Time-series PolSAR Images Considering Time-varying Characteristics first-author 叶家伟，1998年生，男，硕士研究生， 研究方向为时序极化SAR图像处理。E-mail： yejiawei@csu.edu.cn 叶家伟，1998年生，男，硕士研究生， 研究方向为时序极化SAR图像处理。E-mail： yejiawei@csu.edu.cn yejiawei@csu.edu.cn corresp 汪长城，1980年生，男，教授，研究方向为遥感图像处理、SAR/极化SAR图像处理。E-mail： wangchangcheng@csu.edu.cn 汪长城，1980年生，男，教授，研究方向为遥感图像处理、SAR/极化SAR图像处理。E-mail： wangchangcheng@csu.edu.cn wangchangcheng@csu.edu.cn 中南大学 地球科学与信息物理学院, 长沙 410083 中南大学 地球科学与信息物理学院, 长沙 410083 School of Geosciences and Info-Physics, Central South University, Changsha 410083, China School of Geosciences and Info-Physics, Central South University, Changsha 410083, China 超像素生成是对象级数据处理体系的重要预处理步骤，这对于拥有大数据量的多时相、多极化SAR数据的高效处理和应用具有很大的现实意义。针对单时相超像素分割方法没有充分利用地物在时序上完整散射信息的问题，本文综合利用时序极化SAR数据观测充分、可描述时变特性等优势，提出了一种基于简单线性迭代聚类（SLIC）模型的多时相极化SAR影像自适应超像素生成方法。该方法首先联合多个时相的极化协方差矩阵，基于Wishart分布计算时序极化SAR相似性距离；然后利用多时相极化SAR数据进行梯度计算，检测图像边缘；最后提出一种基于多时相极化SAR边缘检测的同质性测度因子，用于自适应平衡极化距离和空间距离的权重关系。本文使用8景Radarsat-2全极化SAR影像，从可视化效果和定量精度两个方面评价方法有效性。相关结果表明，本文方法优于单时相极化SAR超像素生成方法和现有的多时相极化SAR超像素方法，超像素能够有效贴合研究区域地块边界，是一种有效的超像素生成方法。 Objective Superpixel generation is an important pre-processing step in the object-level data processing system, which is of great practical significance for the efficient processing and application of multi-temporal and multi-polarized SAR data. The single-temporal superpixel segmentation method does not fully utilize the complete scattering information of the segmented objects in the time series. To address this problem, this paper proposes a multi-temporal PolSAR Images adaptive cooperative segmentation method based on the Simple Linear Iterative Clustering(SLIC) model, which takes full use of the advantages of fully observed and describable time-varying characteristics of the time-series PolSAR data.Method Firstly, this method calculates the time-series PolSAR similarity distance based on Wishart distribution by uniting the polarization covariance matrix of multi-temporal; then uses multi-temporal polarization SAR data to perform gradient calculation to detect image edges; Finally, a homogeneity measure factor based on multi-temporal polarimetric SAR edge detection is proposed to adaptively balance the weight relationship between polarimetric distance and spatial distance.Result In this paper, we used 8 Radarsat-2 quad-polarization SAR images to evaluate the effectiveness of this method in terms of both visualization effect and quantitative accuracy. The results show that the method in this paper outperforms the single-temporal PolSAR superpixel generation method and the existing traditional multi-temporal PolSAR superpixel method. For example, as for the superpixel generation result with the quad-polarization SAR data (K = 12000), the value of the boundary recall(BR) and the achievable segmentation accuracy(ASA) by the proposed similarity measure and homogeneity factor is about 93.58% and 95.13%, respectively.Conclusion To address the problem that the single-temporal polarization SAR segmentation does not consider the time-varying characteristics of the ground object polarization characteristics, this paper proposes a multi-temporal polarization SAR image adaptive collaborative segmentation method based on the SLIC model. The experimental results show that compared with the segmentation method based on single-temporal data and the traditional multi-temporal polarimetric SAR superpixel segmentation method, the superpixels generated by this paper have obvious advantages in both visualization effect and quantitative accuracy, and can effectively fit the ground truth boundary, which proves that the proposed method is an effective superpixel generation method. 极化SAR 图像分割 SLIC算法 超像素 多时相 PolSAR image segmentation SLIC superpixels multi-temporal 1　引 言 1 极化SAR（Polarimetric Synthetic Aperture Radar， PolSAR）对物体的形态结构和介电常数等特性较为敏感，能够全天时、全天候地工作，已经被广泛应用于目标识别、变化检测和地物分类等领域，是一项极具潜力的遥感监测手段（Lee and Pottier，2009； Zhang等， 2014 Zhang等， 2014 ）。然而，极化SAR图像中特有的斑点噪声增加了图像解译的困难。 与传统像素级图像解译方法相比，基于超像素的对象级极化SAR图像解译由于其计算效率高、受斑点噪声影响小等优点而越来越受到人们的重视（ Niu 和 Ban， 2013 Niu 和 Ban， 2013 ； D'Elia等，2014 D'Elia等，2014 ； Deng等，2014 Deng等，2014 ； Jiao等， 2014 Jiao等， 2014 ）。因此，超像素生成是对象级图像解译的重要预处理步骤。超像素是指图像上具有相似特征的相邻像素构成的具有一定视觉意义的规则像素块（Achanta， Shaji 等，2012），它利用像素之间特征的相似性将像素聚类分组，用少量的超像素代替大量的像素来表达图片特征，很大程度上降低了图像处理的复杂度。 基于单时相极化SAR图像的超像素分割算法已发展得较为充分。按照其分割所用核心技术，可大致分为以下四类：1）基于阈值的分割方法：设置阈值将像素进行划分聚类，如双峰法（ Papamarkos 和 Gatos，1994 Papamarkos 和 Gatos，1994 ）、最大类间方差法（ Gong等，1998 Gong等，1998 ）等。2）基于边缘的分割方法：基于边缘约束对超像素进行聚类，常用的边缘算子包括Canny算子（ Canny ，1987 Canny ，1987 ）、Mallat算子（ Mallat 和 Zhong ，1992 Mallat 和 Zhong ，1992 ）等。3）基于特定理论的分割方法：以能量函数最小或后验概率最大为目标对分割图进行迭代求解以生成最优分割结果，如水平集（Level-set）（ Yin 和 Yang ，2014 Yin 和 Yang ，2014 ）、马尔可夫随机场（Markov random field ， MRF）（ Yu等， 2012 Yu等， 2012 ）的方法等。4）基于区域生长的分割方法：从种子点出发不断向外扩展同质区域，如均值漂移（Mean Shift， MS）（Beaulieu 和 Touzi ，2010）、简单线性迭代聚类（Simple Linear Iterative Clustering， SLIC）（ Achanta等，2012 Achanta等，2012 ）等。其中，SLIC算法最初是针对光学图像提出的一种超像素生成算法，该算法充分利用了图像的空间信息，能够以迭代聚类的方式生成形状规则、大小匀称的超像素，且原理简单、易于实现，在光学图像的分割领域已表现出了良好的性能。 在SLIC模型的基础上，文献（ Feng等，2014 Feng等，2014 ）、文献（ Song等，2015 Song等，2015 ）和文献（ Qin等，2015 Qin等，2015 ）分别使用对称修正Wishart距离（ Cao等，2007 Cao等，2007 ）、Bartlett距离和修正Wishart距离作为相似性度量来生成超像素，顾及了极化SAR数据的统计特性；文献（ Qin等，2015 Qin等，2015 ）通过改进SLIC模型中的初始化及后处理算法，对孤立的超像素进行邻域合并处理，抑制了极化SAR图像中相干斑噪声的影响。文献（ Wang等，2019 Wang等，2019 ）提出一种将改进分水岭算法与SLIC模型相结合的分割方法，通过将梯度分割作为SLIC模型的预处理步骤，为后续的分割提供了高质量的初始化信息。文献（ Yin等，2021 Yin等，2021 ）提出了一种新的SLIC聚类相似度，在该聚类相似度下利用四种经典的极化SAR统计距离进行超像素生成，并对分割效果作了对比分析。文献（ Xiang等，2017 Xiang等，2017 ）采用球不变随机向量积模型估计每个像素的归一化协方差矩阵和纹理，利用高斯边缘检测模板对城市等复杂区域进行梯度计算，并且将梯度值与等效视数相结合，提出了一种基于SLIC模型的自适应极化SAR图像超像素生成算法。 然而，多时相作物类型精准分类和变化检测以及作物高精度时序定量监测依赖于时序上统一的超像素生成结果。上述基于单时相影像的超像素生成方法不能生成时序上统一的分割结果。另外，在单时相影像上，不同种类地物的极化散射特性也可能存在混淆。例如，均处于播种初期的两种农作物可能都表现出相同的极化信息，从而被误分割为同一超像素块；到成熟期后，上述两种农作物又因极化散射信息不同而被分割成不同的超像素块。事实上，在作物的一个生长周期内，地面作物类型并没有发生改变，只是极化散射特性呈现出物候差异。针对这一问题，时序极化SAR影像能够完整表现各种地物类型极化散射特性的时序变化，利用这种时序变化差异能够从时序上正确划分超像素块，从而得到合理统一的分割结果。 目前，基于多时相极化SAR数据进行分割的方法还尚不成熟。文献（ 邵宁远 等， 2019 邵宁远 等， 2019 ）提出一种基于SLIC模型的SAR图像超像素协同分割方法，该方法利用两个时相的融合边缘图以及像素强度相似度进行超像素协同分割，得到了边界准确、空间对应的协同分割结果。但该方法采用的指数加权均值比算子对边缘的检测能力有限，且只适用于SAR影像。针对极化SAR影像，文献（ Bao 等，2017 Bao 等，2017 ）通过将不同图像上相同位置像素的极化相干矩阵按分块对角矩阵的方式联合起来得到多时相极化相干矩阵，以此为基础实现了基于SLIC模型的多时相极化SAR图像超像素分割，得到了统一的超像素分割结果。该方法虽然综合利用了多时相极化SAR数据进行超像素协同分割，但是其生成的超像素不够规则，且没有顾及算法的自适应性，即需要人工调节基于多时相数据的极化距离和空间距离的权重因子。 针对上述问题，本文提出了基于SLIC模型的多时相极化SAR图像自适应协同分割方法。该方法综合利用多个时相的极化协方差矩阵，基于Wishart分布计算时序极化SAR相似性距离，并且能够利用像素梯度值和等效视数值生成同质性测度因子，用以自适应地平衡极化距离和空间距离的权重关系，在顾及了地物时变特性的同时，能够生成基于多时相数据的统一超像素分割结果。为了验证该方法的有效性，本文采用AgriSAR 2009项目提供的8景C波段全极化数据进行图像分割实验，测试本文方法的分割表现，并从可视化效果和定量精度两个方面对本文方法生成的分割效果图进行了精度评定与分析。 2　基于SLIC模型的多时相极化SAR图像自适应分割方法 1 2.1　多时相极化SAR距离度量 2 在极化SAR影像处理领域，基于单时相数据常用的距离量度有Wishart距离（ Qin等，2015 Qin等，2015 ）、假设检验距离（Conradsen等，2003）等距离量度。其中，Wishart距离往往用于基于像素的极化SAR影像分类，而假设检验距离更多被用于面向对象的极化SAR影像分割（ Qin等，2015 Qin等，2015 ）。本文选取假设检验距离与空间距离作为极化SAR图像像素之间的相似性度量，该距离度量将被用于图像分割以及后续的边缘检测算法中。其中，假设检验距离的推导过程如下： 对于单站极化SAR观测下的互易目标，极化散射信息可用复散射矢量表示为： <math display="block" id="M1"><mtable><mtr><mtd><mrow><mi mathvariant="bold-italic">k</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mroot><mrow><mn 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810214&type= 3.21733332 4.23333359 是区域 <math id="M17"><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810110&type= 2.87866688 3.72533321 和 <math id="M18"><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810141&type= 3.04800010 3.72533321 的中心相干矩阵。现设立如下假设检验： <math display="block" id="M19"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>:</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub><mo>≠</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">3</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810122&type= 24.80733490 9.14400005 由于数据集 <math id="M20"><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810137&type= 2.87866688 3.72533321 和 <math id="M21"><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810141&type= 3.04800010 3.72533321 中的样本是独立的，因此 <math id="M22"><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810137&type= 2.87866688 3.72533321 和 <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=24810144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810141&type= 3.04800010 3.72533321 的条件概率密度函数分别可以表示为： <math display="block" id="M24"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mi>p</mi><mfenced separators="|"><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></munderover></mstyle><mrow><mi>p</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">|</mo><mi>L</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub></mrow></mfenced></mrow></mtd></mtr><mtr><mtd><mi>p</mi><mfenced separators="|"><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></munderover></mstyle><mrow><mi>p</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">|</mo><mi>L</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></mrow></mfenced></mrow></mtd></mtr></mtable></mrow></mfenced><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">4</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810150&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810147&type= 46.73600006 21.08200073 式中， N j 和 N i 分别为两个数据集中样本像素的个数， <math id="M25"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810158&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810153&type= 3.38666677 3.72533321 为第 n 个样本的极化相干矩阵， <math id="M26"><mi>p</mi><mfenced separators="|"><mrow><mi>*</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810162&type= 6.68866634 3.80999994 表示概率密度函数。由式（3）和式（4）可得 <math id="M27"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810266&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810263&type= 3.64066648 3.72533321 假设下的似然函数为： <math display="block" id="M28"><msub><mrow><mi>L</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">m</mi></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></munderover></mstyle><mrow><mi>p</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">|</mo><mi>L</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">m</mi></mrow></msub></mrow></mfenced></mrow><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">5</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810174&type= 54.01733780 9.05933285 式中， <math id="M29"><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">m</mi></mrow></msub><mi mathvariant="normal">为</mi><mi mathvariant="normal">区</mi><mi mathvariant="normal">域</mi><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810183&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810180&type= 18.28800011 4.57200003 和 <math id="M30"><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">的</mi><mi mathvariant="normal">联</mi><mi mathvariant="normal">合</mi><mi mathvariant="normal">中</mi><mi mathvariant="normal">心</mi><mi mathvariant="normal">相</mi><mi mathvariant="normal">干</mi><mi mathvariant="normal">矩</mi><mi mathvariant="normal">阵</mi><mo>,</mo><mi mathvariant="normal">在</mi><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="normal">假</mi><mi mathvariant="normal">设</mi><mi mathvariant="normal">下</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">m</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810186&type= 78.14733124 4.65666676 。 <math id="M31"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810192&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810190&type= 3.64066648 3.72533321 假设下的似然函数为： <math display="block" id="M32"><msub><mrow><mi>L</mi></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></munderover></mstyle><mrow><mi>p</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">|</mo><mi>L</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></mrow></mfenced></mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></munderover></mstyle><mrow><mi>p</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">|</mo><mi>L</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub></mrow></mfenced></mrow><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">6</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810198&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810195&type= 75.60733032 9.05933285 由公式（5）和公式（6）可得似然比假设检验统计量为： <math display="block" id="M33"><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></munderover></mstyle><mrow><mi>p</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">|</mo><mi>L</mi><mo>,</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">m</mi></mrow></msub></mrow></mfenced></mrow></mrow><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></munderover></mstyle><mrow><mi>p</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">|</mo><mi>L</mi><mo>,</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub></mrow></mfenced></mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></munderover></mstyle><mrow><mi>p</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">|</mo><mi>L</mi><mo>,</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></mrow></mfenced></mrow></mrow></mfrac><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">7</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810205&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810202&type= 57.40399933 19.38866615 式中， <math id="M34"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810208&type= 3.21733332 5.24933338 为 <math id="M35"><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810214&type= 3.21733332 4.23333359 的极大似然估计值， <math id="M36"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msubsup><mrow><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mo>/</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810224&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810221&type= 27.94000053 5.50333309 。同理， <math id="M37"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810227&type= 3.13266683 5.24933338 和 <math id="M38"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810233&type= 4.14866638 5.24933338 为 <math id="M39"><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810242&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810240&type= 3.13266683 4.23333359 和 <math id="M40"><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810247&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810244&type= 4.14866638 4.23333359 的极大似然估计值。 上述检验统计量 <math id="M41"><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810254&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810251&type= 3.64066648 4.48733330 的取值范围为［0，1］，当统计量 <math id="M42"><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810256&type= 3.64066648 4.48733330 接近于0时，表示拒绝假设 <math id="M43"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810266&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810263&type= 3.64066648 3.72533321 ；当其接近于1时，则表示接受假设 <math id="M44"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810270&type= 3.64066648 3.72533321 。由公式（7）可得到用以度量第 <math id="M45"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810353&type= 0.84666669 3.47133350 个和第 <math id="M46"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810361&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810358&type= 1.10066664 2.62466669 个区域之间相似性的假设检验距离函数为（ Cao等，2007 Cao等，2007 ）： <math display="block" id="M47"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">T</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi></mrow></mfrac><mi>l</mi><mi>n</mi><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mi mathvariant="normal">*</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810285&type= 51.98533249 7.28133297 <math display="block" id="M48"><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">m</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>-</mo><mfenced separators="|"><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>-</mo><mfenced separators="|"><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">8</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810294&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810291&type= 67.98733521 7.11199999 从公式（8）可以看出，假设检验距离函数 <math id="M49"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810300&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810297&type= 4.57200003 3.72533321 是对称的。因此若 <math id="M50"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810306&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810302&type= 10.83733368 5.24933338 ，则 <math id="M51"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">T</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810308&type= 22.18266678 5.16466665 ，否则， <math id="M52"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">H</mi><mi mathvariant="normal">T</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810321&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810316&type= 22.18266678 5.16466665 。 在 <math id="M53"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810324&type= 3.64066648 3.72533321 和 <math id="M54"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810334&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810330&type= 3.64066648 3.72533321 的假设下，当 <math id="M55"><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810336&type= 3.13266683 4.23333359 已知时，上述假设检验则退化为一般的二值假设检验。即对于给定的类别区域 <math id="M56"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810353&type= 0.84666669 3.47133350 ，有： <math display="block" id="M57"><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfrac><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>L</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup></mrow><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>L</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup></mrow></mfrac><mi>e</mi><mi>x</mi><mi>p</mi><mfenced open="{" close="}" separators="|"><mrow><mo>-</mo><mi>L</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi>T</mi><mi>r</mi><mfenced separators="|"><mrow><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></mrow></mfenced><mo>-</mo><mi>q</mi></mrow></mfenced></mrow></mfenced><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">9</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810351&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810348&type= 66.88666534 15.15533447 相应地，第 <math id="M58"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810353&type= 0.84666669 3.47133350 个区域和 <math id="M59"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810361&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810358&type= 1.10066664 2.62466669 个区域之间的假设检验距离则退化为修正的Wishart距离： <math display="block" id="M60"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">W</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>L</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mi>l</mi><mi>n</mi><msub><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810368&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810366&type= 35.05199814 8.38199997 <math display="block" id="M61"><mo>=</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo>+</mo><mi>T</mi><mi>r</mi><mfenced separators="|"><mrow><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></mrow></mfenced><mo>-</mo><mi>q</mi><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">10</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810372&type= 51.47733307 15.15533447 式中， <math id="M62"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810420&type= 1.69333339 3.47133350 为目标散射矢量 k 的维度。在单站全极化情况下， <math id="M63"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810420&type= 1.69333339 3.47133350 等于3。不难看出，如果 <math id="M64"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="bold-italic">j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810389&type= 10.83733368 5.24933338 ，则 <math id="M65"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">W</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810394&type= 16.34066582 5.16466665 取到最小值0；否则其大于0。 对于时序全极化SAR数据，本文根据文献（ Bao 等， 2017 Bao 等， 2017 ）中提出的方法将 N 景影像上同一位置像素的T3矩阵以分块对角矩阵的形式进行组装得到一个3 N *3 N 的联合矩阵 <math id="M66"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi mathvariant="bold-italic">N</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810400&type= 4.74133301 3.72533321 ： <math display="block" id="M67"><mtable><mtr><mtd><mrow><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi mathvariant="bold-italic">N</mi></mrow></msub><mo>=</mo><mtext> </mtext><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">3,1</mn></mrow></msub></mtd><mtd><mo>⋯</mo></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋱</mo></mtd><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>…</mo></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">3</mn><mo>,</mo><mi mathvariant="bold-italic">N</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">11</mn></mrow></mfenced></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810410&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810406&type= 42.75666428 13.71599960 然后将联合矩阵 <math id="M68"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi mathvariant="bold-italic">N</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810419&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810414&type= 4.74133301 3.72533321 代入公式（10）中计算 N 景影像的联合距离度量。值得注意的是，对于单站全极化的 N 景数据而言，公式（10）中的 <math id="M69"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810420&type= 1.69333339 3.47133350 应为3 N 。 2.2　多时相极化SAR影像同质性度量 2 为了使超像素在匀质区域趋于规则而在非匀质区域尽可能的保持目标细节，极化SAR 图像场景均匀性分析就显得尤为重要，对超像素自适应生成有着重要意义。 目前，常用的与图像均匀性相关的参数有两个，分别是边缘检测结果即像素梯度值与等效视数（Equivalent Number of Looks， ENL）。 与光学影像不同，极化SAR影像由于其受相干斑噪声的影响，简单的梯度计算无法得到效果较好的边缘图。因此，需要针对其设计一种边缘检测算法。文献（ Schou 等， 2003 Schou 等， 2003 ）提出了一种基于复Wishart分布的极化SAR图像恒虚警率边缘检测器，该检测器利用一组具有不同方向的边缘检测模板对边缘进行检测。针对非匀质区域，文献（ Xiang 等， 2017 Xiang 等， 2017 ）在传统恒虚警率边缘检测器基础上提出了一种高斯边缘检测模板，取得了良好的效果。如 图1 图1 所示，中心点的梯度由具有一定间隔且大小相同的两个高斯窗口区域之间的相似度来代替。高斯窗口函数为（ Xiang等，2017 Xiang等，2017 ）： <math display="block" id="M70"><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mroot><mrow><mn mathvariant="normal">2</mn><mi>π</mi></mrow><mrow/></mroot><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow></msub><mroot><mrow><mn mathvariant="normal">2</mn><mi>π</mi></mrow><mrow/></mroot><msub><mrow><mi>σ</mi></mrow><mrow><mi>y</mi></mrow></msub></mrow></mfrac><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mtext> </mtext><mfenced separators="|"><mrow><mo>-</mo><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">12</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810425&type= 81.78800201 11.51466751 式中， <math id="M71"><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810436&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810433&type= 3.55599999 3.72533321 和 <math id="M72"><msub><mrow><mi>σ</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810603&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810602&type= 3.55599999 3.72533321 表示窗口的长度和宽度， <math id="M73"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810464&type= 1.69333339 2.62466669 和 <math id="M74"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810469&type= 1.69333339 3.47133350 表示以中心像素为原点的像素位置， <math id="M75"><mi>W</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810461&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810458&type= 2.87866688 2.62466669 表示每个像素的权重。不难看出，当 <math id="M76"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810464&type= 1.69333339 2.62466669 和 <math id="M77"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810469&type= 1.69333339 3.47133350 越小，即越靠近中心像素，拥有的权重值越大，这符合当确定中心像素是否是边缘像素时，靠近中心像素的像素包含的信息比在其他像素处包含的信息更重要的事实。中心像素两边扇形区域的平均中心相干矩阵为： <math display="block" id="M78"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>=</mo><mfrac><mrow><munder><mo movablelimits="false">∑</mo><mrow><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></mrow></munder><mrow><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></mrow></mrow><mrow><munder><mo movablelimits="false">∑</mo><mrow><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></mrow></munder><mrow><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></mrow></mrow></mfrac><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">13</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810473&type= 46.39733505 16.51000023 式中， <math id="M79"><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810480&type= 19.38866615 5.24933338 ，分别表示中心像素两边的高斯窗口区域， <math id="M80"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810490&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810486&type= 12.61533356 4.82600021 表示区域 <math id="M81"><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810491&type= 3.04800010 3.72533321 中 <math id="M82"><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810500&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810497&type= 8.29733276 4.57200003 位置处的极化相干矩阵。 最后，中心像素的梯度计算公式被定义为对称的修正Wishart距离（ Xiang等，2017 Xiang等，2017 ）： <math display="block" id="M83"><mi mathvariant="normal">E</mi><mi mathvariant="normal">D</mi><mi mathvariant="normal">G</mi><mi mathvariant="normal">E</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi mathvariant="normal">T</mi><mi mathvariant="normal">r</mi><mfenced separators="|"><mrow><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>+</mo><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">Σ</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub></mrow></mfenced><mo>-</mo><mn mathvariant="normal">3</mn><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">14</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810516&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810513&type= 64.26200104 7.28133297 从公式（14）可以看出，区域 <math id="M84"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810522&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810519&type= 3.47133350 3.72533321 和 <math id="M85"><msub><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810528&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810524&type= 3.47133350 3.72533321 越相似，则其中心点的梯度值越小，是边缘的概率也就越小，反之依然。 本文在文献（ Xiang 等， 2017 Xiang 等， 2017 ）中方法的基础上，分别对每景影像进行边缘检测，最后对结果取平均得到最后的联合边缘检测图，从而达到对多时相极化SAR图像进行边缘检测的效果。 对于等效视数，文献（ Anfinsen 等， 2009 Anfinsen 等， 2009 ）针对单景SAR图像提出了一种基于Wishart分布的ENL估计方法。文献（ Shen 等， 2020 Shen 等， 2020 ）针对时序极化SAR图像提出了一种ENL估计器，顾及了地面目标的时间特性。其估计公式为： <math display="block" id="M86"><mi mathvariant="normal">E</mi><mi mathvariant="normal">N</mi><mi mathvariant="normal">L</mi><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover></mstyle><mrow><mi>T</mi><mi>r</mi><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">Σ</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover></mstyle><mrow><mfenced open="<" close=">" separators="|"><mrow><mi>T</mi><mi>r</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">T</mi><mi mathvariant="bold-italic">T</mi></mrow></mfenced></mrow></mfenced><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover></mstyle><mrow><mi>T</mi><mi>r</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">Σ</mi><mi mathvariant="bold-italic">Σ</mi></mrow></mfenced></mrow></mrow></mrow></mfrac><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">15</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810532&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810530&type= 61.29866791 17.86466599 式中， N 为时相个数， T 为极化相干矩阵， <math id="M87"><mi mathvariant="bold-italic">Σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810536&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810534&type= 2.45533323 3.13266683 为 T 经7*7的均值滤波处理后的相干矩阵。Tr为求迹符号， < > 为时间平均处理。 本文将基于多时相极化SAR数据的边缘检测结果与文献（ Shen 等， 2020 Shen 等， 2020 ）提出的多时相极化SAR等效视数结合起来得到多时相极化SAR图像均匀性参数 <math id="M88"><mi mathvariant="bold-italic">T</mi><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">H</mi><mi mathvariant="bold-italic">o</mi><mi mathvariant="bold-italic">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810540&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810538&type= 11.34533310 2.62466669 ： <math display="block" id="M89"><mi mathvariant="bold-italic">T</mi><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">H</mi><mi mathvariant="bold-italic">o</mi><mi mathvariant="bold-italic">m</mi><mo>=</mo><mfrac><mrow><mi mathvariant="bold-italic">E</mi><mi mathvariant="bold-italic">N</mi><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="bold-italic">E</mi><mi mathvariant="bold-italic">D</mi><mi mathvariant="bold-italic">G</mi><mi mathvariant="bold-italic">E</mi></mrow></mfrac><mo>#</mo><mo stretchy="false">(</mo><mn mathvariant="normal">16</mn><mo stretchy="false">)</mo><mo>#</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810542&type= 37.50733566 7.28133297 自适应参数 <math id="M90"><msub><mrow><mi>β</mi></mrow><mrow><mi>a</mi><mi>d</mi><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810548&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810546&type= 4.91066647 4.48733330 被定义为： <math display="block" id="M91"><msub><mrow><mi>β</mi></mrow><mrow><mi>a</mi><mi>d</mi><mi>p</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced separators="|"><mrow><mi mathvariant="bold-italic">T</mi><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">H</mi><mi mathvariant="bold-italic">o</mi><mi mathvariant="bold-italic">m</mi><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>+</mo><mi mathvariant="bold-italic">T</mi><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">H</mi><mi mathvariant="bold-italic">o</mi><mi mathvariant="bold-italic">m</mi><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>#</mo><mo stretchy="false">(</mo><mn mathvariant="normal">17</mn><mo stretchy="false">)</mo><mo>#</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810552&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810550&type= 72.72866058 7.28133297 从上式可以看出，自适应参数顾及了两个参加计算的像素的多时相均匀性度量。 最后，完整的分割距离 <math id="M92"><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">I</mi><msup><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810556&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810553&type= 7.19666624 4.06400013 由公式（10）定义的极化距离 <math id="M93"><msub><mrow><mi>d</mi></mrow><mrow><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810557&type= 2.96333337 3.72533321 以及空间距离 <math id="M94"><msub><mrow><mi>d</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810563&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810560&type= 2.70933342 3.72533321 计算得到： <math display="block" id="M95"><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">I</mi><msup><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msub><mo>=</mo><mroot><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><mi>S</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>β</mi></mrow><mrow><mi>a</mi><mi>d</mi><mi>p</mi></mrow></msub></mrow><mrow/></mroot><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">18</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810568&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810565&type= 49.95333481 11.26066685 <math display="block" id="M96"><msub><mrow><mi>d</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>=</mo><mroot><mrow><msup><mrow><mfenced separators="|"><mrow><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></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">19</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810570&type= 52.06999969 7.45066643 公式（18）中，S为超像素宽度，其计算公式为： <math display="block" id="M97"><mi>S</mi><mo>=</mo><mroot><mrow><mfrac><mrow><mi>N</mi></mrow><mrow><mi>k</mi></mrow></mfrac></mrow><mrow/></mroot><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">20</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810572&type= 22.77533340 8.97466660 式中， <math id="M98"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810574&type= 2.53999996 2.62466669 为像素总个数， <math id="M99"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810578&type= 1.69333339 2.62466669 为预期的超像素个数。 2.3　多时相自适应Pol-SLIC分割流程 2 为便于读者理解，本小节将给出本文分割方法的总体流程图。由于极化SAR图像受相干斑噪声影响严重，因此需要先对原始时序全极化SAR数据进行滤波处理。由滤波后的时序数据进行边缘检测以及等效视数的估计，得到统一的边缘图和等效视数图，并由公式（16）和公式（17）计算出同质性测度因子 <math id="M100"><msub><mrow><mi>β</mi></mrow><mrow><mi>a</mi><mi>d</mi><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810584&type= 4.91066647 4.48733330 。最后，利用滤波后的数据以及自适应参数 <math id="M101"><msub><mrow><mi>β</mi></mrow><mrow><mi>a</mi><mi>d</mi><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810584&type= 4.91066647 4.48733330 进行多时相极化SAR图像超像素生成并更新聚类中心，重复上述步骤直至收敛或达到最大迭代次数。当迭代前后的像素变化率小于1%时，可认为算法已收敛。在进行正式实验前，为抑制相干斑噪声的影响，需对多视极化相干矩阵进行滤波处理。本文实验使用的所有极化相干矩阵均为经5*5精致LEE滤波处理后的。具体流程图如 图2 图2 所示。 3　多时相极化SAR真实数据实验 1 本文采用的真实实验数据为AgriSAR 2009项目中RADARSAT-2系统获取的西班牙拉曼查高原Barrax试验区8景星载C波段全极化SAR数据。该8景影像成像日期均在一个生长周期内，即地物类型并没有发生改变。每景影像的像素大小为800*800。该实验区包含多种具有时变极化特征的作物，主要包括苜蓿、大麦、小麦和玉米，因此可以很好地证明使用时序极化SAR数据在精细作物分割与分类和物候监测中的优越潜力。具体卫星参数如表一所示。 图3 图3 （a）为该地区的时序平均PauliRGB图像， 图3 图3 （b）为参考 图3 图3 （a）人工勾画的真实边缘图。 卫星 参数 数值 7 Radarsat-2 日期 20090403，20090427，20090521，20090614，20090708，20090801，20090825，20090918 极化模式 HH+HV+VH+VV 中心入射角 34.25° 波段 C波段 飞行方向 升轨 距离向采样间隔 4.7m 方位向采样间隔 5.1m 3.1　边缘检测与等效视数结果 2 在边缘检测算法中，本文设置的高斯模板参数 <math id="M102"><msub><mrow><mi>d</mi></mrow><mrow><mi>f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810595&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810594&type= 2.62466669 3.72533321 和 <math id="M103"><msub><mrow><mi>θ</mi></mrow><mrow><mi>f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810597&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810596&type= 2.45533323 3.80999994 分别为1和 <math id="M104"><mi>π</mi><mo>/</mo><mn mathvariant="normal">8</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810598&type= 5.41866684 2.70933342 。高斯窗口长度 <math id="M105"><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810601&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810600&type= 3.55599999 3.72533321 和宽度 <math id="M106"><msub><mrow><mi>σ</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810603&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810602&type= 3.55599999 3.72533321 均设置为12.8。联合了8景影像的边缘检测图、等效视数图以及图像均匀性参数TSHom图如 图4 图4 （a）、 图4 图4 （b）以及 图4 图4 （c）所示。从 图4 图4 可知，本文方法检测出来的边缘图轮廓较为清晰，等效视数图以及匀质性参数图能较好地反应出图像的场景均质性，这表明联合了多景影像的边缘检测算法以及等效视数估计器顾及了地物的时变特性，能准确地检测出大麦、小麦和玉米等极化特征随时间变化明显的作物边缘并估计出其等效视数。 3.2　与单时相图像分割方法对比 2 本小节将从定量的角度比较本文方法与单时相分割方法在不同超像素个数下的分割结果。本文采用了两种常用的定量评价指标，分别是文献（ Arbelaez等，2011 Arbelaez等，2011 ）中提出的边界召回率（Boundary Recall，BR）以及文献（ Liu等，2011 Liu等，2011 ）中提出的超像素分割精度（Achievable Segmentation Accuracy， ASA）。为了保证实验的可比较性，采用的单时相分割方法与本文方法类似，即将本文中的公式（10）、边缘检测算法以及等效视数估计算法均改为单时相形式。超像素个数设置为4000到16000，实验结果如 图5 图5 所示。结果表明，本文方法的BR值以及ASA值均高于基于每一景影像的单时相分割方法。这是由于单时相数据只包含地物在某一时刻的极化散射信息，不同种类的农作物等地物由于在某些时刻的物理形态相似，其极化散射特性在这些时刻较为相同，因此容易出现将表征不同种类地物的像素分割成同一超像素的现象。另外，基于单时相数据的边缘检测效果也远不如本文提出的基于多时相数据的边缘检测效果理想，这也影响了最后分割结果的精度。 3.3　与其他多时相图像分割方法对比 2 在本小节中，我们将从可视化效果和定量精度两个层面来比较本文方法和传统多时相分割方法的超像素生成结果。本文采用的传统多时相分割方法为文献（ Bao 等， 2017 Bao 等， 2017 ）提出的方法（下称方法一），该方法使用Wishart距离作为极化距离度量，并提出一种线性相加的SLIC聚类相似度用以生成最后的分割距离。Wishart距离为： <math display="block" id="M107"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">W</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mfenced separators="|"><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>+</mo><mi>T</mi><mi>r</mi><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">Σ</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">21</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810611&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810610&type= 49.61466980 5.92666674 最后的分割距离为： <math display="block" id="M108"><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">C</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>W</mi></mrow></msub></mrow><mrow><mi>m</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow><mrow><mi>S</mi></mrow></mfrac><mo>#</mo><mfenced separators="|"><mrow><mn mathvariant="normal">22</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810612&type= 33.18933105 7.61999989 式中， m为调节极化距离和空间距离的权重因子。 此外，为比较本文方法在不同的SLIC聚类相似度下的分割表现，在其他条件不变的情况下，将本文方法中的SLIC聚类相似度改为式（22）中的形式（下称方法二）。值得注意的是，虽然修正的Wishart距离理论上为非负值，但实际上由于计算精度的取值问题，其会出现负值，这将对后续的分割结果造成一定的影响。因此，在运行方法二时，需对极化距离进行先取实部后取绝对值的操作。三种方法的具体参数如表二所示。 方法 极化相似度 权重因子 SLIC聚类相似度 方法一 <math id="M109"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">W</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810614&type= 3.13266683 2.87866688 0.7 <math id="M110"><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">C</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810617&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810616&type= 5.50333309 2.87866688 方法二 <math id="M111"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">W</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810619&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810618&type= 4.31799984 2.87866688 <math id="M112"><msub><mrow><mi>β</mi></mrow><mrow><mi>a</mi><mi>d</mi><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810621&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810620&type= 4.06400013 3.38666677 <math id="M113"><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">C</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810622&type= 5.50333309 2.87866688 本文方法 <math id="M114"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">W</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810625&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810624&type= 4.31799984 2.87866688 <math id="M115"><msub><mrow><mi>β</mi></mrow><mrow><mi>a</mi><mi>d</mi><mi>p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810627&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810626&type= 4.06400013 3.38666677 <math id="M116"><msub><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">I</mi><msup><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24810628&type= 6.01133299 3.38666677 一方面，从可视化效果上看， 图6 图6 （a）、（b）、（c）分别表示方法一、方法二以及本文方法的分割结果PauliRGB平均图。超像素个数均设置为24000个。 图7 图7 中（a）、 （b）、 （c）分别表示 图6 图6 中红色实线矩形区域的放大图。 图8 图8 中（a）、（b） 、（c）分别为 图7 图7 中对应区域的线性分割图。结果表明，与本文方法相比，方法一的分割效果图中存在着一些孤立的超像素，如 图7 图7 （a）和（c）中的红色矩形虚线区域所示。从 图8 图8 中（a）和（c）中还可以明显看出，相比本文方法产生的超像素，方法一生成的部分超像素形状过于规则，没有本文方法生成的超像素匀称、紧密。这是由于方法一中调节极化距离和空间距离的权重因子是固定的，不能根据图像局部区域场景匀质性的改变而改变，导致在部分区域的权重大小不合理。当空间距离权重过大时，超像素会呈现出明显的矩形形状。另外，如 图7 图7 （b）和（c）中红色圆形虚线区域及 图8 图8 （b）和（c）所示，方法二的分割结果图中存在一些斑点现象，且与本文方法相比，其生成的超像素较为紊乱，这表明本文方法所使用的修正Wishart距离在公式（18）所示的SLIC聚类相似度下表现更好。 另一方面，从定量指标上看， 图9 图9 （a）为方法一、方法二以及本文方法的BR值对比图， 图9 图9 （b）为三种分割方法的ASA值对比图。为充分验证三种方法在不同超像素个数下的分割表现，超像素个数设置为4000到20000。结果表明，本文方法的BR值显著高于方法一，是三种方法中最高的，方法二的BR值略低于本文方法。这是因为BR值反应的是真实边界被正确恢复的程度，而方法一产生的部分矩形超像素形状过于规则，没有正确贴合地面真实边界，从而降低了其BR值。从ASA值对比情况上看，三种方法的ASA值均较高，差距并不大，总体上本文方法及方法二略高于方法一。这是因为本文方法及方法二中调节极化距离和空间距离的权重因子是自适应的，具有较好的分割性能。综合不同超像素个数下两种指标的对比情况来看，本文方法的分割精度是优于方法一及方法二的。 通过本文方法与单时相分割方法和传统多时相分割方法的分割结果对比，综合可视化效果和定量指标两方面来看，本文提出的基于多时相数据的自适应协同分割方法结果更好，得到的超像素形状规则、大小均匀，且能较好地贴合真实地面边缘。 4　结 论 1 针对单时相极化SAR分割没有考虑地物极化特征具有时变特性的问题，本文提出了一种基于SLIC模型的多时相极化SAR影像自适应协同分割方法。首先，通过联合多个时相的极化协方差矩阵计算得到联合多时相数据的时序极化SAR相似性距离，顾及了地物的时变特性，其次。利用统一的像素梯度值和等效视数值生成同质性测度因子，用以自适应地平衡极化距离和空间距离的权重关系。最后，在SLIC框架下，利用联合的时序极化SAR相似性距离、空间距离以及同质性测度因子生成时序上统一的超像素分割结果。本文采用AgriSAR 2009项目提供的8景C波段西班牙Barrax实验区的全极化SAR数据进行了图像分割实验。该实验区包含多种具有时变极化特征的作物，能够很好地对本文方法进行验证。实验结果表明，与基于单时相数据的分割方法和现有的多时相极化SAR超像素分割方法相比，本文方法无论是在可视化效果上还是在定量精度上都具有明显优势，体现了方法在对具有时变特性的地物进行分割时的潜力。 参考文献 Achanta R , Shaji A , Smith K , Lucchi A , Fua P and Susstrunk S . 2012 . 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National Remote Sensing Bulletin， XX（XX）：1-11 国家自然科学基金项目（编号： 42030112和41671356） Supported by Supported by National Natural Science Foundation of China (No. 42030112 and 41671356) 2022-03-03 遥感学报