联合超像素降维和后处理优化的高光谱图像分类方法

联合超像素降维和后处理优化的高光谱图像分类方法 XX（XX）：1-19 1 2 first-author 黄媛，1998年生，女，硕士研究生，研究方向为高光谱影像处理与分析。E-mail： huangyuan@hunnu.edu.cn 黄媛，1998年生，女，硕士研究生，研究方向为高光谱影像处理与分析。E-mail： huangyuan@hunnu.edu.cn huangyuan@hunnu.edu.cn 1 2 corresp 贺新光，1973年生，男，教授，研究方向为地理时空大数据挖掘与应用。E-mail： xghe@hunnu.edu.cn 贺新光，1973年生，男，教授，研究方向为地理时空大数据挖掘与应用。E-mail： xghe@hunnu.edu.cn xghe@hunnu.edu.cn 1 2 湖南师范大学地理科学学院,湖南 长沙 410081 湖南师范大学地理科学学院,湖南 长沙 410081 湖南师范大学地理空间大数据挖掘与应用湖南省重点实验室,长沙 410081 湖南师范大学地理空间大数据挖掘与应用湖南省重点实验室,长沙 410081 针对高光谱图像样本标签量少且空间-光谱信息利用不充分而导致图像分类精度较低的问题，本文提出一种联合超像素降维和后验概率优化的高光谱图像分类方法。该方法首先基于高光谱图像的空间-光谱信息为每个样本构建局部邻域集合，并从局部邻域集合中提取超像素稀疏混合特征来充分表征图像的空谱信息和相关变化信息，然后将全局稀疏混合特征输入支持向量机分类器中生成像素的类别概率向量，最后采用后验概率模型优化类别概率向量，并依据概率最大值得到分类标签图。在三组常用的小规模数据集Indian Pines、Pavia University和Salinas，以及一组大规模数据集HoustonU上的实验结果表明：所提出的分类方法能够自适应地充分提取高光谱图像的高判别性特征信息，且在少量样本标签情形下，该方法在四组实验数据集上分别获得了98.58%、96.88%、98.54%和91.01%的总体分类精度，并优于目前几种先进的分类方法。 Objective Hyperspectral image (HSI) classification is one of the fundamental tasks in the field of applied remote sensing. As technological advances have increased the spatial and spectral resolutions available for data acquisition, the problem of achieving accurate HSI classification is becoming more challenging, especially for the HSI data with small labeled training samples and the insufficient utilization of spatial-spectral information in the HSI classification models. Aiming at these problems, this paper proposes a new HSI classification method (expressed as SKERW_SVM) by combining the superpixel dimension reduction (SDR) with post-processing optimization.Method First, we develop a superpixel sparse linear discriminant analysis (SSLDA) method by combining regional clustering (RC) with SLDA. In the SSLDA method, the RC is applied to construct a homogeneous local neighbourhood set with high spatial correlation and spectral similarity for each pixel of the HSI, and the SLDA to extract superpixel sparse mixture features that can fully characterize spatial-spectral information and related change information of the HSI based on the constructed homogeneous regions. Then, the extracted sparse mixture features are inputted into the support vector machine to generate the class probabilities of all pixels. Finally, the original class probabilities are optimized in the post-processing step by the extended random walker that can express the spatial relationship among adjacent pixels quantitatively, and the classification map is gotten according to the maximum probability.Result In order to assess the performance of the proposed method, a series of experiments are conducted on three small-scale HSI datasets including Indian Pines, University of Pavia, and Salinas, as well as a large-scale HSI dataset HoustonU. The proposed SKERW_SVM obtains overall accuracy of 98.58%, 96.88%, 98.54%, and 91.01%, respectively, on Indian Pines, University of Pavia, Salinas, and HoustonU. Experimental results demonstrate that our SKERW_SVM can fully mine the joint spatial-spectral features of HSI and achieve higher classification accuracy under the case of a small labeled training samples in comparison to several related advanced methods. Moreover, the operation time consuming by the SKERW_SVM is appropriate compared to other methods.Conclusion Therefore, under the lack of the labeled HSI pixels condition, the proposed HSI classification method by combining the SDR with post-processing optimization can efficiently extract the high discrimination mixture features information of HSI and significantly enhance classification performance. The SDR based on the homogeneous local regions, one of the components of the SKERW_SVM classification model, can greatly reduce the data redundancy and fully extract the information of spatial and spectral signatures in comparison to the pixel-wise dimension reduction methods. Meanwhile, the extended random walker in the post-processing step can fully use spatial information of HSI by constructing relationship graph to optimizing the original class probabilities, which further improves the classification performance. 高光谱图像分类 超像素降维 混合特征提取 后处理优化 支持向量机 hyperspectral image classification superpixel dimension reduction mixture feature extraction post-processing optimization support vector machine 1　引 言 1 高光谱图像HSI（Hyperspectral Image）是由成像光谱仪在数十甚至上百个波段内对地面物体成像的结果，与多光谱图像相比，HSI包含了大量的地物信息，有效提高了在遥感图像分类方面的识别精度，被广泛地应用于农业（ Liang等，2015 Liang等，2015 ）、军事（ 徐栋等，2020 徐栋等，2020 ）和矿产资源开发（ Carrino等，2018 Carrino等，2018 ； 董新丰等，2020 董新丰等，2020 ）等领域。然而由于HSI的高维特性，导致图像在分类过程中产生Huge现象，致使分类精度随着参与运算波段数目的增加而先升后降（ 董安国等，2019 董安国等，2019 ）。因此，如何从HSI中提取高识别力特征以促进分类精度的提高仍是一个非常值得研究的问题。 早期为提高HSI分类的准确率，提出了包括支持向量机SVM（Support Vector Machine）（ Melgani等， 2004 Melgani等， 2004 ）、极限学习机（ Li等，2015 Li等，2015 ）、K-近邻法（ Blanzieri和Melgani，2008 Blanzieri和Melgani，2008 ）和决策树（ 华晔等，2012 华晔等，2012 ）等在内的许多基于谱特征的机器学习分类方法。由于SVM具有能够以稳健有效的方式处理高维数据和小训练样本数据的特性，从而在高光谱图像分类任务中脱颖而出（ 闫敬文等，2019 闫敬文等，2019 ； Camps-Valls和Bruzzone，2005 Camps-Valls和Bruzzone，2005 ）。到目前为止，SVM仍被广泛作为模型分类器而应用于HSI的分类任务（ Jiang等，2018 Jiang等，2018 ； Duan等，2020 Duan等，2020 ； 李丹等，2021 李丹等，2021 ）。已有文献表明（ 李非燕等，2019 李非燕等，2019 ； 付青等，2020 付青等，2020 ； 魏祥坡等，2020 魏祥坡等，2020 ），仅依靠高光谱图像的谱特征，而忽略空间信息的分类方法，其分类效果往往不是很理想。对此，众多学者（ Duan等，2020 Duan等，2020 ； 程文娟和陈文强，2020 程文娟和陈文强，2020 ； 张祥东等，2021 张祥东等，2021 ； 高奎亮等，2021 高奎亮等，2021 ）相继提出了多种联合空间和光谱信息的特征提取方法，致力于高精度地识别出HSI的地物类型。例如， Duan等（2020） Duan等（2020） 在光谱信息的基础上，构建了一种基于自适应纹理平滑的双通道空间特征提取算法； 程文娟和陈文强（2020） 程文娟和陈文强（2020） 利用卷积核提取HSI的空间-光谱特征并跨层融合，发展了一种多尺度跨层特征融合注意力机制的分类方法； 张祥东等（2021） 张祥东等（2021） 设计了一种能同步提取HSI空间信息和光谱信息的卷积神经网络模型。然而，这些基于空-谱联合特征的分类方法并没有很好的关注到HSI的空域与谱域信息之间的联系。值得注意的是，HSI的同一类别像素通常以不同的区域聚集在一起，导致图像是由许多均匀的同质区域组成。目前，已经有学者（ 李丹等，2021 李丹等，2021 ； Zheng等，2019 Zheng等，2019 ； Jiang等，2018 Jiang等，2018 ）关注到了这个现象。如 李丹等（2021） 李丹等（2021） 基于局部邻域集合提取图像的高斯混合特征，充分捕获了HSI的空间纹理结构信息、不同谱段间联系和空-谱变化信息； Zheng等（2019） Zheng等（2019） 根据超像素分割和距离加权线性回归分类器设计了一种分类方法，并应用于HSI的分类得到了良好的结果。尤其是， Jiang等（2018） Jiang等（2018） 提出的一种以均匀同质区域为混合特征提取单元的超像素主成分分析SPCA（Superpixelwise Principal Component Analysis）方法，该方法可有效提取HSI的高区分性特征信息，并在小样本标签下具有优良的分类性能。但是，SPCA方法中采用的熵率超像素分割算法（ 宋熙煜等，2015 宋熙煜等，2015 ）生成的超像素紧密度和边缘贴合度一般，不能很好的形成均匀区域。然而， Xu等（2018） Xu等（2018） 基于简单线性迭代聚类SLIC（Simple Linear Iterative Clustering）发展了一种区域聚类RC（Regional Clustering）算法。该方法可自然平衡HSI的空间信息和光谱信息，是一种适合HSI分类的空间预处理方法。另一方面， Li等（2019） Li等（2019） 在传统线性判别分析的基础上考虑了巴氏误差界优化和非凸问题，提出了一种鲁棒的稀疏线性判别分析SLDA（Sparse Linear Discriminant Analysis）方法。受SPCA的启发，本文通过有效结合RC和SLDA发展了一种超像素稀疏线性判别分析SSLDA（Superpixelwise SLDA）方法。然而，预实验表明，基于SSLDA方法提取的HSI空谱特征信息尽管能够很好地表征高光谱图像信息，但在部分异质边缘区域的像元识别率欠佳。为了优化HSI的初始类别概率向量， Kang等（2014） Kang等（2014） 基于扩展的随机游走ERW（Extended Random Walker）构建了一种分类方法，并应用于三组常用的高光谱数据集，获得了良好的分类结果。为了弥补SSLDA在部分异质边缘区域像元识别率欠佳的问题，本文应用ERW进一步优化基于SSLDA获得的初始类别概率向量。相较于 Kang等（2014） Kang等（2014） ，本文首先利用SSLDA提取HSI的高判别性稀疏混合特征，然后输入SVM获得HSI的初始类别概率向量，以期提高分类方法的精度和效率。 综上所述，为了充分利用高光谱图像的空间-光谱信息，且在样本标签量少的情况下提高图像的分类精度，本文提出一种联合超像素降维和后验概率优化的高光谱图像分类方法。方法首先基于HSI的空间-光谱信息，利用已发展的超像素稀疏线性判别分析法在空间邻域信息的基础上为每个样本分配不同的局部邻域，并在局部邻域集合中提取超像素稀疏混合特征；然后将全局稀疏混合特征代表HSI的空间-光谱特征输入支持向量机分类器中，生成像素的类别概率向量；最后采用后验概率模型-随机游走法刻画HSI相邻像素之间的空间相关性，并优化类别概率向量且依据概率最大值得到分类标签图，从而完成图像分类任务。为了验证所提出的方法是否可以在HSI的图像分类任务中取得优良的识别性能，将其应用到三组小规模数据集Indian Pines、Pavia University和Salinas，以及一组大规模数据集HoustonU上进行分类实验，并与几种先进方法进行对比分析。 2　研究方法 1 2.1　超像素稀疏线性判别分析法 2 为了有效提取高判别性的HSI特征信息和减少模型的整体运行时间，本文首先发展一种超像素稀疏线性判别分析（SSLDA）方法，致力于获取一组超像素稀疏混合特征来充分表征HSI的空间-光谱信息及其相关变化信息。相比于归一化切割算法（ Shi和Malik，2000 Shi和Malik，2000 ）、分水岭分割算法（ Vincent和Soille，1991 Vincent和Soille，1991 ）、基于图的分割算法（ Felzenszwalb和Huttenlocher，2004 Felzenszwalb和Huttenlocher，2004 ）和SLIC（ Achanta等，2012 Achanta等，2012 ），SSLDA方法中采用的区域聚类不仅降低了计算复杂度，还同时集成了HSI的空间-光谱信息，得到一组既表现出空间相关性又表现出光谱相似性的聚类分区。SSLDA方法在这些聚类子块集合中应用稀疏线性判别法降低HSI的光谱维度，并提取混合特征。因此，超像素降维方法主要分为局部邻域集合的提炼和稀疏混合特征的提取两大步骤，具体操作如下： 令 <math id="M1"><mi mathvariant="bold-italic">X</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><msub><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>×</mo><mi mathvariant="normal">d</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705849&type= 19.38866615 3.21733332 表示HSI数据集，其中 <math id="M2"><msub><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705851&type= 12.19200039 3.72533321 为图像尺寸， <math id="M3"><mi mathvariant="normal">d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705853&type= 1.86266661 2.70933342 为原始光谱维度。把3D立方体矩阵 <math id="M4"><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705855&type= 2.45533323 2.62466669 重塑成2D矩阵为 <math id="M5"><mi mathvariant="bold-italic">X</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi mathvariant="normal">N</mi></mrow></msub></mrow></mfenced><mo>∈</mo><msup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi mathvariant="normal">d</mi><mo>×</mo><mi mathvariant="normal">N</mi></mrow></msup><mo>,</mo><mi mathvariant="normal">N</mi><mo>=</mo><msub><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705858&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705857&type= 75.60733032 5.24933338 ，其中 <math id="M6"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mi mathvariant="normal">d</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705859&type= 45.88933182 5.84200001 是列向量，表示HSI中的第 <math id="M7"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706142&type= 1.10066664 2.62466669 个像素向量。 1）初始化聚类。根据预先设置的分区数 <math id="M8"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706237&type= 1.86266661 2.62466669 ，等距离初始化聚类中心 <math id="M9"><mi mathvariant="bold-italic">C</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></mfenced><mo>∈</mo><msup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi mathvariant="normal">d</mi><mo>×</mo><mi>S</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705865&type= 53.34000015 5.24933338 ，其中 <math id="M10"><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>j</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mi mathvariant="normal">d</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705868&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705867&type= 50.12266922 6.43466663 ；根据聚类中心初始化局部邻域集合 <math id="M11"><mi mathvariant="bold-italic">Ζ</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705870&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705869&type= 45.38133240 5.24933338 ，其中 <math id="M12"><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mo>,</mo><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705872&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705871&type= 31.75000000 4.40266657 ， <math id="M13"><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705874&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705873&type= 2.87866688 4.06400013 表示局部邻域集合 <math id="M14"><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705975&type= 3.47133350 3.72533321 包含的第 <math id="M15"><mi mathvariant="normal">o</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705878&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705877&type= 1.69333339 2.70933342 个像元。 2）计算空间距离和光谱距离。定义像元 <math id="M16"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705939&type= 2.53999996 3.72533321 和聚类中心 <math id="M17"><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705941&type= 2.70933342 3.72533321 的空间坐标分别为 <math id="M18"><mfenced separators="|"><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>s</mi><mi>p</mi><mi>a</mi></mrow></msubsup><mo>,</mo><mtext> </mtext><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>s</mi><mi>p</mi><mi>a</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705884&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705883&type= 14.73200035 4.57200003 和 <math id="M19"><mfenced separators="|"><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mi>s</mi><mi>p</mi><mi>a</mi></mrow></msubsup><mo>,</mo><mtext> </mtext><msubsup><mrow><mi>y</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mi>s</mi><mi>p</mi><mi>a</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705886&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705885&type= 14.73200035 5.84200001 ，在 <math id="M20"><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705887&type= 2.70933342 3.72533321 的 <math id="M21"><msub><mrow><mi mathvariant="normal">N</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="normal">N</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705937&type= 11.59933376 3.80999994 矩形搜索范围内计算像元 <math id="M22"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705892&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705891&type= 2.53999996 3.72533321 到 <math id="M23"><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705941&type= 2.70933342 3.72533321 的空间距离 <math id="M24"><msub><mrow><mi>D</mi></mrow><mrow><mi>S</mi><mi>p</mi><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705923&type= 5.50333309 3.72533321 和光谱距离 <math id="M25"><msub><mrow><mi>D</mi></mrow><mrow><mi>S</mi><mi>p</mi><mi>e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705925&type= 5.33400011 3.72533321 。其中 <math id="M26"><msub><mrow><mi>D</mi></mrow><mrow><mi>S</mi><mi>p</mi><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705899&type= 5.50333309 3.72533321 利用欧氏距离度量获得，而 <math id="M27"><msub><mrow><mi>D</mi></mrow><mrow><mi>S</mi><mi>p</mi><mi>e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705925&type= 5.33400011 3.72533321 采用光谱信息散度-光谱角获得。具体定义为（ Xu等，2018 Xu等，2018 ）： <math display="block" id="M28"><msub><mrow><mi>D</mi></mrow><mrow><mi>S</mi><mi>p</mi><mi>a</mi></mrow></msub><mo>=</mo><mrow><mrow><mroot><mrow><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>s</mi><mi>p</mi><mi>a</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mi>s</mi><mi>p</mi><mi>a</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>s</mi><mi>p</mi><mi>a</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>y</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mi>s</mi><mi>p</mi><mi>a</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow><mo>/</mo><mrow><mi>r</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705903&type= 58.33533096 8.55133343 <math display="block" id="M29"><msub><mrow><mi>D</mi></mrow><mrow><mi>S</mi><mi>p</mi><mi>e</mi></mrow></msub><mo>=</mo><mi>S</mi><mi>I</mi><mi>D</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><mo>×</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mi>S</mi><mi>A</mi><mi>M</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705906&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705905&type= 56.47266769 5.92666674 其中： <math id="M30"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705908&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705907&type= 1.26999998 2.62466669 表示归一化搜索范围的对角线长度；且 <math display="block" id="M31"><mi>S</mi><mi>I</mi><mi>D</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">d</mi></mrow></munderover></mstyle><mrow><mfenced separators="|"><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705909&type= 53.42466354 10.49866676 <math display="block" id="M32"><mi>S</mi><mi>A</mi><mi>M</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><mo>=</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mfrac><mrow><mfenced open="<" close=">" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mtext> </mtext><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></mrow><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mtext> </mtext><mo>·</mo><mtext> </mtext><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705912&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705911&type= 48.76800156 12.78466606 式中： <math id="M33"><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>/</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">d</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705914&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705913&type= 19.30400085 8.80533314 ， <math id="M34"><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>j</mi></mrow></msub><mo>/</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">d</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>j</mi></mrow></msub></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705915&type= 23.28333473 8.80533314 ； <math id="M35"><mfenced open="<" close=">" separators="|"><mrow><mo>·</mo><mtext> </mtext><mo>,</mo><mtext> </mtext><mo>·</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705918&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705917&type= 9.31333351 3.89466691 表示向量的内积； <math id="M36"><mfenced open="‖" close="‖" separators="|"><mrow><mtext> </mtext><mo>·</mo><mtext> </mtext></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705920&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705919&type= 6.94266701 3.80999994 表示向量的模。 3）计算空-谱距离 <math id="M37"><mi>D</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705922&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705921&type= 14.05466747 5.16466665 。根据空间距离 <math id="M38"><msub><mrow><mi>D</mi></mrow><mrow><mi>S</mi><mi>p</mi><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705923&type= 5.50333309 3.72533321 和光谱距离 <math id="M39"><msub><mrow><mi>D</mi></mrow><mrow><mi>S</mi><mi>p</mi><mi>e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705925&type= 5.33400011 3.72533321 计算像元 <math id="M40"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705939&type= 2.53999996 3.72533321 与聚类中心 <math id="M41"><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705941&type= 2.70933342 3.72533321 的空-谱距离（ Xu等，2018 Xu等，2018 ）： <math display="block" id="M42"><mi>D</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>σ</mi></mrow></mfenced><mo>×</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>S</mi><mi>p</mi><mi>e</mi></mrow></msub><mo>+</mo><mi>σ</mi><mo>×</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>S</mi><mi>p</mi><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705932&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705931&type= 55.79533768 5.16466665 式中： <math id="M43"><mi>σ</mi><mo>∈</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705934&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705933&type= 15.83266735 3.89466691 表示平衡空间项和光谱项的权重因子。 4）更新聚类中心。根据式（ 1 1 ）~（ 5 5 ）计算各聚类中心 <math id="M44"><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705941&type= 2.70933342 3.72533321 的 <math id="M45"><msub><mrow><mi mathvariant="normal">N</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="normal">N</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705937&type= 11.59933376 3.80999994 搜索范围内所有像元 <math id="M46"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705939&type= 2.53999996 3.72533321 与 <math id="M47"><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705941&type= 2.70933342 3.72533321 的距离，以各聚类中心子块内距离最小值对应的像元更新聚类中心，为 <math id="M48"><msup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>u</mi></mrow></msup><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>u</mi></mrow></msubsup><mo>,</mo><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>u</mi></mrow></msubsup><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>u</mi></mrow></msubsup><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>S</mi></mrow><mrow><mi>u</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705943&type= 43.26466751 4.31799984 ，其中 <math id="M49"><msubsup><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>u</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705945&type= 2.79399991 4.06400013 的表达式为： <math display="block" id="M50"><msubsup><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>u</mi></mrow></msubsup><mo>=</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></munder><mfenced open="{" close="}" separators="|"><mrow><mi mathvariant="bold-italic">D</mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>u</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705947&type= 33.78200150 6.77333355 式中： <math id="M51"><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705975&type= 3.47133350 3.72533321 表示第 <math id="M52"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705952&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705951&type= 1.69333339 2.62466669 个局部邻域集合； <math id="M53"><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705954&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705953&type= 1.86266661 2.62466669 表示更新迭代次数。 5）终止迭代。一般迭代次数设置为10次，完成迭代后输出最终局部邻域集合 <math id="M54"><mi mathvariant="bold-italic">Z</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705956&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705955&type= 45.29666519 5.24933338 和中心 <math id="M55"><mi mathvariant="bold-italic">C</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706094&type= 41.99466705 5.24933338 ，否则重复步骤2~4直到完成迭代。 6）降低局部邻域集合 <math id="M56"><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705959&type= 3.47133350 3.72533321 的光谱维度。由于SLDA方法不仅能最大程度减少数据冗余信息，还能提取出最具判别性的稀疏特征，并对图像中包含的异常值和噪声具有一定的鲁棒性。因此，为了准确表征HSI的空间-光谱特征，利用SLDA提取邻域集合 <math id="M57"><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705975&type= 3.47133350 3.72533321 的稀疏混合特征。为此，求解以下目标函数得到线性变换矩阵 <math id="M58"><msubsup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705963&type= 3.55599999 4.06400013 （ Li等，2019 Li等，2019 ）： <math display="block" id="M59"><msubsup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><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">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></munder><mfenced open="{" close="}" separators="|"><mrow><mi>ϑ</mi><msub><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>δ</mi><msubsup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>w</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705966&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705965&type= 51.64666748 8.55133343 式中： <math id="M60"><msub><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>≠</mo><mtext> </mtext><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705967&type= 9.90600014 3.89466691 ； <math id="M61"><mi>ϑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705969&type= 2.53999996 2.70933342 和 <math id="M62"><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705971&type= 1.60866666 2.62466669 为调整参数，值均大于0； <math id="M63"><msub><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">o</mi></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">o</mi></mrow></munderover></mstyle><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705973&type= 43.34933472 8.38199997 为类内散度矩阵。然后，通过以下公式得到局部邻域集合 <math id="M64"><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705975&type= 3.47133350 3.72533321 的稀疏混合特征 <math display="block" id="M65"><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">*</mi><mi mathvariant="normal">T</mi></mrow></msubsup><mo>×</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705977&type= 20.40466690 4.06400013 7）输出HSI的所有邻域集合的稀疏混合特征 <math id="M66"><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705980&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705979&type= 3.38666677 3.72533321 ，并表示为 <math id="M67"><mi mathvariant="bold-italic">Y</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext>  </mtext><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705982&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705981&type= 45.21200180 5.24933338 <math id="M68"><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">y</mi></mrow><mrow><mi mathvariant="normal">N</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705983&type= 39.11600113 5.24933338 。 2.2　核主成分分析法 2 基于同质局部邻域集合的SSLDA方法所提取的超像素稀疏混合特征，虽然能够充分挖掘HSI的纹理结构信息和判别性光谱特征，但是却忽略了不同局部邻域集合之间存在的内在联系，从而导致得到的稀疏混合特征仍存在一定的数据冗余。考虑到核主成分分析KPCA（Kernel Principal Component Analysis）（ Yu等，2021 Yu等，2021 ）利用核函数，不仅获得数据的主成分信息，还能使非线性数据线性可分（Schölkopf等，1997）。因此，本文对超像素稀疏混合特征应用KPCA，以至于能够得到更加精简和高区分性的特征信息。KPCA的主要步骤如下： 1）针对HSI的全局稀疏混合特征 <math id="M69"><mi mathvariant="bold-italic">Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706106&type= 2.28600001 2.62466669 ，利用非线性映射函数 <math id="M70"><mi>ϕ</mi><mfenced separators="|"><mrow><mo>•</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705991&type= 6.94266701 3.80999994 把 <math id="M71"><mi mathvariant="bold-italic">Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706106&type= 2.28600001 2.62466669 映射到高维空间。由于 <math id="M72"><mi>ϕ</mi><mfenced separators="|"><mrow><mo>•</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705991&type= 6.94266701 3.80999994 不能显性给出，因此令 <math id="M73"><mi mathvariant="bold-italic">K</mi><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mi>ϕ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">Y</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi>ϕ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">Y</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705993&type= 28.53266525 5.07999992 ，则 <math id="M74"><mi mathvariant="bold-italic">Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706106&type= 2.28600001 2.62466669 的主成分特征 <math id="M75"><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705997&type= 2.03200006 3.72533321 计算如下（ Yu等，2021 Yu等，2021 ）： <math display="block" id="M76"><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub><mo>=</mo><mtext> </mtext><msubsup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mi mathvariant="bold-italic">K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706000&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24705999&type= 12.86933327 4.14866638 式中： <math id="M77"><mi mathvariant="bold-italic">K</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi mathvariant="normal">N</mi><mo>×</mo><mi mathvariant="normal">N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706001&type= 14.39333248 2.96333337 表示通过高斯核函数得出的矩阵， <math id="M78"><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706003&type= 3.04800010 3.72533321 表示 <math id="M79"><mi mathvariant="bold-italic">K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706006&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706005&type= 2.45533323 2.62466669 的第 <math id="M80"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706142&type= 1.10066664 2.62466669 个特征值 <math id="M81"><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706010&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706009&type= 2.96333337 3.72533321 对应的特征向量。 2）根据 <math id="M82"><mi mathvariant="bold-italic">K</mi><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706011&type= 16.17133331 3.72533321 计算出特征值 <math id="M83"><msub><mrow><mi mathvariant="normal">λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≥</mo><msub><mrow><mi mathvariant="normal">λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msub><mrow><mi mathvariant="normal">λ</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706013&type= 27.26266861 3.72533321 ，并取前 <math id="M84"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706239&type= 2.28600001 2.62466669 个特征值对应的特征向量 <math id="M85"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>K</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706017&type= 25.39999962 5.24933338 。 3）利用特征向量 <math id="M86"><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706020&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706019&type= 3.04800010 3.72533321 和（9）式计算出 <math id="M87"><mi mathvariant="bold-italic">Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706106&type= 2.28600001 2.62466669 的前 <math id="M88"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706239&type= 2.28600001 2.62466669 个主成分组成的特征矩阵 <math id="M89"><mi mathvariant="bold-italic">T</mi><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><mi>K</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>∈</mo><msup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>K</mi><mo>×</mo><mi mathvariant="normal">N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706025&type= 42.50266647 5.84200001 。 2.3　扩展的随机游走法 2 由于超像素降维提取的空谱特征容易导致分类方法在部分异质边缘区域的错分率较高，然而扩展的随机游走算法（ Grady，2006 Grady，2006 ）具有计算简单、搜索速度快、不受噪声影响和能够同时识别多个对象的特点。因此，本文采用ERW编码HSI的空间信息至加权图中，以刻画相邻像素之间的空间相关性，并用之优化由SVM得到的初始类别概率向量。 首先，利用已计算的主成分特征矩阵 <math id="M90"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706113&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706112&type= 2.20133328 2.62466669 代表高光谱图像的空间-光谱特征输入SVM分类器中，得到的初始类别概率向量 <math id="M91"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi mathvariant="normal">N</mi></mrow></msub></mrow></mfenced><mo>∈</mo><msup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi mathvariant="normal">M</mi><mo>×</mo><mi mathvariant="normal">N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706030&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706029&type= 54.35599899 5.24933338 ，其中 <math id="M92"><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><msubsup><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msubsup><mo>,</mo><mtext> </mtext><msubsup><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msubsup><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">M</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706031&type= 30.98800087 4.99533367 ，且 <math id="M93"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">M</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>m</mi></mrow></msubsup></mrow><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706034&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706033&type= 14.39333248 8.29733276 ， <math id="M94"><mi mathvariant="normal">M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706124&type= 2.87866688 2.62466669 表示类别数量。ERW对不同类别 <math id="M95"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706051&type= 2.70933342 2.62466669 的空间概率优化通过以下能量函数（ Duan等，2020 Duan等，2020 ）来实现： <math display="block" id="M96"><msup><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>E</mi></mrow><mrow><mi>S</mi><mi>p</mi><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced><mo>+</mo><mi>γ</mi><msubsup><mrow><mi>E</mi></mrow><mrow><mi>a</mi><mi>s</mi><mi>p</mi><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706039&type= 48.09066391 5.16466665 其中： <math id="M97"><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706041&type= 4.14866638 4.48733330 表示根据训练样本的位置索引和标签值形成的类别 <math id="M98"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706051&type= 2.70933342 2.62466669 的区域像元集合； <math id="M99"><msubsup><mrow><mi>E</mi></mrow><mrow><mi>s</mi><mi>p</mi><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706045&type= 12.36133385 5.16466665 表示第 <math id="M100"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706051&type= 2.70933342 2.62466669 类像元集合相邻像素之间的空间相关性； <math id="M101"><msubsup><mrow><mi>E</mi></mrow><mrow><mi>a</mi><mi>s</mi><mi>p</mi><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706049&type= 13.29266548 5.16466665 表示第 <math id="M102"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706051&type= 2.70933342 2.62466669 类像元初始类别概率的拉普拉斯-贝尔特拉米算子； <math id="M103"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706053&type= 2.03200006 3.38666677 表示权重因子；且有： <math display="block" id="M104"><msubsup><mrow><mi>E</mi></mrow><mrow><mi>s</mi><mi>p</mi><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mi mathvariant="bold-italic">L</mi><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706055&type= 27.43199921 5.16466665 <math display="block" id="M105"><msubsup><mrow><mi>E</mi></mrow><mrow><mi>a</mi><mi>s</mi><mi>p</mi><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced><mo>=</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>i</mi><mo>≠</mo><mi>m</mi></mrow></munder><mrow><msubsup><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></mrow><msub><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Λ</mi></mrow><mrow><mi>m</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706057&type= 75.52266693 7.78933382 其中： <math id="M106"><mi mathvariant="bold-italic">Λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706059&type= 2.70933342 2.79399991 表示由初始类别概率组成的对角线矩阵； <math id="M107"><mi mathvariant="bold-italic">L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706062&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706061&type= 2.03200006 2.62466669 表示对原始HSI利用KPCA计算的第一主成分特征的拉普拉斯矩阵（ Grady，2006 Grady，2006 ）： <math display="block" id="M108"><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>-</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mtext>  </mtext><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">和</mi><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mi mathvariant="normal">是</mi><mi mathvariant="normal">相</mi><mi mathvariant="normal">邻</mi><mi mathvariant="normal">节</mi><mi mathvariant="normal">点</mi></mtd></mtr><mtr><mtd><mtext>  </mtext><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mtext>  </mtext><mi>i</mi><mo>=</mo><mi>j</mi></mtd></mtr><mtr><mtd><mtext>  </mtext><mn mathvariant="normal">0</mn><mo>,</mo><mtext>  </mtext><mi mathvariant="normal">其</mi><mi mathvariant="normal">他</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706063&type= 44.70399857 16.00200081 式中： <math id="M109"><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706079&type= 2.20133328 3.72533321 和 <math id="M110"><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706067&type= 2.11666679 3.72533321 表示图的顶点，其中图是依据图像尺寸构建而成。 <math id="M111"><msub><mrow><mi>ω</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced separators="|"><mrow><mo>-</mo><mi>β</mi><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706070&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706069&type= 35.81399918 7.19666624 表示连接 <math id="M112"><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706079&type= 2.20133328 3.72533321 和 <math id="M113"><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706073&type= 2.11666679 3.72533321 两个顶点的边界的权重值，其中 <math id="M114"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706075&type= 1.86266661 3.38666677 为自由参数； <math id="M115"><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mo>∑</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706077&type= 14.81666565 4.82600021 表示顶点 <math id="M116"><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706079&type= 2.20133328 3.72533321 的阶数。 然后，利用（10）式对不同类别的初始概率向量优化后得到所有类别的后验概率向量 <math id="M117"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706082&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706081&type= 3.55599999 3.72533321 。最后，根据概率最大值获得HSI的分类标签图。 2.4　本文的算法 2 联合上述的超像素稀疏线性判别分析、核主成分分析和扩展的随机游走方法，本文提出一种联合超像素降维和后验概率优化的高光谱图像分类模型，为方便表达简称为SKERW_SVM。其模型框架如 图1 图1 所示，算法的详细流程如下： 输入：高光谱图像 <math id="M118"><mi mathvariant="bold-italic">X</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><msub><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>×</mo><mi mathvariant="normal">d</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706086&type= 19.38866615 3.21733332 ，降维参数 <math id="M119"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706235&type= 1.10066664 2.62466669 ，超像素数量 <math id="M120"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706237&type= 1.86266661 2.62466669 和核主成分数量 <math id="M121"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706239&type= 2.28600001 2.62466669 。 1）初始化聚类中心 <math id="M122"><mi mathvariant="bold-italic">C</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">c</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706094&type= 41.99466705 5.24933338 和局部邻域集合 <math id="M123"><mi mathvariant="bold-italic">Z</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706097&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706096&type= 45.29666519 5.24933338 ； 2）基于（5）~（6）式和迭代次数不断更新 <math id="M124"><mi mathvariant="bold-italic">C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706099&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706098&type= 2.37066650 2.62466669 和 <math id="M125"><mi mathvariant="bold-italic">Z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706103&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706102&type= 2.45533323 2.62466669 ，并获得最终的局部邻域集合； 3）利用（7）~（8）式计算 <math id="M126"><mi mathvariant="bold-italic">Z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706103&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706102&type= 2.45533323 2.62466669 中各邻域集合的超像素稀疏混合特征，即得 <math id="M127"><mi mathvariant="bold-italic">Y</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706105&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706104&type= 44.45000076 5.24933338 ； 4）采用（9）式获得 <math id="M128"><mi mathvariant="bold-italic">Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706106&type= 2.28600001 2.62466669 的前 <math id="M129"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706239&type= 2.28600001 2.62466669 个主成分特征组成的矩阵 <math id="M130"><mi mathvariant="bold-italic">T</mi><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">t</mi></mrow><mrow><mi>K</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706110&type= 30.64933395 5.84200001 ； 5）针对 <math id="M131"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706113&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706112&type= 2.20133328 2.62466669 并按照一定比例选取训练样本和测试样本； 6）将训练样本和测试样本输入SVM分类器中，得到初始类别概率向量 <math id="M132"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><mo>⋯</mo><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi mathvariant="normal">N</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706115&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706114&type= 42.24866486 5.24933338 ； 7）应用（10）式来优化初始类别概率，得到 <math id="M133"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706117&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706116&type= 3.55599999 3.72533321 ，并依据概率最大值获得高光谱图像的类别标签图。 输出：HSI的类别标签图。 2.5　评价指标 2 为了定量评估不同算法的分类性能，本文采用总体精度OA（Overall Accuracy）、平均精度AA（Average Accuracy）和Kappa系数作为评价指标。OA是指正确识别像素占图像总像数的百分比。AA是指每类正确识别像素百分比和的平均值。Kappa系数是利用混淆矩阵计算出的统计量，用来衡量预测标签和真实标签之间的一致性。 各评价指标的具体定义为： <math display="block" id="M134"><mi>O</mi><mi>A</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 mathvariant="normal">M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub></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 mathvariant="normal">M</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mrow></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706119&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706118&type= 21.67466736 18.28800011 <math display="block" id="M135"><mi>A</mi><mi>A</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 mathvariant="normal">M</mi></mrow></munderover></mstyle><mrow><mi>C</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></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 mathvariant="normal">M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706120&type= 18.71133423 17.86466599 <math display="block" id="M136"><mi>K</mi><mi>a</mi><mi>p</mi><mi>p</mi><mi>a</mi><mo>=</mo><mfrac><mrow><mi>O</mi><mi>A</mi><mo>-</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>e</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>e</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706123&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706122&type= 26.75466537 8.72066593 式中： <math id="M137"><mi mathvariant="normal">M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706124&type= 2.87866688 2.62466669 表示高光谱图像的类别数量； <math id="M138"><mi mathvariant="bold-italic">F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706126&type= 2.45533323 2.62466669 表示混淆矩阵； <math id="M139"><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706129&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706128&type= 3.21733332 3.72533321 表示图像第 <math id="M140"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706142&type= 1.10066664 2.62466669 类测试样本总数；且 <math id="M141"><mi>C</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub></mrow><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=24706133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706132&type= 22.35199928 8.29733276 和 <math id="M142"><msub><mrow><mi>F</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mo>·</mo></mrow></msub><msub><mrow><mi>F</mi></mrow><mrow><mo>·</mo><mi>i</mi></mrow></msub></mrow><mo>/</mo><msup><mrow><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">M</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706135&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706134&type= 34.96733475 8.29733276 ；其中 <math id="M143"><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mo>·</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706136&type= 3.97933316 3.72533321 和 <math id="M144"><msub><mrow><mi>F</mi></mrow><mrow><mo>·</mo><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706138&type= 3.97933316 3.72533321 表示混淆矩阵第 <math id="M145"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706142&type= 1.10066664 2.62466669 行和第 <math id="M146"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706142&type= 1.10066664 2.62466669 列的和。OA、AA和Kappa系数的值通常在0~1之间，三者值越高，模型的分类性能越好。 3　实验与分析 1 实验在Windows10平台，内存为16GB的Intel（R） Core（TM） i7-10750H CPU 2.60GHz计算机硬件环境中运行。软件环境为MATLAB 2018b。 3.1　数据集介绍 2 本文采用三组常用的小规模数据集Indian Pines、Pavia University和Salinas，以及一组大规模数据集HoustonU来评估分类方法的性能。 图2 图2 绘制了这四组数据集的假彩色图像和标准地物分类图像，并给出了合成各数据集假彩色图像的波段序号。 （1）Indian Pines是由AVIRIS（Airborne Visible Infrared Imaging Spectrometer）传感器在1992年美国印第安纳州西北部的印第安松树场上空采集，有 <math id="M147"><mn mathvariant="normal">145</mn><mo>×</mo><mn mathvariant="normal">145</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706147&type= 15.66333294 2.70933342 个像素和224个光谱带，波长范围为0.4~2.5微米。实验中选择去除24个吸水通道后的200个波段图像作为输入数据。它包含16类地物，如 图2 图2 （a）所示，其中大部分地物为处于不同生长阶段的作物，由于不同作物之间具有相似的光谱反射率，因此具有一定的挑战性（ Mou等，2020 Mou等，2020 ）。 （2）Pavia University是2003年在Pavia大学上空由ROSIS（Reflective Optics System Imaging Spectrometer）传感器采集的 <math id="M148"><mn mathvariant="normal">610</mn><mo>×</mo><mn mathvariant="normal">340</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706150&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706149&type= 15.66333294 2.70933342 图像大小的场景影像，波长范围为0.43~0.86微米。将12个噪声波段筛出后，剩余103个波段图像应用于本次实验。图像包含沥青（Asphalt）、碎石（Gravel）和裸露土壤（Bare Soil）等9类地物，如 图2 图2 （b）所示。 （3）Salinas是由AVIRIS传感器在加利福尼亚州的Salinas山谷采集的224个波段图像组成的数据集，各波段图像尺寸为 <math id="M149"><mn mathvariant="normal">512</mn><mo>×</mo><mn mathvariant="normal">217</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706151&type= 15.66333294 2.70933342 像素，其中包含了16个地物类别。除未分类像素外，标准地物分类图像共54129标签像素可用，如 图2 图2 （c）所示。在本文的分类实验中，选择去除了20个吸水带后的204个波段图像作为实验数据。 （4）HoustonU是2018年IEEE GRASS 数据融合分类挑战大赛发布的竞赛数据集，由搭载在飞机上的CASI高光谱成像仪于2017年2月16日在休斯顿大学校园及其附近区域拍摄的场景图像，包括50个波段和20类地物，波长范围为0.38~1.05微米，图像的尺寸为 <math id="M150"><mn mathvariant="normal">601</mn><mo>×</mo><mn mathvariant="normal">2384</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706153&type= 17.52599907 2.70933342 像素，有标签的像元共504856个，如 图2 图2 （d）所示。与上述三组常用数据集相比，HoustonU的地物类型和有标签样本量多，空间范围广，分类任务更加复杂。有关本文数据集的详细介绍可参考网站 http：//www.ehu.eus/ccwintco/index.php？title=Hyperspectral http://www.ehu.eus/ccwintco/index.php?title=Hyperspectral _Remote_Sensing_Scenes和 http：//hyperspectral.ee.uh.edu/？page_id=1075 http://hyperspectral.ee.uh.edu/?page_id=1075 。 3.2　参数分析 2 本节对SKERW_SVM方法中降维后的光谱维度 <math id="M151"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706235&type= 1.10066664 2.62466669 、超像素数量 <math id="M152"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706237&type= 1.86266661 2.62466669 和用于分类的核主成分数量 <math id="M153"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706239&type= 2.28600001 2.62466669 三个重要参数进行优化分析。为检验所提出的方法在少量标签样本下的分类能力，本文将三组常用的Indian Pines、Pavia University和Salinas数据集的有标签样本分别随机划分为训练、验证和测试样本集，具体的划分样本数如表 1 1 ~ 3 3 所示。在这一节，选取训练样本训练模型，而依据验证样本集上的总体分类精度确定最优参数值，然后，在下一节利用测试集评估模型的分类性能。 图3 图3 显示了 <math id="M154"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706235&type= 1.10066664 2.62466669 、 <math id="M155"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706237&type= 1.86266661 2.62466669 和 <math id="M156"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706239&type= 2.28600001 2.62466669 的不同取值对SKERW_SVM方法在三组常用数据集上分类性能的影响。首先，初始化 <math id="M157"><mi>S</mi><mo>=</mo><mn mathvariant="normal">100</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706167&type= 11.93799973 2.70933342 和 <math id="M158"><mi>K</mi><mo>=</mo><mn mathvariant="normal">20</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706169&type= 10.49866676 2.70933342 ，取 <math id="M159"><mi>l</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">3</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">5</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">10</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">15</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">20</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">25</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">30</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">35</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">40</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706171&type= 60.62133789 3.13266683 和 <math id="M160"><mn mathvariant="normal">50</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706173&type= 3.72533321 2.70933342 分别对三组数据集进行分类实验，以识别参数 <math id="M161"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706235&type= 1.10066664 2.62466669 的最优值，如 图3 图3 （a~c）表明，当 <math id="M162"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706235&type= 1.10066664 2.62466669 为30、20和15（红色虚线处）时，对应在Indian Pines、Pavia University和Salinas上获得最大OA值。然后，在 <math id="M163"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706235&type= 1.10066664 2.62466669 为最优值的条件下，讨论参数 <math id="M164"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706237&type= 1.86266661 2.62466669 的最优值。由于无论KPCA中的主成分数 <math id="M165"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706239&type= 2.28600001 2.62466669 取何值，与未使用KPCA的分类方法相比，其分类效果均有不同程度的提高。因此，在分析参数 <math id="M166"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706237&type= 1.86266661 2.62466669 时，不考虑KPCA的使用，以强调 <math id="M167"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706237&type= 1.86266661 2.62466669 对分类结果的影响。取 <math id="M168"><mi>S</mi><mo>=</mo><mn mathvariant="normal">10</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">15</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">20</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">25</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">30</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">35</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">40</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">50</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">60</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">80</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">100</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706189&type= 75.09933472 3.13266683 和 <math id="M169"><mn mathvariant="normal">120</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706192&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706191&type= 5.58799982 2.70933342 ，分别对三组数据集进行分类实验， 图3 图3 （d~f）呈现了所获得的总体分类精度。从 图3 图3 （d~f）可知，在设置的参数范围内，Indian Pines、Pavia University和Salinas数据集对应的最优 <math id="M170"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706237&type= 1.86266661 2.62466669 值分别为50、20和15。由于参数 <math id="M171"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706237&type= 1.86266661 2.62466669 的最优取值与图像的纹理结构关系密切，合适的超像素数量有利于充分挖掘同质区域的高区分性特征。然而，无论 <math id="M172"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706237&type= 1.86266661 2.62466669 的大小如何，SKERW_SVM的分类性能总是高于未进行超像素分割的方法，这与 Jiang等（2018） Jiang等（2018） 中的结论是一致的。因此，对于Salinas数据集，尽管SKERW_SVM的最优参数 <math id="M173"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706237&type= 1.86266661 2.62466669 值稍小于该数据集的类别数，但本文提出的SSLDA方法仍能对模型的分类性能起到提高的作用。最后，在 <math id="M174"><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">3</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">5</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">10</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">15</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">20</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">25</mn><mo>,</mo><mtext> </mtext><mn mathvariant="normal">30</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706202&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706201&type= 44.95800018 3.89466691 的集合中，当 <math id="M175"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706235&type= 1.10066664 2.62466669 和 <math id="M176"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706237&type= 1.86266661 2.62466669 均为最优值的条件下，讨论三组数据集上的最优 <math id="M177"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706239&type= 2.28600001 2.62466669 值。由 图3 图3 （g~i）知，在Indian Pines、Pavia University和Salinas数据集上的最优参数 <math id="M178"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706239&type= 2.28600001 2.62466669 值依次为20、25和20。 类别号 类名 训练集 验证集 测试集 1 Alfalfa 5 10 31 2 Corn-notill 70 80 1278 3 Corn-mintill 50 60 720 4 Corn 20 40 177 5 Grass-pasture 40 50 393 6 Grass-trees 40 50 640 7 Grass-pasture-mowed 5 5 18 8 Hay-windrowed 40 40 398 9 Oats 5 5 10 10 Soybean-notill 50 60 862 11 Soybean-mintill 80 90 2285 12 Soybean-clean 50 60 483 13 Wheat 20 40 145 14 Woods 50 60 1155 15 Buildings-Grass-Trees-Drives 30 40 316 16 Stone-Steel-Towers 5 10 78 2 总计 560 700 8989 类别号 类名 训练集 验证集 测试集 1 Asphalt 66 130 6435 2 Meadows 186 300 18163 3 Gravel 21 40 2038 4 Trees 31 60 2973 5 Painted metal sheets 13 25 1307 6 Bare Soil 50 100 4879 7 Bitumen 13 25 1292 8 Self-Blocking Bricks 37 80 3565 9 Shadows 9 20 918 2 总计 426 780 41570 类别号 类名 训练集 验证集 测试集 1 Brocoli_green_weeds_1 20 40 1949 2 Brocoli_green_weeds_2 30 60 3636 3 Fallow 20 40 1916 4 Fallow_rough_plow 15 30 1349 5 Fallow_smooth 30 60 2588 6 Stubble 40 80 3839 7 Celery 40 80 3459 8 Grapes_untrained 112 220 10939 9 Soil_vinyard_develop 62 120 6021 10 Corn_senesced_green_weeds 32 60 3186 11 Lettuce_romaine_4wk 10 20 1038 12 Lettuce_romaine_5wk 20 40 1867 13 Lettuce_romaine_6wk 10 20 886 14 Lettuce_romaine_7wk 10 20 1040 15 Vinyard_untrained 72 170 7026 16 Vinyard_vertical_trellis 20 40 1747 2 总计 543 1100 52486 3.3　实验设置 2 为评估SKERW_SVM在HSI分类任务上是否具有优良的分类性能，本文实验选择5个相关分类方法与SKERW_SVM在已选用的高光谱数据集上进行仿真对比实验。对比方法分别是：（1）传统的支持向量机分类器（ Melgani和Bruzzone，2004 Melgani和Bruzzone，2004 ），简称为SVM；（2）利用主成分分析（PCA）提取HSI的光谱特征并将其作为HSI的图像信息输入SVM中的方法（ Prasad和Bruce，2008 Prasad和Bruce，2008 ），简称为PCA_SVM；（3）采用一种结构轮廓SP（Structural Profile）自适应地提取HSI的空-谱特征并输入SVM的方法（ Duan等，2020 Duan等，2020 ），简称为SP_SVM；（4）一种以均质区域为混合特征提取单元的SPCA并结合SVM的方法（ Jiang等，2018 Jiang等，2018 ），简称为SPCA_SVM；（5）应用ERW编码HSI的空间信息至加权图中，并用之优化由SVM获得的类别概率向量的方法（ Kang等，2014 Kang等，2014 ），简称为ERW_SVM。此外，为评估SSLDA、KPCA和ERW对SKERW_SVM分类结果的阶段性贡献，我们也进行了消融实验。其对比方法包括：（1）仅利用SSLDA提取混合特征即输入SVM的方法，简称为SSLDA_SVM；（2）利用SSLDA和KPCA提取混合特征并输入SVM的方法，简称为SKPCA_SVM。在所有对比方法的分类实验中，涉及与本文提出的SKERW_SVM方法同样的参数时，均设置一致。其中，SVM分类器采用径向基核函数，通过五次交叉验证方法选择分类模型参数。 由于训练样本和测试样本是按照一定比例随机分布，从而导致分类精度容易产生较小的误差，因此对所有方法均重复10次，取平均值作为最终的分类精度。 3.4　结果与分析 2 首先分析SKERW_SVM与所选对比方法分别在小规模数据集Indian Pines、Pavia University和Salinas上的分类效果。根据表 1 1 ~ 3 3 所示的各数据集训练样本和测试样本的数量随机生成训练集和测试集，各分类模型的OA、AA和Kappa，及单个类别的精度分别如表 4 4 ~ 6 6 所示，可视化各模型的分类标签图分别如图 4 4 ~ 6 6 所示。 类别号 SVM PCA_SVM SP_SVM SPCA_SVM ERW_SVM SSLDA_SVM SKPCA_SVM SKERW_SVM 1 76.19 59.09 100.00 50.00 100.00 88.45 71.43 100.00 2 75.12 58.79 96.61 90.13 96.00 87.65 91.30 99.06 3 59.71 59.63 98.89 85.87 98.91 81.40 84.11 100.00 4 54.92 38.16 98.59 73.22 96.64 88.63 79.92 98.34 5 83.49 81.50 99.58 97.74 99.57 93.09 93.98 99.79 6 88.63 87.98 100.00 100.00 99.86 94.97 94.28 99.73 7 92.31 88.89 100.00 0.00 100.00 94.95 100.00 100.00 8 92.50 90.43 100.00 100.00 100.00 99.75 95.98 99.58 9 73.68 50.00 100.00 100.00 100.00 83.55 70.59 100.00 10 76.28 64.21 91.93 93.94 98.98 88.79 90.67 98.00 11 75.52 72.00 96.36 94.81 96.67 92.26 93.77 97.85 12 57.00 49.75 96.31 71.33 97.67 85.06 76.99 94.27 13 94.53 94.09 100.00 99.51 100.00 95.81 97.84 100.00 14 94.33 90.95 99.92 99.14 99.29 98.84 99.92 99.14 15 61.34 49.36 100.00 99.22 99.48 98.33 93.66 98.97 16 100.00 97.18 98.80 94.59 96.84 97.54 100.00 100.00 OA（%） 76.69 70.30 97.47 92.41 98.06 91.45 91.77 98.58 AA（%） 78.47 70.75 98.56 84.34 98.75 91.82 89.65 99.05 Kappa（%） 73.50 66.16 97.12 91.34 97.78 90.23 90.62 98.38 类别号 SVM PCA_SVM SP_SVM SPCA_SVM ERW_SVM SSLDA_SVM SKPCA_SVM SKERW_SVM 1 80.20 76.18 90.21 75.20 91.12 85.52 85.61 92.59 2 91.17 88.12 99.06 96.02 96.61 91.59 92.54 97.24 3 67.97 65.11 90.94 89.70 99.95 75.89 68.83 99.95 4 96.65 94.84 91.84 85.26 99.58 86.40 96.76 99.70 5 98.81 100.00 86.54 95.63 99.93 98.34 99.26 99.70 6 85.30 87.47 99.58 90.99 97.80 88.04 90.66 96.86 7 100.00 35.00 94.66 86.22 100.00 61.23 74.20 100.00 8 80.42 68.71 92.36 79.87 98.25 80.02 71.88 97.19 9 100.00 100.00 87.01 90.02 99.88 100.00 99.89 99.89 OA（%） 87.19 84.01 95.51 89.13 96.53 87.89 88.62 96.88 AA（%） 88.95 79.49 92.47 87.66 98.12 85.23 86.63 98.13 Kappa（%） 82.74 78.26 94.03 85.50 95.35 83.76 84.70 95.83 类别号 SVM PCA_SVM SP_SVM SPCA_SVM ERW_SVM SSLDA_SVM SKPCA_SVM SKERW_SVM 1 100.00 99.95 100.00 100.00 100.00 95.05 98.43 100.00 2 99.22 99.68 100.00 100.00 100.00 99.52 97.04 100.00 3 86.12 93.73 99.55 95.60 99.55 98.40 81.38 99.76 4 97.17 99.13 97.56 99.05 97.34 97.20 97.96 92.45 5 97.63 93.33 91.71 93.91 99.96 96.33 97.97 100.00 6 100.00 100.00 100.00 100.00 100.00 99.74 99.95 100.00 7 96.92 100.00 100.00 99.92 99.94 97.36 96.41 99.89 8 71.42 78.69 96.49 95.31 94.68 81.14 78.29 96.98 9 99.36 98.94 100.00 98.66 99.98 92.98 94.89 99.37 10 80.17 93.96 96.95 98.66 99.20 71.22 80.13 95.89 11 97.20 96.53 97.23 100.00 97.36 93.48 89.63 97.09 12 90.73 98.31 97.47 98.27 100.00 95.04 95.49 96.15 13 95.01 97.49 91.16 96.03 100.00 86.76 89.37 98.46 14 92.20 98.43 100.00 94.93 99.34 82.08 97.35 100.00 15 57.19 77.47 98.00 95.90 96.68 84.11 84.23 99.73 16 97.79 99.12 100.00 100.00 100.00 99.68 99.48 100.00 OA（%） 85.40 91.24 97.98 97.52 98.22 89.43 89.28 98.54 AA（%） 91.13 95.30 97.88 97.90 99.00 91.88 92.38 98.49 Kappa（%） 83.77 90.23 97.75 97.23 98.02 88.20 88.02 98.38 （d） SPCA_SVM， （e） ERW_SVM， （f） SSLDA_SVM， （g） SKPCA_SVM， （h） SKERW_SVM. （c） SP_SVM， （d） SPCA_SVM， （e） ERW_SVM， （f） SSLDA_SVM， （g） SKPCA_SVM， （h） SKERW_SVM. （d） SPCA_SVM， （e） ERW_SVM， （f） SSLDA_SVM， （g） SKPCA_SVM， （h） SKERW_SVM. 从表 4 4 ~ 6 6 可以看到，所提出的SKERW_SVM方法在三组小规模数据集的少训练样本分类实验中都获得了最高的OA和Kappa值，以及多个单类别精度的最高值。这是由组成所提方法的SSLDA、KPCA和ERW共同作用产生的结果，因方法依次利用了高判别性的稀疏混合特征信息、不同局部邻域之间的内在联系以及相邻像素之间的空间相关性，从而在训练样本有限的条件下，SKERW_SVM仍能表现出优良的分类性能。如在Indian Pines和Salinas数据集的分类实验中，SKERW_SVM的OA值和Kappa系数都以超过98%的分类精度，成为对比方法中的最优模型；在Pavia University的实验中，SKERW_SVM取得了高达98.13%的平均精度。其次，在Salinas的实验中，与SVM相比，由于PCA_SVM、SP_SVM、SPCA_SVM、SSLDA_SVM、SKPCA_SVM和SKERW_SVM均考虑了对原始谱域降低维度，提取判别性的特征信息后再输入分类器进行标签识别，从而减少了数据冗余和避免了休斯现象，因此他们的分类精度较SVM有很大幅度的提升。如 表6 表6 所示，SKERW_SVM在Salinas数据集上的OA、AA和Kappa值比SVM分别提高了13.14%、7.63%和14.61%。然而，在Indian Pines和Pavia University数据集上，除PCA_SVM外，其余六种方法均比SVM的识别率高，这可能是由于本文确定的降维参数值不是PCA_SVM方法的最优参数，从而导致提取的谱特征信息不够充分。然后，从表4~6可以明显发现，联合空谱信息的SKERW_SVM等HSI分类方法明显高于仅考虑光谱信息的PCA_SVM，正如 闫敬文等（2019） 闫敬文等（2019） 和 Duan等（2020） Duan等（2020） 所指出：仅利用光谱信息的方法难以实现HSI的精确分类。最后，观察在Indian Pines和Pavia University数据集上的实验结果，SKERW_SVM的OA值较SP_SVM分别提升了1.11%和1.37%的正确率，较SPCA_SVM分别提高了6.17%和7.75%的正确率。且在Salinas的分类实验中，SKERW_SVM较SP_SVM和SPCA_SVM分别提升了0.56%和1.02%的正确率。这是由于，虽然SP_SVM关注到光谱信息和空间信息的重要性，却未考虑两者之间的相关性，而SKERW_SVM不仅利用了HSI的空间-光谱特征，还注意到HSI图像存在均匀同质区域，并基于这些局部邻域集合结合稀疏算法提取混合特征，有效提高了属于不同目标的像素可分性；另一方面，由于图像分割生成的聚类区域并不能与所有地物的边缘贴合一致，导致在部分超像素分割边缘的目标像素中识别率普遍偏低，而SKERW_SVM利用HSI相邻像素之间的空间相关性弥补了这一问题，从而使得SKERW_SVM在分类精度上大于SPCA_SVM。 观察图 4 4 ~ 6 6 ，并对比 图2 图2 中的标准地物分类图像可以发现，SVM和PCA_SVM得到的分类结果图像产生类似于“椒盐噪声”的现象，SKERW_SVM、SP_SVM和ERW_SVM得到的分类标签图最接近标准地物分类图像，这与表 4 4 ~ 6 6 得到的结果一致。受制于光谱信息的局限性，SVM和PCA_SVM对HSI的分类能力不高。由于SPCA_SVM、SSLDA_SVM和SKPCA_SVM考虑了高光谱图像的空间结构信息、光谱信息和空-谱变化信息，因此分类精度较仅基于谱域的SVM有一定程度的提高，这种现象在Indian Pines数据集上尤为明显。但因为SPCA_SVM、SSLDA_SVM和SKPCA_SVM方法在部分分割区域边缘贴合度一般，导致基于这些区域的光谱降维容易在异质边缘产生“噪声”。而SKERW_SVM利用了后验概率模型-扩展的随机游走算法弥补了超像素降维带来的“后遗症”，从而提高了模型整体的识别能力。 在消融实验中，对比SKPCA_SVM和SSLDA_SVM可以发现，KPCA整体上提高了模型的分类性能，这是由于KPCA把不同区域的稀疏混合特征融合在一起，进一步增强了不同类别像元之间的区分度和降低了数据冗余。观察SKERW_SVM和SKPCA_SVM的分类精度（表 4 4 ~ 6 6 ）和分类结果图像（图 4 4 ~ 6 6 ）可以发现，ERW解决了基于局部邻域集合提取的稀疏混合特征输入SVM后在异质边缘区域产生的“噪声”问题，显著提高了模型的性能。如在Indian Pines数据集上，SKERW_SVM的OA值比SKPCA_SVM提升了6.81的百分点。此外，本文提出的SKERW_SVM在三个数据集上的分类性能均较优于ERW_SVM方法，这间接体现了所提出的SSLDA方法可有效提高模型的分类精度。这是由于SSLDA在利用高光谱图像空间纹理信息构建的局部邻域集合的基础上，挖掘了HSI的高判别性稀疏混合特征，那充分表达了HSI的空谱信息和相关变化信息，从而减少了光谱维度，以及在分类的过程中加大了样本间的可分性，从而提高了SKERW_SVM的分类性能。 为了进一步验证本文提出的SKERW_SVM在大规模数据集上是否仍然具有优良的分类性能，本文选取HoustonU数据集作为实验对象，分析SKERW_SVM与五种对比方法的分类效果。为确定SKERW_SVM在HoustonU上的最优参数，把该数据集的有标签样本随机划分为训练集、验证集和测试集，具体的划分样本数量分布如 表7 表7 所示。首先，利用训练集和验证集获得SKERW_SVM的参数 <math id="M179"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706235&type= 1.10066664 2.62466669 、 <math id="M180"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706237&type= 1.86266661 2.62466669 和 <math id="M181"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=24706239&type= 2.28600001 2.62466669 的最优值分别为25、15和25。然后，依据划分的训练集和测试集开展对比实验， 表8 表8 列出了各分类方法的分类精度，且 图7 图7 展示了他们的分类标签图。从 表8 表8 可以看到，本文的SKERW_SVM在HoustonU数据集上以最高的OA、AA和Kappa值成为对比方法中的最优方法，尤其是相较于SPCA_SVM和PCA_SVM，SKERW_SVM的OA值分别提高了3.56%和9.32%的正确率，这表明本文提出的HSI分类方法在大规模数据集上具有一定的有效性。观察 图7 图7 可以发现，利用SKERW_SVM、SP_SVM和ERW_SVM获得的分类标签图比应用SVM和PCA_SVM得到的结果图像更不容易产生类似于“椒盐噪声”的现象，且更接近于HoustonU的标准地物分类图像，这也进一步验证SKERW_SVM具有优良的分类性能。 类别号 类名 训练集 验证集 测试集 1 Healthy Grass 500 500 8799 2 Stressed Grass 500 500 31502 3 Artificial Turf 68 68 548 4 Evergreen Trees 500 500 12595 5 Deciduous Trees 500 500 4021 6 Bare Earth 451 451 3614 7 Water 26 26 214 8 Residential 500 500 38772 9 Commercial 2237 500 221015 10 Roads 500 500 44866 11 Sidewalks 500 500 33029 12 Crosswalks 151 151 1216 13 Major Thoroughfares 500 500 45348 14 Highways 500 500 8865 15 Railways 500 500 5937 16 Paved Parking Lots 500 500 10500 17 Gravel Parking Lots 14 14 118 18 Cars 500 500 5547 19 Trains 500 500 4369 20 Stadium Seats 500 500 5824 2 总计 9947 8210 486699 类别号 SVM PCA_SVM SP_SVM SPCA_SVM ERW_SVM SKERW_SVM 1 88.96 75.34 57.89 70.96 56.98 82.95 2 95.59 91.20 91.07 92.41 89.03 81.87 3 100.00 100.00 100.00 100.00 100.00 100.00 4 86.69 84.76 79.55 85.12 78.81 91.62 5 66.63 60.88 75.66 68.10 70.72 93.68 6 94.08 90.19 100.00 95.37 97.22 99.34 7 99.62 100.00 100.00 100.00 100.00 100.00 8 88.39 74.04 97.91 91.55 97.22 96.98 9 94.01 93.98 96.91 96.03 97.43 85.10 10 68.17 64.63 80.88 76.20 85.83 76.94 11 68.06 62.39 65.39 68.97 74.66 81.85 12 26.72 22.93 27.42 20.07 25.72 80.16 13 75.32 69.80 91.21 78.32 90.23 86.10 14 64.37 54.35 94.84 64.01 88.60 97.04 15 86.26 83.90 99.11 97.45 98.22 97.85 16 84.76 75.04 94.96 89.07 91.82 87.30 17 94.63 97.87 100.00 98.50 92.99 100.00 18 71.75 59.24 73.51 79.99 94.39 99.30 19 72.46 74.16 98.69 97.75 97.99 100.00 20 95.16 90.32 99.80 95.14 99.69 100.00 OA（%） 85.64 81.69 90.03 87.45 90.97 91.01 AA（%） 81.08 76.25 86.24 83.25 86.38 91.90 Kappa（%） 81.41 76.40 87.05 83.77 88.29 90.45 （d） SPCA_SVM， （e） ERW_SVM， （f） SKERW_SVM. 与此同时，为了更好的说明本文提出方法的分类性能，实验记录了各对比方法在四组数据集上的仿真运行时间，如 表9 表9 所示。通过比较发现，PCA_SVM在Pavia University数据集上的运行时间最少，而SPCA_SVM在另外三组数据集上耗时最短，但PCA_SVM和SPCA_SVM的分类精度较SP_SVM、ERW_SVM和SKERW_SVM方法却相对较低。其次，尽管ERW_SVM的分类精度仅次于SKERW_SVM，然而它的实验运行时间在所有对比方法中最长，尤其是在Indian Pines数据集上，ERW_SVM的时间成本约为SKERW_SVM的四倍。因此，综合考虑分类精度和时间成本，本文提出的方法在高光谱图像分类任务上是更可取的。 数据集 SVM PCA_SVM SP_SVM SPCA_SVM ERW_SVM SKERW_SVM Indian Pines 35.47 8.03 6.81 5.85 36.91 9.52 Pavia University 13.51 6.32 11.39 7.52 16.93 12.28 Salinas 35.26 6.10 6.85 4.52 37.27 23.01 HoustonU 1922.56 1547.94 1546.84 1167.47 1953.24 1706.53 4　结论 1 本文提出了一种联合超像素降维和后验概率优化的高光谱图像分类方法，旨在解决小样本标签条件下空间-光谱信息利用不充分导致的图像分类精度较低的问题。提出的方法采用区域聚类的图像分割方法，为HSI的像元构建局部邻域集合，这准确地利用了高光谱图像的空间信息。其次，基于同质的局部邻域集合应用降维方法提取高判别性的稀疏混合特征，深度挖掘了HSI的空间纹理信息、不同维度光谱间的联系以及空间-光谱变化信息。同时，应用的后验概率模型有效降低了因超像素降维导致的部分分割边缘像素错分率较高的现象，这在很大程度上提高了分类精度。在四组高光谱数据集上的实验结果充分证明了所提方法的有效性，结果表明：本文方法能够自适应地提取高光谱图像的高判别性特征信息，并在少量的样本标签下，分类精度均优于目前几种先进的分类方法，且时间损耗适中。 参考文献（ References ） Achanta R , Shaji A , Smith K , Lucchi A , Fua P and Süsstrunk S . 2012 . SLIC superpixels compared to state-of-the-art superpixel methods . IEEE transactions on pattern analysis and machine intelligence , 34 ( 11 ): 2274 - 2282 [ DOI: 10.1109/TPAMI.2012.120 http://dx.doi.org/10.1109/TPAMI.2012.120 ] SLIC superpixels compared to state-of-the-art superpixel methods Blanzieri E and Melgani F . 2008 . Nearest neighbor classification of remote sensing images with the maximal margin principle . IEEE Transactions on geoscience and remote sensing , 46 ( 6 ): 1804 - 1811 [ DOI: 10.1109/TGRS.2008.916090 http://dx.doi.org/10.1109/TGRS.2008.916090 ] Nearest neighbor classification of remote sensing images with the maximal margin principle Carrino T A , Crósta A P , Toledo C L B and Silva A M . 2018 . Hyperspectral remote sensing applied to mineral exploration in southern Peru: A multiple data integration approach in the Chapi Chiara gold prospect . International journal of applied earth observation and geoinformation , 64 : 287 - 300 [ DOI: 10.1016/j.jag.2017.05.004 http://dx.doi.org/10.1016/j.jag.2017.05.004 ] Hyperspectral remote sensing applied to mineral exploration in southern Peru: A multiple data integration approach in the Chapi Chiara gold prospect Camps-Valls G and Bruzzone L . 2005 . Kernel-based methods for hyperspectral image classification . IEEE Transactions on Geoscience and Remote Sensing , 43 ( 6 ): 1351 - 1362 [ DOI: 10.1109/TGRS.2005.846154 http://dx.doi.org/10.1109/TGRS.2005.846154 ] Kernel-based methods for hyperspectral image classification Cheng W J and Chen W Q . 2020 . Hyperspectral image classification based on MCFFN-Attention . Computer Engineering and Applications , 56 ( 24 ): 201 - 206 Hyperspectral image classification based on MCFFN-Attention 程文娟 , 陈文强 . 2020 . 基于MCFFN-Attention的高光谱图像分类 . 计算机工程与应用 , 56 ( 24 ): 201 - 206 [ DOI: 10.3778/j.issn.1002-8331.1911-0102 http://dx.doi.org/10.3778/j.issn.1002-8331.1911-0102 ] 基于MCFFN-Attention的高光谱图像分类 Dong X F , Gan F P , Li N , Yan B K , Zhang L , Zhao J Q , Yu J C , Liu R Y and Ma Y N . 2020 . Fine mineral identification of GF-5 hyperspectral image . Journal of Remote Sensing , 24 ( 4 ): 454 - 464 Fine mineral identification of GF-5 hyperspectral image 董新丰 , 甘甫平 , 李娜 , 闫柏琨 , 张磊 , 赵佳琪 , 于峻川 , 刘镕源 , 马燕妮 . 2020 . 高分五号高光谱影像矿物精细识别 . 遥感学报 , 24 ( 4 ): 454 - 464 [ DOI: 10.11834/jrs.20209194 http://dx.doi.org/10.11834/jrs.20209194 ] 高分五号高光谱影像矿物精细识别 Dong A G , Zhang Q , Liu H C and Liang M M . 2019 . Hyperspectral image classification based on TSNE and multiscale sparse auto-encoder . Computer Engineering and Applications , 55 ( 21 ): 177 - 182+219 Hyperspectral image classification based on TSNE and multiscale sparse auto-encoder 董安国 , 张倩 , 刘洪超 , 梁苗苗 . 2019 . 基于TSNE和多尺度稀疏自编码的高光谱图像分类 . 计算机工程与应用 , 55 ( 21 ): 177 - 182+219 [ DOI: 10.3778/j.issn.1002-8331.1903-0155 http://dx.doi.org/10.3778/j.issn.1002-8331.1903-0155 ] 基于TSNE和多尺度稀疏自编码的高光谱图像分类 Duan P , Ghamisi P , Kang X , Rasti B , Li S and Gloaguen R . 2020 . Fusion of dual spatial information for hyperspectral image classification . IEEE Transactions on Geoscience and Remote Sensing [ DOI: 10.1109/TGRS.2020.3031928 http://dx.doi.org/10.1109/TGRS.2020.3031928 ] Fusion of dual spatial information for hyperspectral image classification Fu Q , Guo C and Luo W L . 2020 . A Hyperspectral image classification method based on spectral-spatial features . Laser & Optoelectronics Progress , 57 ( 20 ): 380 - 386 A Hyperspectral image classification method based on spectral-spatial features 付青 , 郭晨 , 罗文浪 . 2020 . 一种利用空谱联合特征的高光谱图像分类方法 . 激光与光电子学进展 , 57 ( 20 ): 380 - 386 [ DOI: 10.3788/LOP57.202803 http://dx.doi.org/10.3788/LOP57.202803 ] 一种利用空谱联合特征的高光谱图像分类方法 Felzenszwalb P F and Huttenlocher D P . 2004 . Efficient graph-based image segmentation . International journal of computer vision , 59 ( 2 ): 167 - 181 [ DOI: 10.1023/B:VISI.0000022288.19776.77 http://dx.doi.org/10.1023/B:VISI.0000022288.19776.77 ] Efficient graph-based image segmentation Gao K L , Yu X C , Song Z H , Zhang J , Liu B and Sun Y F . 2021 . Deep capsule network combined with spatial-spectral information for hyperspectral image classification . Journal of Remote Sensing , 25 ( 06 ): 1257 - 1269 Deep capsule network combined with spatial-spectral information for hyperspectral image classification 高奎亮 , 余旭初 , 宋治杭 , 张晋 , 刘冰 , 孙一帆 . 2021 . 联合空谱信息的高光谱影像深度胶囊网络分类 . 遥感学报 , 25 ( 06 ): 1257 - 1269 [ DOI: 10.11834/jrs.20210309 http://dx.doi.org/10.11834/jrs.20210309 ] 联合空谱信息的高光谱影像深度胶囊网络分类 Grady L . 2006 . Random walks for image segmentation . IEEE transactions on pattern analysis and machine intelligence , 28 ( 11 ): 1768 - 1783 [ DOI: 10.1109/TPAMI.2006.233 http://dx.doi.org/10.1109/TPAMI.2006.233 ] Random walks for image segmentation Hua Y , Zhang T , Xi H W , Wang Y F and Huang X L . 2012 . Research on method of hyperspectral remote sensing image classification based on decision tree . Computer Technology and Development , 22 ( 06 ): 198 - 202 Research on method of hyperspectral remote sensing image classification based on decision tree 华晔 , 张涛 , 奚后玮 , 王玉斐 , 黄秀丽 . 2012 . 基于决策树的高光谱遥感影像分类方法研究 . 计算机技术与发展 , 22 ( 06 ): 198 - 202 [ DOI: 10.3969/j.issn.1673-629X.2012.06.051 http://dx.doi.org/10.3969/j.issn.1673-629X.2012.06.051 ] 基于决策树的高光谱遥感影像分类方法研究 Jiang J , Ma J , Chen C , Wang Z , Cai Z and Wang L . 2018 . SuperPCA: A superpixelwise PCA approach for unsupervised feature extraction of hyperspectral imagery . IEEE Transactions on Geoscience and Remote Sensing , 56 ( 8 ): 4581 - 4593 [ DOI: 10.1109/TGRS.2018.2828029 http://dx.doi.org/10.1109/TGRS.2018.2828029 ] SuperPCA: A superpixelwise PCA approach for unsupervised feature extraction of hyperspectral imagery Kang X D , Li S T , Fang L Y , Li M X and Benediktsson J A . 2014 . Extended random walker-based classification of hyperspectral images . IEEE Transactions on Geoscience and Remote Sensing , 53 ( 1 ): 144 - 153 [ DOI: 10.1109/TGRS.2014.2319373 http://dx.doi.org/10.1109/TGRS.2014.2319373 ] Extended random walker-based classification of hyperspectral images Liang L , Di L , Zhang L , Deng M , Qin Z , Zhao S and Lin H . 2015 . Estimation of crop LAI using hyperspectral vegetation indices and a hybrid inversion method . Remote Sensing of Environment , 165 : 123 - 134 [ DOI: 10.1016/j.rse.2015.04.032 http://dx.doi.org/10.1016/j.rse.2015.04.032 ] Estimation of crop LAI using hyperspectral vegetation indices and a hybrid inversion method Li F Y , Huo H T , Li J and Bai J . 2019 . Hyperspectral image classification via multiple-feature-based improved sparse representation . Acta Optica Sinica , 39 ( 05 ): 351 - 359 Hyperspectral image classification via multiple-feature-based improved sparse representation 李非燕 , 霍宏涛 , 李静 , 白杰 . 2019 . 基于多特征和改进稀疏表示的高光谱图像分类 . 光学学报 , 39 ( 05 ): 351 - 359 [ DOI: 10.3788/AOS201939.0528004 http://dx.doi.org/10.3788/AOS201939.0528004 ] 基于多特征和改进稀疏表示的高光谱图像分类 Li D , Kong F Q and Zhu D Y . 2021 . Hyperspectral image classification based on local Gaussian mixture feature extraction . Acta Optica Sinica , 41 ( 06 ): 78 - 89 Hyperspectral image classification based on local Gaussian mixture feature extraction 李丹 , 孔繁锵 , 朱德燕 . 2021 . 基于局部高斯混合特征提取的高光谱图像分类 . 光学学报 , 41 ( 06 ): 78 - 89 [ DOI: 10.3788/AOS202141.0610001 http://dx.doi.org/10.3788/AOS202141.0610001 ] 基于局部高斯混合特征提取的高光谱图像分类 Li W , Chen C , Su H J and Du Q . 2015 . Local binary patterns and extreme learning machine for hyperspectral imagery classification . IEEE Transactions on Geoscience and Remote Sensing , 53 ( 7 ): 3681 - 3693 [ DOI: 10.1109/TGRS.2014.2381602 http://dx.doi.org/10.1109/TGRS.2014.2381602 ] Local binary patterns and extreme learning machine for hyperspectral imagery classification Li C N , Shao Y H , Yin W and Liu M Z . 2019 . Robust and sparse linear discriminant analysis via an alternating direction method of multipliers . IEEE transactions on neural networks and learning systems , 31 ( 3 ): 915 - 926 [ DOI: 10.1109/TNNLS.2019.2910991 http://dx.doi.org/10.1109/TNNLS.2019.2910991 ] Robust and sparse linear discriminant analysis via an alternating direction method of multipliers Mou L , Lu X , Li X and Zhu X X . 2020 . Nonlocal graph convolutional networks for hyperspectral image classification . IEEE Transactions on Geoscience and Remote Sensing , 58 ( 12 ): 8246 - 8257 [ DOI: 10.1109/TGRS.2020.2973363 http://dx.doi.org/10.1109/TGRS.2020.2973363 ] Nonlocal graph convolutional networks for hyperspectral image classification Melgani F and Bruzzone L . 2004 . Classification of hyperspectral remote sensing images with support vector machines . IEEE Transactions on geoscience and remote sensing , 42 ( 8 ): 1778 - 1790 [ DOI: 10.1109/TGRS.2004.831865 http://dx.doi.org/10.1109/TGRS.2004.831865 ] Classification of hyperspectral remote sensing images with support vector machines Prasad S and Bruce L M . 2008 . Limitations of principal components analysis for hyperspectral target recognition . IEEE Geoscience and Remote Sensing Letters , 5 ( 4 ): 625 - 629 [ DOI: 10.1109/LGRS.2008.2001282 http://dx.doi.org/10.1109/LGRS.2008.2001282 ] Limitations of principal components analysis for hyperspectral target recognition Song X Y , Zhou L L , Li Z G , Chen J , Zeng L and Yan B . 2015 . Review on superpixel methods in image segmentation . Journal of Image and Graphics , 20 ( 05 ): 599 - 608 Review on superpixel methods in image segmentation 宋熙煜 , 周利莉 , 李中国 , 陈健 , 曾磊 , 闫镔 . 2015 . 图像分割中的超像素方法研究综述 . 中国图象图形学报 , 20 ( 05 ): 599 - 608 [ DOI: 10.11834/jig.20150502 http://dx.doi.org/10.11834/jig.20150502 ] 图像分割中的超像素方法研究综述 Shi J and Malik J . 2000 . Normalized cuts and image segmentation . IEEE Transactions on pattern analysis and machine intelligence , 22 ( 8 ): 888 - 905 [ DOI: 10.1109/34.868688 http://dx.doi.org/10.1109/34.868688 ] Normalized cuts and image segmentation Schölkopf B , Smola A and Müller K R . 1997 , October . Kernel principal component analysis . In International conference on artificial neural networks (pp . 583 -588). Springer, , HeidelbergBerlin. Kernel principal component analysis Vincent L and Soille P . 1991 . Watersheds in digital spaces: an efficient algorithm based on immersion simulations . IEEE Transactions on Pattern Analysis & Machine Intelligence , 13 ( 06 ): 583 - 598 [ DOI: 10.1109/34.87344 http://dx.doi.org/10.1109/34.87344 ] Watersheds in digital spaces: an efficient algorithm based on immersion simulations Wei X P , Yu X C , Zhang P Q , Zhi L and Yang F . 2020 . CNN with local binary patterns for hyperspectral images classification . Journal of Remote Sensing , 24 ( 08 ): 1000 - 1009 CNN with local binary patterns for hyperspectral images classification 魏祥坡 , 余旭初 , 张鹏强 , 职露 , 杨帆 . 2020 . 联合局部二值模式的CNN高光谱图像分类 . 遥感学报 , 24 ( 08 ): 1000 - 1009 [ DOI: 10.11834/jrs.20208333 http://dx.doi.org/10.11834/jrs.20208333 ] 联合局部二值模式的CNN高光谱图像分类 Xu D , Yang M , Miao Y H , Yan J and Sun M . 2020 . Research and application of accurate extraction method of island shoreline based on hyperspectral remote sensing of unmanned aerial vehicle . Marine Sciences , 44 ( 12 ): 54 - 60 Research and application of accurate extraction method of island shoreline based on hyperspectral remote sensing of unmanned aerial vehicle 徐栋 , 杨敏 , 苗宇宏 , 严晋 , 孙苗 . 2020 . 基于无人机高光谱遥感的海岛岸线精准提取方法研究与应用 . 海洋科学 , 44 ( 12 ): 54 - 60 [ DOI: 10.11759/hykx20200529002 http://dx.doi.org/10.11759/hykx20200529002 ] 基于无人机高光谱遥感的海岛岸线精准提取方法研究与应用 Xu X , Li J , Wu C and Plaza A . 2018 . Regional clustering-based spatial preprocessing for hyperspectral unmixing . Remote Sensing of Environment , 204 : 333 - 346 [ DOI: 10.1016/j.rse.2017.10.020 http://dx.doi.org/10.1016/j.rse.2017.10.020 ] Regional clustering-based spatial preprocessing for hyperspectral unmixing Yan J W , Guo H D and Liu L . 2019 . Overview of hyperspectral image classification . Optics and Precision Engineering , 27 ( 03 ): 680 - 693 Overview of hyperspectral image classification 闫敬文 , 陈宏达 , 刘蕾 . 2019 . 高光谱图像分类的研究进展 . 光学精密工程 , 27 ( 03 ): 680 - 693 [ DOI: 10.3788/OPE.20192703.0680 http://dx.doi.org/10.3788/OPE.20192703.0680 ] 高光谱图像分类的研究进展 Yu H , Xu Z , Wang Y , Jiao T and Guo Q . 2021 . The use of KPCA over subspaces for cross-scale superpixel based hyperspectral image classification . Remote Sensing Letters , 12 ( 5 ): 470 - 477 [ DOI: 10.1080/2150704X.2021.1897180 http://dx.doi.org/10.1080/2150704X.2021.1897180 ] The use of KPCA over subspaces for cross-scale superpixel based hyperspectral image classification Zhang X D , Wang T J , Zhu S X and Yang Y . 2021 . Hyperspectral image classification based on dilated convolutional attention neural network . Acta Optica Sinica , 41 ( 03 ): 49 - 59 Hyperspectral image classification based on dilated convolutional attention neural network 张祥东 , 王腾军 , 朱劭俊 , 杨耘 . 2021 . 基于扩张卷积注意力神经网络的高光谱图像分类 . 光学学报 , 41 ( 03 ): 49 - 59 [ DOI: 10.3788/AOS202141.0310001 http://dx.doi.org/10.3788/AOS202141.0310001 ] 基于扩张卷积注意力神经网络的高光谱图像分类 Zheng C , Wang N and Cui J . 2019 . Hyperspectral image classification with small training sample size using superpixel-guided training sample enlargement . IEEE Transactions on Geoscience and Remote Sensing , 57 ( 10 ): 7307 - 7316 [ DOI: 10.1109/TGRS.2019.2912330 http://dx.doi.org/10.1109/TGRS.2019.2912330 ]Hyperspectral image classification method combining superpixel dimension reduction with post-processing optimization Hyperspectral image classification with small training sample size using superpixel-guided training sample enlargement HUANG Yuan 1 , 2 , HE Xinguang 1, 2* ,WAN Yiliang 1 , 2 2 , HE Xinguang1, 2* ,WAN Yiliang 1 College of Geographic Sciences , Hunan Normal University , Changsha 410081 , China ; China 2 Key Laboratory of Geospatial Big Data Mining and Application , Hunan Province , Changsha 410081 , China China 致谢:由衷的感谢巴斯克大学计算情报小组分享了开源可用的Indian Pines、Pavia University和Salinas数据集，以及IEEE GRSS图像分析和数据融合技术委员会与休斯顿大学的高光谱图像分析实验室分发的HoustonU数据集！ 10.11834/jrs.20221497 TP701 A 黄媛，贺新光，万义良.XXXX.联合超像素降维和后处理优化的高光谱图像分类方法.遥感学报，XX（XX）： 1-19 黄媛，贺新光，万义良.XXXX.联合超像素降维和后处理优化的高光谱图像分类方法.遥感学报，XX（XX）： 1-19 湖南省自然资源厅科技项目经费资助（编号： 2021-45） Supported by Supported by the Science and Technology Project of Department of Natural Resources of Hunan Province (No.2021-45) 2022-03-01 遥感学报