CN102122352A

CN102122352A - Characteristic value distribution statistical property-based polarized SAR image classification method

Info

Publication number: CN102122352A
Application number: CN 201110049008
Authority: CN
Inventors: 缑水平; 焦李成; 乔鑫; 王爽; 吴建设; 朱虎明; 李阳阳; 费全花
Original assignee: Xidian University
Current assignee: Xidian University
Priority date: 2011-03-01
Filing date: 2011-03-01
Publication date: 2011-07-13
Anticipated expiration: 2031-03-01
Also published as: CN102122352B

Abstract

The invention discloses a polarization SAR image classification method based on Gaussian statistic characteristics of eigenvalues, which mainly solves the problems of lack of cognition of characteristic distribution characteristics in the prior art and the need for artificial determination of category judgment boundaries. The implementation process is as follows: 1) Decompose the eigenvalues of all pixels in the polarimetric SAR image to be classified; 2) Select different homogeneous regions as the most basic category representative regions, and extract the eigenvalues representing the homogeneous regions; 3) Use The EM algorithm estimates the Gaussian mixture model parameters of various homogeneous region eigenvalues λ ₁ , λ ₂ , λ ₃ respectively, and obtains the probability density distribution function of each eigenvalue; 4) Solve the three eigenvalues of various homogeneous regions 5) Perform Bayesian classification on the pixels in the homogeneous area, and output the classification result. The invention has the advantage of remarkable classification effect on polarimetric SAR images, and can be used for polarimetric SAR image target detection and target recognition.

Description

A Polarization SAR Image Classification Method Based on Statistical Characteristics of Eigenvalue Distribution

技术领域technical field

本发明属于图像处理技术领域，涉及极化SAR图像分类，可用于雷达目标检测和目标识别。The invention belongs to the technical field of image processing, relates to polarization SAR image classification, and can be used for radar target detection and target recognition.

背景技术Background technique

合成孔径雷达SAR是利用合成孔径原理提高方位分辨率，利用脉冲压缩技术提高距离分辨率，从而获得比真是孔径雷达更好的性能。极化SAR属于SAR的范畴，相比于传统的SAR，它利用多通道收发电磁波技术，通过对其不同通道的解译，可以获得对目标更全面的认识。对极化SAR图像的理解和解译属于图像处理范畴，还涉及信号处理，模式识别及机器学习等众多学科。极化SAR图像分类作为极化SAR图像处理的关键环节之一，在国防和民用领域正受到越来越广泛的关注。现有的极化SAR图像分类方法大致可以分为基于散射特性的方法和基于统计特性方法，如基于freeman分解的分类方法和基于散射熵、散射角的分类方法等。Synthetic aperture radar SAR uses the principle of synthetic aperture to improve the azimuth resolution, and uses pulse compression technology to improve the range resolution, so as to obtain better performance than real aperture radar. Polarized SAR belongs to the category of SAR. Compared with traditional SAR, it uses multi-channel transceiver electromagnetic wave technology, and can obtain a more comprehensive understanding of the target through the interpretation of its different channels. The understanding and interpretation of polarimetric SAR images belongs to the category of image processing, and also involves many disciplines such as signal processing, pattern recognition and machine learning. As one of the key links in polarimetric SAR image processing, polarimetric SAR image classification is receiving more and more attention in the fields of national defense and civilian use. The existing polarization SAR image classification methods can be roughly divided into methods based on scattering characteristics and methods based on statistical characteristics, such as classification methods based on freeman decomposition and classification methods based on scattering entropy and scattering angle.

其中，基于统计特性的极化SAR图像分类方法，是将极化SAR图像中具有某方面相似特征的像素点划分为一类，已经有很多成熟的分类算法被用到极化SAR图像分类中。作为一门新兴学科，目前主要的研究工作集中在散射特征的提取和适合极化SAR图像分类的分类器设计上。不同极化散射特征的提取方法得到不同的特征，不同的特征又适合于不同性能的分类器。现有的特征提取方法主要有freeman分解和cloud分解方法；典型的分类器有Wishart分类器和H/alpha分类器等。其中，利用freeman分解方法得到特征对极化SAR图像进行分类时，存在对特征分布特性认知上的不足；利用H/alpha分类器进行类别判定时，存在类别判决界限需要人为确定的不足，这些不足限制了其在极化SAR图像分类上的广泛应用。Among them, the polarimetric SAR image classification method based on statistical characteristics is to divide the pixels with similar characteristics in the polarimetric SAR image into one category, and many mature classification algorithms have been used in the polarimetric SAR image classification. As a new subject, the current main research work is focused on the extraction of scattering features and the design of classifiers suitable for polarimetric SAR image classification. The extraction methods of different polarized scattering features get different features, and different features are suitable for classifiers with different performances. The existing feature extraction methods mainly include freeman decomposition and cloud decomposition methods; typical classifiers include Wishart classifier and H/alpha classifier. Among them, when using the freeman decomposition method to obtain features to classify polarimetric SAR images, there is a lack of cognition of the feature distribution characteristics; when using the H/alpha classifier for category determination, there is a deficiency that the category decision boundary needs to be determined manually. Insufficiency limits its wide application in polarimetric SAR image classification.

发明内容Contents of the invention

本发明的目的在于克服上述已有技术的不足，提出了一种基于特征值分布特性的极化SAR图像分类方法，以明确特征的分布特性，避免类别判决界限需要人为确定的问题。The purpose of the present invention is to overcome the deficiencies of the above-mentioned prior art, and propose a polarization SAR image classification method based on the distribution characteristics of eigenvalues, so as to clarify the distribution characteristics of the characteristics and avoid the problem that the category judgment boundary needs to be determined manually.

实现本发明目的个技术方案是通过研究分析特征值的分布特性，结合贝叶斯分类对极化SAR图像进行分类，其步骤包括如下：Achieving the purpose of the present invention is a technical solution to classify polarimetric SAR images in conjunction with Bayesian classification by studying and analyzing the distribution characteristics of eigenvalues, and the steps include the following:

(1)对待分类极化SAR图像的所有像素点进行特征值分解；(1) Decompose the eigenvalues of all pixels of the polarimetric SAR image to be classified;

(2)分别对每个像素点分解得到的特征值λ₁，λ₂，λ₃进行排序，使λ₁≥λ₂≥λ₃；(2) Sorting the eigenvalues λ ₁ , λ ₂ , and λ ₃ obtained by decomposing each pixel, so that λ ₁ ≥ λ ₂ ≥ _{λ 3} ;

(3)从待分类极化SAR图像中依据实际地物分布，人工构造待分类极化SAR图像中k类匀质区域集合：C＝{C₁，...，C_k}，并提取出所选取k类匀质区域的特征值；(3) From the polarimetric SAR image to be classified according to the actual distribution of ground objects, artificially construct a set of k-type homogeneous regions in the polarimetric SAR image to be classified: C={C ₁ ,...,C _k }, and extract all Select the eigenvalues of k-type homogeneous regions;

(4)采用EM算法分别对k类匀质区域的特征值λ₁，λ₂，λ₃的高斯混合模型参数进行估计，得到每类匀质区域的各个特征值的概率密度分布函数：其中λ为某类匀质区域的某个特征值、ω为高斯函数的混合权重、μ为均值参数、σ为方差参数、m为高斯函数的混合个数、μ_i为第i个高斯函数的均值，σ_i为第i个高斯函数的方差，i＝1，2，...，k；(4) Using the EM algorithm to estimate the Gaussian mixture model parameters of the eigenvalues λ ₁ , λ ₂ , λ ₃ of k types of homogeneous regions, and obtain the probability density distribution function of each eigenvalue of each type of homogeneous regions: Among them, λ is a certain eigenvalue of a certain type of homogeneous region, ω is the mixture weight of Gaussian function, μ is the mean parameter, σ is the variance parameter, m is the mixture number of Gaussian function, μ _i is the i-th Gaussian function Mean value, σ _i is the variance of the i-th Gaussian function, i=1, 2,..., k;

(5)利用步骤4求得的每类匀质区域的各个特征值的概率密度分布函数，计算每类匀质区域的特征值联合概率密度分布函数：(5) Using the probability density distribution function of each eigenvalue of each type of homogeneous area obtained in step 4, calculate the joint probability density distribution function of the eigenvalues of each type of homogeneous area:

$p p ((x x | | x x &Element; &Element; {C C}_{i i})) = = {Π Π}_{j j = = 11}^{33} p p (({λ λ}_{j j} | | μ μ,, σ σ)),,$

其中，x为匀质区域C_i中的像素点，λ_j为像素点x的第j个特征值，i＝1，2，...，k，j＝1，2，3；Wherein, x is the pixel point in the homogeneous region _Ci , and λ _j is the jth eigenvalue of the pixel point x, i=1,2,...,k,j=1,2,3;

(6)根据MAP原则，依据表达式：x∈C_i st.p(x|C_i)＝maxp(x|x∈C_k)，i＝1，2，...，k，对待分类极化SAR图像中所选匀质区域的全部像素点进行分类，并输出分类结果。(6) According to the MAP principle, according to the expression: x∈C _i st.p(x|C _i )=maxp(x|x∈C _k ), i=1, 2,..., k, the classification pole Classify all the pixels in the selected homogeneous area in the SAR image, and output the classification results.

本发明与现有的技术相比具有以下优点：Compared with the prior art, the present invention has the following advantages:

1)本发明由于对极化SAR图像中不同匀质区域的特征值分布特性进行了充分的研究，以不同的概率密度函数来表达不同匀质区域之间特征值分布特性的差异性，明确了各类匀质区域特征值分布特性之间的差异性；1) Since the present invention has fully studied the eigenvalue distribution characteristics of different homogeneous regions in polarimetric SAR images, different probability density functions are used to express the differences in eigenvalue distribution characteristics between different homogeneous regions, and it is clear that The difference between the eigenvalue distribution characteristics of various homogeneous regions;

2)本发明相对已有的极化SAR图像分类技术，从统计角度入手，以贝叶斯分类方法实现分类，避免类别判决界限需要人为确定的问题。2) Compared with the existing polarimetric SAR image classification technology, the present invention starts from the statistical point of view and realizes the classification with the Bayesian classification method, so as to avoid the problem that the category judgment boundary needs to be manually determined.

仿真实验结果表明，本发明提出的基于特征值分布统计特性的极化SAR图像分类方法能够有效地应用于极化SAR图像分类，并进一步应用于雷达目标检测和目标识别。The simulation experiment results show that the polarimetric SAR image classification method based on the eigenvalue distribution statistical properties proposed by the present invention can be effectively applied to the polarimetric SAR image classification, and further applied to radar target detection and target recognition.

附图说明Description of drawings

图1是本发明的实现流程图；Fig. 1 is the realization flowchart of the present invention;

图2是本发明仿真采用的极化SAR图像的RGB合成图；Fig. 2 is the RGB synthetic figure of the polarization SAR image that simulation of the present invention adopts;

图3是本发明仿真在极化SAR图像中所提取的匀质区域示意图；Fig. 3 is the schematic diagram of the homogeneous region extracted in the polarimetric SAR image by the simulation of the present invention;

图4是本发明仿真某匀质区域的特征值实际分布和和估计分布示意图。Fig. 4 is a schematic diagram of the actual distribution and the estimated distribution of the eigenvalues of a simulated homogeneous region according to the present invention.

具体实施方式Detailed ways

参照图1，本发明的具体实施如下：With reference to Fig. 1, concrete implementation of the present invention is as follows:

步骤1.对待分类极化SAR图像的所有像素点进行特征值分解。Step 1. Perform eigenvalue decomposition on all pixels of the polarimetric SAR image to be classified.

极化SAR图像包含丰富的幅度信息和相位信息，其每个像素的信息由赫米特矩阵T来表示，矩阵大小为3×3。由于特征值最能代表矩阵所包含的信息，因此选用特征值作为极化SAR图像的特征，利用MATLAB的eigs函数对每个像素的矩阵T进行特征分解，分解表达式如下：Polarimetric SAR images contain rich amplitude information and phase information, and the information of each pixel is represented by a Hermitian matrix T with a size of 3×3. Since the eigenvalue can best represent the information contained in the matrix, the eigenvalue is selected as the feature of the polarimetric SAR image, and the matrix T of each pixel is decomposed using the eigs function of MATLAB. The decomposition expression is as follows:

$[[T T]] = = [[{U u}_{33}]] [\begin{matrix} {λ λ}_{11} & 00 & 00 \\ 00 & {λ λ}_{22} & 00 \\ 00 & 00 & {λ λ}_{33} \end{matrix}] {[[{U u}_{33}]]}^{* *};;$

其中，U₃为像素矩阵T特征值分解的特征向量，λ₁，λ₂，λ₃为像素矩阵T进行特征值分解所得的大小不同的特征值。Among them, U ₃ is the eigenvector of the eigenvalue decomposition of the pixel matrix T, and λ ₁ , λ ₂ , λ ₃ are the eigenvalues of different sizes obtained from the eigenvalue decomposition of the pixel matrix T.

步骤2.分别对每个像素点分解得到的大小不同的特征值λ₁，λ₂，λ₃按照从大到小的原则进行排序，使λ₁≥λ₂≥λ₃。Step 2. Sorting the eigenvalues λ ₁ , λ ₂ , and λ ₃ of different sizes obtained by decomposing each pixel according to the principle from large to small, so that λ ₁ ≥ λ ₂ ≥ λ ₃ .

步骤3.选取匀质区域，构造匀质区域的训练数据集和测试数据集。Step 3. Select a homogeneous area, and construct a training data set and a test data set for the homogeneous area.

3a)从待分类极化SAR图像中依据实际地物分布，人工构造待分类极化SAR图像中k类匀质区域集合：3a) From the polarimetric SAR image to be classified, according to the actual distribution of ground objects, artificially construct a set of k-type homogeneous regions in the polarimetric SAR image to be classified:

C＝{C₁，...，C_k}，k＝2，3，...，n，其中：n为待分类极化SAR图像的像素点个数；C={C ₁ ,...,C _k }, k=2, 3,..., n, where: n is the number of pixels of the polarimetric SAR image to be classified;

3b)提取出所选取k类匀质区域中像素点作为算法的数据集，其中每类匀质区域中10％～50％像素点作为每类匀质区域的训练数据集，其余作为该匀质区域的测试数据集。3b) Extract the pixels in the selected k types of homogeneous regions as the data set of the algorithm, in which 10% to 50% of the pixels in each type of homogeneous region are used as the training data set for each type of homogeneous region, and the rest are used as the homogeneous region test data set.

步骤4.采用EM算法求解训练数据集的各个特征值的概率密度分布函数。Step 4. Using the EM algorithm to solve the probability density distribution function of each eigenvalue of the training data set.

4a)将各类匀质区域训练数据集中像素点的特征值λ₁，λ₂，λ₃分别作为EM算法的输入数据；4a) Taking the eigenvalues λ ₁ , λ ₂ , and λ ₃ of the pixels in the training data sets of various homogeneous regions as the input data of the EM algorithm;

4b)设定EM算法所求解的高斯混合模型中高斯函数的混合个数m值为1、混合权重ω为1，EM算法迭代终止条件为算法的迭代误差小于1.0E-06，初始化均值μ和方差σ为随机值；4b) Set the mixed number m of Gaussian functions in the Gaussian mixture model solved by the EM algorithm to 1, the mixed weight ω to 1, the EM algorithm iteration termination condition is that the algorithm iteration error is less than 1.0E-06, and the initial mean value μ and The variance σ is a random value;

4c)通过EM算法中的期望最大化流程对均值μ和方差σ进行更新，每次更新后计算算法的迭代误差并判断是否满足迭代终止条件，满足迭代终止条件时迭代停止，输出得到输入特征值的均值μ和方差σ；4c) Update the mean value μ and variance σ through the expectation maximization process in the EM algorithm. After each update, calculate the iteration error of the algorithm and judge whether the iteration termination condition is met. When the iteration termination condition is satisfied, the iteration stops, and the output is the input feature value The mean μ and variance σ of ;

4d)将混合个数m、混合权重ω和输出的均值μ、方差σ代入式：4d) Substitute the mixing number m, mixing weight ω, and output mean μ and variance σ into the formula:

$p p ((λ λ | | ω ω,, μ μ,, σ σ)) = = {Σ Σ}_{i i = = 11}^{m m} {ω ω}_{i i} \times \times \frac{11}{\sqrt{22 π π} {σ σ}_{i i}} exp exp ((- - \frac{{((x x - - {μ μ}_{i i}))}^{22}}{22 {σ σ}_{i i}^{22}}))$

得到输入特征值的概率密度分布函数。Gets the probability density distribution function of the input eigenvalues.

步骤5.构造匀质区域的特征值的联合概率函数分布表达式。Step 5. Construct the joint probability function distribution expression of the eigenvalues of the homogeneous region.

用步骤4求得的每类匀质区域的各个特征值的概率密度分布函数，计算每类匀质区域的特征值联合概率密度分布函数：Using the probability density distribution function of each eigenvalue of each type of homogeneous area obtained in step 4, calculate the joint probability density distribution function of the eigenvalues of each type of homogeneous area:

其中，x为匀质区域C_i中的像素点，λ_j为像素点x的第j个特征值，i＝1，2，...，k，j＝1，2，3。Wherein, x is the pixel point in the homogeneous region C _i , λ _j is the jth eigenvalue of the pixel point x, i=1, 2, ..., k, j=1, 2, 3.

步骤6.对测试数据集中的像素点进行分类，计算像素点的分类准确率。Step 6. Classify the pixels in the test data set, and calculate the classification accuracy of the pixels.

4a)根据MAP原则，依据表达式：x∈C_i st.p(x|C_i)＝maxp(x|x∈C_k)，i＝1，2，...，k，对测试数据集中的像素点进行分类，并输出分类结果；4a) According to the MAP principle, according to the expression: x∈C _i st.p(x|C _i )=maxp(x|x∈C _k ), i=1, 2, ..., k, for the test data set The pixels are classified, and the classification results are output;

4b)将测试数据集中像素点的类别和步骤4a)中输出的类别进行比较，类别相同则为分类正确，否则为分类错误；根据分类正确数和分类错误数计算分类准确率并输出。4b) Compare the category of the pixel points in the test data set with the category output in step 4a), if the category is the same, it is classified as correct, otherwise it is a classification error; calculate the classification accuracy rate according to the number of correct classifications and the number of classification errors and output.

本发明的效果可以通过如下仿真实验进行验证。The effects of the present invention can be verified by the following simulation experiments.

1.实验条件1. Experimental conditions

1.1)实验中选取的极化SAR图像如图2所示，该图为荷兰Flevoland地区某农田区域的L波段极化SAR的RGB合成图像，其中R代表|HH-VV|，G代表|HV|+|VH|，B代表|HH+VV|，图像尺寸大小为1024×750。该极化SAR图像中区域的地表覆盖物为农田和水域。1.1) The polarimetric SAR image selected in the experiment is shown in Figure 2, which is an RGB synthetic image of the L-band polarimetric SAR in a farmland area in Flevoland, the Netherlands, where R stands for |HH-VV|, and G stands for |HV| +|VH|, B stands for |HH+VV|, and the image size is 1024×750. The land cover of the area in this polarimetric SAR image is farmland and water.

1.2)实验中选取的五类匀质区域如图3所示，其中图3a、图3b，图3c、图3d和图3e为五类不同种类的农作物区域。1.2) The five types of homogeneous areas selected in the experiment are shown in Figure 3, where Figure 3a, Figure 3b, Figure 3c, Figure 3d and Figure 3e are five types of different types of crop areas.

2.实验内容2. Experimental content

2.1)对上述匀质区域的各个特征值实际分布和估计分布进行实验，其结果如图4所示，其中：图4a、图4b和图4c依次分别为特征值λ₁，λ₂，λ₃的实际分布和估计分布示意图。图4中的实线为特征值的实际分布曲线，虚线为特征值的估计分布曲线。从图4a、图4b和图4c可以看出，特征值λ₁，λ₂，λ₃实际分布曲线和估计分布曲线之间的偏差很小，表明该区域的特征值分布特性通过高斯混合模型得到很好的表达。2.1) Experiments were carried out on the actual distribution and estimated distribution of each eigenvalue of the above-mentioned homogeneous region, and the results are shown in Figure 4, in which: Figure 4a, Figure 4b and Figure 4c are the eigenvalues λ ₁ , λ ₂ , λ ₃ respectively in sequence Schematic diagram of the actual and estimated distributions of . The solid line in Figure 4 is the actual distribution curve of the eigenvalues, and the dashed line is the estimated distribution curve of the eigenvalues. From Figure 4a, Figure 4b and Figure 4c, it can be seen that the deviation between the actual distribution curve and the estimated distribution curve of eigenvalues λ ₁ , λ ₂ , λ ₃ is very small, indicating that the eigenvalue distribution characteristics of this region are obtained through the Gaussian mixture model Very well expressed.

对上述五类区域各个特征值进行估计得到的高斯混合模型均值参数如表一所示，方差参数如表二所示：The mean parameters of the Gaussian mixture model obtained by estimating each eigenvalue of the above five types of regions are shown in Table 1, and the variance parameters are shown in Table 2:

表一均值参数Table 1 Mean parameter

表二方差参数Table 2 Variance parameter

从表一和表二中可以看出，不同匀质区域的特征值服从的高斯混合模型参数的均值和方差不同，表明了不同匀质区域之间特征值分布的差异性。It can be seen from Table 1 and Table 2 that the mean and variance of the Gaussian mixture model parameters obeyed by the eigenvalues of different homogeneous regions are different, indicating the difference in the distribution of eigenvalues between different homogeneous regions.

2.2)仿真分类准确率结果如表三所示：2.2) The simulation classification accuracy results are shown in Table 3:

表三仿真实验的分类准确率Table 3 Classification accuracy of simulation experiments

从表三可以看出本发明对于极化SAR图像的分类准确率基本为90％以上，表明本发明可以有效解决极化SAR图像分类问题。It can be seen from Table 3 that the classification accuracy rate of the present invention for polarimetric SAR images is basically over 90%, indicating that the present invention can effectively solve the classification problem of polarimetric SAR images.

Claims

1. The polarimetric SAR image classification method based on eigenvalue distribution characteristics, comprising the following steps:

(1) Decompose the eigenvalues of all pixels of the polarimetric SAR image to be classified;

(2) Sorting the eigenvalues λ ₁ , λ ₂ , and λ ₃ of different sizes obtained by decomposing each pixel point in order from large to small, so that λ ₁ ≥ λ ₂ ≥ _{λ 3} ;

(3) From the polarimetric SAR image to be classified according to the actual distribution of ground objects, artificially construct a set of k-type homogeneous regions in the polarimetric SAR image to be classified: C={C ₁ ,...,C _k }, and extract all Select the eigenvalues of k-type homogeneous regions;

(4) Using the EM algorithm to estimate the Gaussian mixture model parameters of the eigenvalues λ ₁ , λ ₂ , λ ₃ of k types of homogeneous regions, and obtain the probability density distribution function of each eigenvalue of each type of homogeneous regions:

Among them, λ is a certain eigenvalue of a certain type of homogeneous region, ω is the mixture weight of Gaussian function, μ is the mean parameter, σ is the variance parameter, m is the mixture number of Gaussian function, μ _i is the i-th Gaussian function Mean value, σ _i is the variance of the i-th Gaussian function, i=1, 2,..., k;

(5) Using the probability density distribution function of each eigenvalue of each type of homogeneous area obtained in step 4, calculate the joint probability density distribution function of the eigenvalues of each type of homogeneous area:

p p ((x x | | x x &Element; &Element; {C C}_{i i})) = = {Π Π}_{j j = = 11}^{33} p p (({λ λ}_{j j} | | μ μ,, σ σ)),,

Wherein, x is the pixel point in the homogeneous region _Ci , and λ _j is the jth eigenvalue of the pixel point x, i=1,2,...,k,j=1,2,3;

(6) According to the MAP principle, according to the expression: x∈C _i st.p(x|C _i )=maxp(x|x∈C _k ), i=1, 2,..., k, the classification pole Classify all the pixels in the selected homogeneous area in the SAR image, and output the classification results.

2. polarimetric SAR image classification method according to claim 1, wherein said in step (1) carries out eigenvalue decomposition to all pixels of polarimetric SAR image to be classified, carries out according to the following steps:

2a) The T matrix of the pixels of the polarimetric SAR image to be classified is used as input data;

2b) Using the eigs function in MATLAB software to perform eigenvalue decomposition on the input T matrix of each pixel to obtain eigenvalues.

3. polarimetric SAR image classification method according to claim 1, wherein the adopting EM algorithm described in step (4) is respectively to the eigenvalue λ ₁ of k class homogeneous region, λ ₂ , the Gaussian mixture model parameter of λ ₃ To estimate, proceed as follows:

3a) The eigenvalues λ ₁ , λ ₂ , and λ ₃ of various types of homogeneous regions are respectively used as input data of the EM algorithm;

3b) Set the mixed number m of Gaussian functions in the Gaussian mixture model solved by the EM algorithm to be 1, the mixed weight ω to be 1, the termination condition of the EM algorithm iteration is that the iteration error of the algorithm is less than 1.0E-06, and the initial mean value μ and The variance σ is a random value;

3c) Update the mean value μ and variance σ through the expectation maximization process in the EM algorithm. After each update, calculate the iteration error of the algorithm and judge whether the iteration termination condition is met. When the iteration termination condition is satisfied, the iteration stops, and the output is the input feature value The mean μ and variance σ of ;

3d) Substitute the mixing number m, mixing weight ω, and the output mean μ and variance σ into the formula:

p p ((λ λ | | ω ω,, μ μ,, σ σ)) = = {Σ Σ}_{i i = = 11}^{m m} {ω ω}_{i i} \times \times \frac{11}{\sqrt{22 π π} {σ σ}_{i i}} exp exp ((- - \frac{{((x x - - {μ μ}_{i i}))}^{22}}{22 {σ σ}_{i i}^{22}}))

Gets the probability density distribution function of the input eigenvalues.