PCA人脸识别

Love The Way You Lie 2022-08-03 14:54 164阅读 0赞

PCA方法由于其在降维和特征提取方面的有效性，在人脸识别领域得到了广泛的应用。  
其基本原理是:利用K-L变换抽取人脸的主要成分，构成特征脸空间，识别时将测试图像投影到此空间，得到一组投影系数，通过与各个人脸图像比较进行识别。  
进行人脸识别的过程，主要由训练阶段和识别阶段组成：

## 训练阶段 ##

第一步：写出训练样本矩阵，其中向量xi为由第i个图像的每一列向量堆叠成一列的MN维列向量,即把矩阵向量化。假设训练集有200个样本,，由灰度图组成，每个样本大小为M\*N。  
![这里写图片描述][20150810191535268]  
第二步：计算平均脸

Ψ=1200∑*i*=1*i*=200*x**i*

第三步：计算差值脸,计算每一张人脸与平均脸的差值

*d**i*=*x**i*−Ψ,*i*=1,2,...,200

第四步：构建协方差矩阵

*C*=1200∑*i*=1200*d**i**d**i**T*=1200*A**A**T*

第五步：求协方差矩阵的特征值和特征向量，构造特征脸空间  
求出  *A**T**A* 的特征值  *λ**i* 及其正交归一化特征向量  *ν**i* ，根据特征值的贡献率选取前p个最大特征值及其对应的特征向量，贡献率是指选取的特征值的和与占所有特征值的和比，即：

*ϕ*=∑*i*=1*i*=*p**λ**i*∑*i*=1*i*=200*λ**i*≥*a*

若选取前  *p* 个最大的特征值，则“特征脸”空间为：

*w*=(*u*1,*u*2,...*u**p*)

第六步：将每一幅人脸与平均脸的差值脸矢量投影到“特征脸”空间，即

Ω*i*=*w**T**d**i*(*i*=1,2,...,200)

## 识别阶段 ##

第一步：将待识别的人脸图像Γ与平均脸的差值脸投影到特征脸空间，得到其特征向量表示：

ΩΓ=*w**T*(Γ−Ψ)

第二布：采用欧式距离来计算  ΩΓ 与每个人脸的距离  *ε**i*

*ε**i*2=∥∥Ω*i*−ΩΓ∥∥2(*i*=1,2,...,200)

求最小值对应的训练集合中的标签号作为识别结果  
需要说明的是协方差矩阵  *A**A**T* 的维数为MN\*MN，其维数是比较较大的，而我们在这里的训练样本个数为200，  *A**T**A* 的维数为200\*200小了许多，实际情况中，采用奇异值分解(SingularValue Decomposition ,SVD)定理，通过求解  *A**T**A* 的特征值和特征向量来组成特征脸空间的。必须明白的是特征脸空间是由  *A**T**A* 的子空间构成，我们的识别任务也是将原始  *A**T**A* 所构成的空间投影到我们选取前p个最大的特征值对应的特征向量组成的子空间里，进行比较，选取最近的训练样本为标号。

## 代码 ##

训练过程代码如下：

function [m, A, Eigenfaces] = EigenfaceCore(T)
    % Use Principle Component Analysis (PCA) to determine the most 
    % discriminating features between images of faces.
    %
    % Description: This function gets a 2D matrix, containing all training image vectors
    % and returns 3 outputs which are extracted from training database.
    %
    % Argument:      T                      - A 2D matrix, containing all 1D image vectors.
    %                                         Suppose all P images in the training database 
    %                                         have the same size of MxN. So the length of 1D 
    %                                         column vectors is M*N and 'T' will be a MNxP 2D matrix.
    % 
    % Returns:       m                      - (M*Nx1) Mean of the training database
    %                Eigenfaces             - (M*Nx(P-1)) Eigen vectors of the covariance matrix of the training database
    %                A                      - (M*NxP) Matrix of centered image vectors
    %
    % See also: EIG
    
    % Original version by Amir Hossein Omidvarnia, October 2007
    %                     Email: aomidvar@ece.ut.ac.ir                  
    
    %%%%%%%%%%%%%%%%%%%%%%%% Calculating the mean image 
    m = mean(T,2); % Computing the average face image m = (1/P)*sum(Tj's)    (j = 1 : P)
    Train_Number = size(T,2);
    
    %%%%%%%%%%%%%%%%%%%%%%%% Calculating the deviation of each image from mean image
    A = [];  
    for i = 1 : Train_Number
        temp = double(T(:,i)) - m; % Computing the difference image for each image in the training set Ai = Ti - m
        A = [A temp]; % Merging all centered images
    end
    
    %%%%%%%%%%%%%%%%%%%%%%%% Snapshot method of Eigenface methos
    % We know from linear algebra theory that for a PxQ matrix, the maximum
    % number of non-zero eigenvalues that the matrix can have is min(P-1,Q-1).
    % Since the number of training images (P) is usually less than the number
    % of pixels (M*N), the most non-zero eigenvalues that can be found are equal
    % to P-1. So we can calculate eigenvalues of A'*A (a PxP matrix) instead of
    % A*A' (a M*NxM*N matrix). It is clear that the dimensions of A*A' is much
    % larger that A'*A. So the dimensionality will decrease.
    
    L = A'*A; % L is the surrogate of covariance matrix C=A*A'.
    [V D] = eig(L); % Diagonal elements of D are the eigenvalues for both L=A'*A and C=A*A'.
    
    %%%%%%%%%%%%%%%%%%%%%%%% Sorting and eliminating eigenvalues
    % All eigenvalues of matrix L are sorted and those who are less than a
    % specified threshold, are eliminated. So the number of non-zero
    % eigenvectors may be less than (P-1).
    
    L_eig_vec = [];
    for i = 1 : size(V,2) 
        if( D(i,i)>4e+07)     
            L_eig_vec = [L_eig_vec V(:,i)];
        end
    end
    
    %%%%%%%%%%%%%%%%%%%%%%%% Calculating the eigenvectors of covariance matrix 'C'
    % Eigenvectors of covariance matrix C (or so-called "Eigenfaces")
    % can be recovered from L's eiegnvectors.
    Eigenfaces = A * L_eig_vec; % A: centered image vectors

识别过程代码如下：

function OutputName = Recognition(TestImage, m, A, Eigenfaces)
    % Recognizing step....
    %
    % Description: This function compares two faces by projecting the images into facespace and 
    % measuring the Euclidean distance between them.
    %
    % Argument:      TestImage              - Path of the input test image
    %
    %                m                      - (M*Nx1) Mean of the training
    %                                         database, which is output of 'EigenfaceCore' function.
    %
    %                Eigenfaces             - (M*Nx(P-1)) Eigen vectors of the
    %                                         covariance matrix of the training
    %                                         database, which is output of 'EigenfaceCore' function.
    %
    %                A                      - (M*NxP) Matrix of centered image
    %                                         vectors, which is output of 'EigenfaceCore' function.
    % 
    % Returns:       OutputName             - Name of the recognized image in the training database.
    %
    % See also: RESHAPE, STRCAT
    
    % Original version by Amir Hossein Omidvarnia, October 2007
    %                     Email: aomidvar@ece.ut.ac.ir                  
    
    %%%%%%%%%%%%%%%%%%%%%%%% Projecting centered image vectors into facespace
    % All centered images are projected into facespace by multiplying in
    % Eigenface basis's. Projected vector of each face will be its corresponding
    % feature vector.
    
    ProjectedImages = [];
    Train_Number = size(Eigenfaces,2);
    for i = 1 : Train_Number
        temp = Eigenfaces'*A(:,i); % Projection of centered images into facespace
        ProjectedImages = [ProjectedImages temp]; 
    end
    
    %%%%%%%%%%%%%%%%%%%%%%%% Extracting the PCA features from test image
    InputImage = imread(TestImage);
    temp = InputImage(:,:,1);
    
    [irow icol] = size(temp);
    InImage = reshape(temp',irow*icol,1);
    Difference = double(InImage)-m; % Centered test image
    ProjectedTestImage = Eigenfaces'*Difference; % Test image feature vector
    
    %%%%%%%%%%%%%%%%%%%%%%%% Calculating Euclidean distances 
    % Euclidean distances between the projected test image and the projection
    % of all centered training images are calculated. Test image is
    % supposed to have minimum distance with its corresponding image in the
    % training database.
    
    Euc_dist = [];
    for i = 1 : Train_Number
        q = ProjectedImages(:,i);
        temp = ( norm( ProjectedTestImage - q ) )^2;
        Euc_dist = [Euc_dist temp];
    end
    
    [Euc_dist_min , Recognized_index] = min(Euc_dist);
    OutputName = strcat(int2str(Recognized_index),'.jpg');

其中训练样本的最后一行代码：

Eigenfaces = A * L_eig_vec; % A: centered image vectors

和识别过程的

temp = Eigenfaces'*A(:,i); % Projection of centered images into facespace

写成公式也就是如下

*T*=(*A**V*)*T**A*=*V**T*(*A**T**A*)

这里  *V**T* 是新坐标系下的基，投影的结果也就是新坐标系下的系数。基于PCA的人脸识别也就是我们在新的坐标系下比较两个向量的距离。完整代码下载地址点击 [这里][Link 1]。

## Licenses ##

<table> 
 <thead> 
  <tr> 
   <th>作者</th> 
   <th align="center">日期</th> 
   <th align="right">联系方式</th> 
  </tr> 
 </thead> 
 <tbody> 
  <tr> 
   <td>风吹夏天</td> 
   <td align="center">2015年8月10日</td> 
   <td align="right">wincoder@qq.com</td> 
  </tr> 
 </tbody> 
</table>

[20150810191535268]: /images/20220731/64d410f7be4d4b2fa7c04968f5372b71.png
[Link 1]: http://download.csdn.net/detail/bluecol/8987259