PCA人脸识别

    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
    
    
![](https://ad.itadn.com/c/weblog/blog-img/images/2025-07-12/R0gtAxOqLNXdMWYKSwu8TjvVko94.png)

    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');
    
    
![](https://ad.itadn.com/c/weblog/blog-img/images/2025-07-12/e9OJ4sZvzHEjQLm07WXIuCNnUkt6.png)

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

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