close all;
% clear all;
%%================================================================
%% Step 0a: Load data
% Here we provide the code to load natural image data into x.
% x will be a 144 * 10000 matrix, where the kth column x(:, k) corresponds to
% the raw image data from the kth 12x12 image patch sampled.
% You do not need to change the code below.

x = sampleIMAGESRAW();
figure(‘name‘,‘Raw images‘);
randsel = randi(size(x,2),200,1); % A random selection of samples for visualization
display_network(x(:,randsel));

%%================================================================
%% Step 0b: Zero-mean the data (by row)
% You can make use of the mean and repmat/bsxfun functions.

% -------------------- YOUR CODE HERE --------------------
avg = mean(x, 1);                 %x的每一列的均值
x = x - repmat(avg, size(x, 1), 1);
%%================================================================
%% Step 1a: Implement PCA to obtain xRot
% Implement PCA to obtain xRot, the matrix in which the data is expressed
% with respect to the eigenbasis of sigma, which is the matrix U.

 
% -------------------- YOUR CODE HERE --------------------
xRot = zeros(size(x)); % You need to compute this
sigma = x * x‘ / size(x, 2);
[U,S,V]=svd(sigma);
xRot=U‘*x;
 
%%================================================================
%% Step 1b: Check your implementation of PCA
% The covariance matrix for the data expressed with respect to the basis U
% should be a diagonal matrix with non-zero entries only along the main
% diagonal. We will verify this here.
% Write code to compute the covariance matrix, covar.
% When visualised as an image, you should see a straight line across the
% diagonal (non-zero entries) against a blue background (zero entries).
 
% -------------------- YOUR CODE HERE --------------------
covar = zeros(size(x, 1)); % You need to compute this
covar = xRot * xRot‘ / size(xRot, 2);
% Visualise the covariance matrix. You should see a line across the
% diagonal against a blue background.
figure(‘name‘,‘Visualisation of covariance matrix‘);
imagesc(covar);
 
%%================================================================
%% Step 2: Find k, the number of components to retain
% Write code to determine k, the number of components to retain in order
% to retain at least 99% of the variance.
 
% -------------------- YOUR CODE HERE --------------------
k = 0; % Set k accordingly
sum_k=0;
sum=trace(S);
for k=1:size(S,1)
        sum_k=sum_k+S(k,k);
        if(sum_k/sum>=0.99) %0.9
               break;
       end
end
 
%%================================================================
%% Step 3: Implement PCA with dimension reduction
% Now that you have found k, you can reduce the dimension of the data by
% discarding the remaining dimensions. In this way, you can represent the
% data in k dimensions instead of the original 144, which will save you
% computational time when running learning algorithms on the reduced
% representation.
%
% Following the dimension reduction, invert the PCA transformation to produce
% the matrix xHat, the dimension-reduced data with respect to the original basis.
% Visualise the data and compare it to the raw data. You will observe that
% there is little loss due to throwing away the principal components that
% correspond to dimensions with low variation.

% -------------------- YOUR CODE HERE --------------------
xHat = zeros(size(x));% You need to compute this
xTilde = U(:,1:k)‘ * x;
xHat(1:k,:)=xTilde;
xHat=U*xHat;
 
% Visualise the data, and compare it to the raw data
% You should observe that the raw and processed data are of comparable quality.
% For comparison, you may wish to generate a PCA reduced image which
% retains only 90% of the variance.
 
figure(‘name‘,[‘PCA processed images ‘,sprintf(‘(%d / %d dimensions)‘, k, size(x, 1)),‘‘]);
display_network(xHat(:,randsel));
figure(‘name‘,‘Raw images‘);
display_network(x(:,randsel));
 
%%================================================================
%% Step 4a: Implement PCA with whitening and regularisation
% Implement PCA with whitening and regularisation to produce the matrix
% xPCAWhite.
 
epsilon = 0.1;
xPCAWhite = zeros(size(x));
 
% -------------------- YOUR CODE HERE --------------------
xPCAWhite = diag(1./sqrt(diag(S) + epsilon)) * U‘ * x;

figure(‘name‘,‘PCA whitened images‘);
display_network(xPCAWhite(:,randsel));

%%================================================================
%% Step 4b: Check your implementation of PCA whitening
% 检查PCA白化是否规整化。如未规整化，则协方差矩阵是一个恒等矩阵；如已规整化，则其协方差矩阵的对角线上的值接近于1且依次变小。
% Check your implementation of PCA whitening with and without regularisation.
% PCA whitening without regularisation results a covariance matrix
% that is equal to the identity matrix. PCA whitening with regularisation
% results in a covariance matrix with diagonal entries starting close to
% 1 and gradually becoming smaller. We will verify these properties here.
% Write code to compute the covariance matrix, covar.
%
% 如未规整化，把epsilon置为0或接近于0，就会得到一条红线对角穿过蓝色背景图片。
% 如已规整化，就会得到就会得到一条从红色渐变到蓝色的线对角穿过蓝色背景的图片。
% Without regularisation (set epsilon to 0 or close to 0),
% when visualised as an image, you should see a red line across the
% diagonal (one entries) against a blue background (zero entries).
% With regularisation, you should see a red line that slowly turns
% blue across the diagonal, corresponding to the one entries slowly
% becoming smaller.
 
% -------------------- YOUR CODE HERE --------------------
covar = zeros(size(xPCAWhite, 1));
covar = xPCAWhite * xPCAWhite‘ / size(xPCAWhite, 2);
% Visualise the covariance matrix. You should see a red line across the
% diagonal against a blue background.
figure(‘name‘,‘Visualisation of covariance matrix‘);
imagesc(covar);
 
%%================================================================
%% Step 5: Implement ZCA whitening
% Now implement ZCA whitening to produce the matrix xZCAWhite.
% Visualise the data and compare it to the raw data. You should observe
% that whitening results in, among other things, enhanced edges.
 
xZCAWhite = zeros(size(x));
 
% -------------------- YOUR CODE HERE --------------------
xZCAWhite=U * diag(1./sqrt(diag(S) + epsilon)) * U‘ * x;
% Visualise the data, and compare it to the raw data.
% You should observe that the whitened images have enhanced edges.
figure(‘name‘,‘ZCA whitened images‘);
display_network(xZCAWhite(:,randsel));
figure(‘name‘,‘Raw images‘);
display_network(x(:,randsel));