Deep learning: 12 (PCA 와 whitening 이 자연 이미지 에서 의 연습)

본문http://www.cnblogs.com/tornadomeet/archive/2013/03/22/2975456.html
선언:
이제 PCA, PCA Whitening 으로 자연 이미 지 를 처리 합 니 다.이러한 이론 지식 은 앞의 박문: Deep learning: 10 (PCA 와 whitening) 을 참고 한다.이번 시험의 데이터, 절차, 요구 등 참고 페이지:http://deeplearning.stanford.edu/wiki/index.php/UFLDL_Tutorial 。실험 데 이 터 는 자연 이미지 에서 12 * 12 의 패 치 10000 개 를 무 작위 로 선택 한 다음 에 이 패 치 를 99% 의 분산 보존 PCA 로 계산 하고 마지막 으로 이 패 치 를 PCA Whitening 과 ZCA Whitening 으로 비교 하 는 것 이다.
실험 환경: matlab 2012 a
실험 과정 및 결과:
10000 개의 패 치 를 무 작위 로 선택 하고 204 개의 패 치 를 표시 합 니 다. 다음 그림 과 같 습 니 다.
　　
그리고 이 패 치 에 대해 평균 값 을 0 으로 조작 하면 다음 과 같은 그림 을 얻 을 수 있 습 니 다.
　　
선택 한 patch 를 PCA 로 변환 하여 새로운 견본 데 이 터 를 얻 을 수 있 습 니 다. 새로운 견본 데이터 의 협 방 차 행렬 은 다음 그림 과 같 습 니 다.
　　
99% 의 분산 을 유지 한 후의 PCA 는 원시 데 이 터 를 복원 합 니 다. 다음 과 같 습 니 다.
　　
PCA Whitening 후의 그림 은 다음 과 같 습 니 다.
　　
이때 샘플 패 치 의 협 방 차 행렬 은 다음 과 같다.
　　
ZCA Whitening 의 결 과 는 다음 과 같 습 니 다.
　　
실험 코드 및 설명:

%%================================================================
%% 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),204,1); % A random selection of samples for visualization
display_network(x(:,randsel));%   x        ？

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

% -------------------- YOUR CODE HERE -------------------- 
x = x-repmat(mean(x,1),size(x,1),1);%         
%x = x-repmat(mean(x,2),1,size(x,2));

%%================================================================
%% 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
[n m] = size(x);
sigma = (1.0/m)*x*x';
[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 = (1./m)*xRot*xRot';

% 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
ss = diag(s);
% for k=1:m
%    if sum(s(1:k))./sum(ss) < 0.99
%        continue;
% end
%  cumsum(ss)          ，    ss       
%  (cumsum(ss)/sum(ss))<=0.99     ，  0  1   ， 1        
k = length(ss((cumsum(ss)/sum(ss))<=0.99));

%%================================================================
%% 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
xHat = u*[u(:,1:k)'*x;zeros(n-k,m)];

% 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 
%  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. 
%
%  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 = (1./m)*xPCAWhite*xPCAWhite';

% 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*xPCAWhite;

% 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));

참고 자료:
     Deep learning: 10 (PCA 와 whitening)
     http://deeplearning.stanford.edu/wiki/index.php/UFLDL_Tutorial