在 PCA 详细算法介绍 （http://blog.csdn.net/watkinsong/article/details/38536463） 中， 因为篇幅问题 没有给出详细的代码示例， 这里给出代码示例。

通过对人脸图像进行降维深入了解PCA算得使用。

首先看一下数据集， 我们有12张人脸图像， 用10张人脸训练PCA降维矩阵， 剩下的两张可以用作测试。 

需要特别注意： 只能使用训练集样本进行所有的PCA训练过程。

这里所有的代码都采用 octave实现， 跟matlab应该是一致的。

1. 加载图像

%% Initialization
clear ; close all; clc

fprintf('this code will load 12 images and do PCA for each face.\n');
fprintf('10 images are used to train PCA and the other 2 images are used to test PCA.\n');

trainset = zeros(10, 32 * 32); % image size is : 32 * 32
m = 10; % number of samples

for i = 1 : m
	img = imread(strcat(int2str(i), '.bmp'));
	img = double(img);
	trainset(i, :) = img(:);
end

%% before training PCA, do feature normalization
mu = mean(trainset);
trainset_norm = bsxfun(@minus, trainset, mu);

sigma = std(trainset_norm);
trainset_norm = bsxfun(@rdivide, trainset_norm, sigma);

比如这里可能需要保存mu 和 sigma， 这里我们已 mean face 的方式保存 mu， 因为本示例比较小， 所以没有保存 sigma， 这里保存mu 的目的也仅仅是 为了让大家看一下平均脸的样子。 如果在做项目的过程中， 可能训练PCA是分开进行的， 以后需要进行降维， 那么就需要保存这两个归一化参数。 

%% we could save the mean face mu to take a look the mean face
imwrite(uint8(reshape(mu, 32, 32)), 'mf.bmp');

是不是比较丑？  因为人脸太少了， 再来看看由5000个人脸图像生成的平均脸： 

4. 计算降维矩阵

%% compute reduce matrix
X = trainset_norm; % just for convience
[m, n] = size(X);

U = zeros(n);
S = zeros(n);

Cov = 1 / m * X' * X;
[U, S, V] = svd(Cov);
fprintf('compute cov done.\n');

降维矩阵U中的特征向量， 在关于人脸的降维中，又被称为特征脸，  U 中的每个特征向量相当于找到的降维空间的一个方向。 利用U可以将特征映射到这个空间中。

这里我们把的U中的前几个特征向量保存下来，  看一下特征脸。 U 中的特征向量是按照特征值进行由大到小排序的， 这个排序的顺序也决定了对于降维的影响最大的向量放在最前面。 

这里的 eigen face 和人脸的相似性比较高， 因为我们的样本数量比较少， 就10个样本。。。所以会出现这种相似度比较高的情况。

补充： 给出几张用5000张人脸图像训练得到的eigen face, 如下所示：

6. 降维

%% dimension reduction
k = 100; % reduce to 100 dimension
test = zeros(2, 32 * 32);
for i = 1:2
	img = imread(strcat(int2str(i + 10), '.bmp'));
	img = double(img);
	test(i, :) = img(:);
end

% test set need to do normalization
test_norm = bsxfun(@minus, test, mu);
test_norm = bsxfun(@rdivide, test_norm, sigma);

% reduction
Uk = U(:, 1:k);
Z = test_norm * Uk;
fprintf('reduce done.\n');

%% reconstruction
%% for the test set images, we only minus the mean face,
% so in the reconstruct process, we need add the mean face back
Xp = Z * Uk';
% show reconstructed face
for i = 1:5
	face = Xp(i, :) + mu;
	face = reshape((face), 32, 32);
	imwrite(uint8(face), strcat('./reconstruct/', int2str(4000 + i), '.bmp'));
end

%% for the train set reconstruction, we minus the mean face and divide by standard deviation during the train
% so in the reconstruction process, we need to multiby standard deviation first, 
% and then add the mean face back
trainset_re = trainset_norm * Uk; % reduction
trainset_re = trainset_re * Uk'; % reconstruction
for i = 1:5
	train = trainset_re(i, :);
	train = train .* sigma;
	train = train + mu;
	train = reshape(train, 32, 32);
	imwrite(uint8(train), strcat('./reconstruct/', int2str(i), 'train.bmp'));
end

看一下特征还原的效果： 左边为原始图像， 右边为还原的图像

对于测试样本还原， 测试样本在降维之前减去了mean face, 所以在还原之后还要加上mean face才是真正的还原的图像。

对于训练样本还原， 因为训练样本即减去了mean face 还除以了 standard deviation， 所以在计算得到还原的样本特征后， 还首先要将特征 按元素乘上 standard deviation, 也就是 .* ，  然后再加上mean face才是最后得到的真实的还原的数据。 

看以下几个关于训练样本的还原： 同样左边为原始图像， 右边为还原之后的图像

整个工程的全部代码： 

%% Initialization
clear ; close all; clc

fprintf('this code will load 12 images and do PCA for each face.\n');
fprintf('10 images are used to train PCA and the other 2 images are used to test PCA.\n');

m = 4000; % number of samples
trainset = zeros(m, 32 * 32); % image size is : 32 * 32

for i = 1 : m
	img = imread(strcat('./img/', int2str(i), '.bmp'));
	img = double(img);
	trainset(i, :) = img(:);
end


%% before training PCA, do feature normalization
mu = mean(trainset);
trainset_norm = bsxfun(@minus, trainset, mu);

sigma = std(trainset_norm);
trainset_norm = bsxfun(@rdivide, trainset_norm, sigma);

%% we could save the mean face mu to take a look the mean face
imwrite(uint8(reshape(mu, 32, 32)), 'meanface.bmp');
fprintf('mean face saved. paused\n');
pause;

%% compute reduce matrix
X = trainset_norm; % just for convience
[m, n] = size(X);

U = zeros(n);
S = zeros(n);

Cov = 1 / m * X' * X;
[U, S, V] = svd(Cov);
fprintf('compute cov done.\n');

%% save eigen face
for i = 1:10
	ef = U(:, i)';
	img = ef;
	minVal = min(img);
	img = img - minVal;
	max_val = max(abs(img));
	img = img / max_val;
	img = reshape(img, 32, 32);
	imwrite(img, strcat('eigenface', int2str(i), '.bmp'));
end

fprintf('eigen face saved, paused.\n');
pause;

%% dimension reduction
k = 100; % reduce to 100 dimension
test = zeros(10, 32 * 32);
for i = 4001:4010
	img = imread(strcat('./img/', int2str(i), '.bmp'));
	img = double(img);
	test(i - 4000, :) = img(:);
end

% test set need to do normalization
test = bsxfun(@minus, test, mu);

% reduction
Uk = U(:, 1:k);
Z = test * Uk;
fprintf('reduce done.\n');

%% reconstruction
%% for the test set images, we only minus the mean face,
% so in the reconstruct process, we need add the mean face back
Xp = Z * Uk';
% show reconstructed face
for i = 1:5
	face = Xp(i, :) + mu;
	face = reshape((face), 32, 32);
	imwrite(uint8(face), strcat('./reconstruct/', int2str(4000 + i), '.bmp'));
end

%% for the train set reconstruction, we minus the mean face and divide by standard deviation during the train
% so in the reconstruction process, we need to multiby standard deviation first, 
% and then add the mean face back
trainset_re = trainset_norm * Uk; % reduction
trainset_re = trainset_re * Uk'; % reconstruction
for i = 1:5
	train = trainset_re(i, :);
	train = train .* sigma;
	train = train + mu;
	train = reshape(train, 32, 32);
	imwrite(uint8(train), strcat('./reconstruct/', int2str(i), 'train.bmp'));
end

fprintf('job done.\n');