Coursera-Machine Learning-Andrew Ng-Programming Exercise 7

【Exercise 7 K-means Clustering and Principal Component Analysis】
【代码】【第一部分】
ex7.m
K-means基础 -> 在图像上的应用 
转换(行 列 RGB)=>(idx RGB) -> 初始化 -> K-means -> 每个像素用中心点替代 -> 展示图片
X_recovered = centroids(idx,:);
%% Machine Learning Online Class
% Exercise 7 | Principle Component Analysis and K-Means Clustering
%
% Instructions
% ------------
%
% This file contains code that helps you get started on the
% exercise. You will need to complete the following functions:
%
% pca.m
% projectData.m
% recoverData.m
% computeCentroids.m
% findClosestCentroids.m
% kMeansInitCentroids.m
%
% For this exercise, you will not need to change any code in this file,
% or any other files other than those mentioned above.
%

%% Initialization
clear ; close all; clc

%% ================= Part 1: Find Closest Centroids ====================
% To help you implement K-Means, we have divided the learning algorithm
% into two functions -- findClosestCentroids and computeCentroids. In this
% part, you should complete the code in the findClosestCentroids function.
%
fprintf('Finding closest centroids.\n\n');

% Load an example dataset that we will be using
load('ex7data2.mat');

% Select an initial set of centroids
K = 3; % 3 Centroids
initial_centroids = [3 3; 6 2; 8 5];

% Find the closest centroids for the examples using the
% initial_centroids
idx = findClosestCentroids(X, initial_centroids);

fprintf('Closest centroids for the first 3 examples: \n')
fprintf(' %d', idx(1:3));
fprintf('\n(the closest centroids should be 1, 3, 2 respectively)\n');

fprintf('Program paused. Press enter to continue.\n');
pause;

%% ===================== Part 2: Compute Means =========================
% After implementing the closest centroids function, you should now
% complete the computeCentroids function.
%
fprintf('\nComputing centroids means.\n\n');

% Compute means based on the closest centroids found in the previous part.
centroids = computeCentroids(X, idx, K);

fprintf('Centroids computed after initial finding of closest centroids: \n')
fprintf(' %f %f \n' , centroids');
fprintf('\n(the centroids should be\n');
fprintf(' [ 2.428301 3.157924 ]\n');
fprintf(' [ 5.813503 2.633656 ]\n');
fprintf(' [ 7.119387 3.616684 ]\n\n');

fprintf('Program paused. Press enter to continue.\n');
pause;

%% =================== Part 3: K-Means Clustering ======================
% After you have completed the two functions computeCentroids and
% findClosestCentroids, you have all the necessary pieces to run the
% kMeans algorithm. In this part, you will run the K-Means algorithm on
% the example dataset we have provided.
%
fprintf('\nRunning K-Means clustering on example dataset.\n\n');

% Load an example dataset
load('ex7data2.mat');

% Settings for running K-Means
K = 3;
max_iters = 10;

% For consistency, here we set centroids to specific values
% but in practice you want to generate them automatically, such as by
% settings them to be random examples (as can be seen in
% kMeansInitCentroids).
initial_centroids = [3 3; 6 2; 8 5];

% Run K-Means algorithm. The 'true' at the end tells our function to plot
% the progress of K-Means
[centroids, idx] = runkMeans(X, initial_centroids, max_iters, true);
fprintf('\nK-Means Done.\n\n');

fprintf('Program paused. Press enter to continue.\n');
pause;

%% ============= Part 4: K-Means Clustering on Pixels ===============
% In this exercise, you will use K-Means to compress an image. To do this,
% you will first run K-Means on the colors of the pixels in the image and
% then you will map each pixel onto its closest centroid.
%
% You should now complete the code in kMeansInitCentroids.m
%

fprintf('\nRunning K-Means clustering on pixels from an image.\n\n');

% Load an image of a bird
A = double(imread('bird_small.png'));

% If imread does not work for you, you can try instead
% load ('bird_small.mat');

A = A / 255; % Divide by 255 so that all values are in the range 0 - 1

% Size of the image
img_size = size(A);

% Reshape the image into an Nx3 matrix where N = number of pixels.
% Each row will contain the Red, Green and Blue pixel values
% This gives us our dataset matrix X that we will use K-Means on.
X = reshape(A, img_size(1) * img_size(2), 3);

% Run your K-Means algorithm on this data
% You should try different values of K and max_iters here
K = 9;
max_iters = 10;

% When using K-Means, it is important the initialize the centroids
% randomly.
% You should complete the code in kMeansInitCentroids.m before proceeding
initial_centroids = kMeansInitCentroids(X, K);

% Run K-Means
[centroids, idx] = runkMeans(X, initial_centroids, max_iters);

fprintf('Program paused. Press enter to continue.\n');
pause;

%% ================= Part 5: Image Compression ======================
% In this part of the exercise, you will use the clusters of K-Means to
% compress an image. To do this, we first find the closest clusters for
% each example. After that, we

fprintf('\nApplying K-Means to compress an image.\n\n');

% Find closest cluster members
idx = findClosestCentroids(X, centroids);

% Essentially, now we have represented the image X as in terms of the
% indices in idx.

% We can now recover the image from the indices (idx) by mapping each pixel
% (specified by its index in idx) to the centroid value
X_recovered = centroids(idx,:);

% Reshape the recovered image into proper dimensions
X_recovered = reshape(X_recovered, img_size(1), img_size(2), 3);

% Display the original image
subplot(1, 2, 1);
imagesc(A);
title('Original');

% Display compressed image side by side
subplot(1, 2, 2);
imagesc(X_recovered)
title(sprintf('Compressed, with %d colors.', K));

fprintf('Program paused. Press enter to continue.\n');
pause;


findClosestCentroids.m
assign centroids 注意向量化function idx = findClosestCentroids(X, centroids)
%FINDCLOSESTCENTROIDS computes the centroid memberships for every example
% idx = FINDCLOSESTCENTROIDS (X, centroids) returns the closest centroids
% in idx for a dataset X where each row is a single example. idx = m x 1
% vector of centroid assignments (i.e. each entry in range [1..K])
%

% Set K
K = size(centroids, 1);

% You need to return the following variables correctly.
idx = zeros(size(X,1), 1);

% ====================== YOUR CODE HERE ======================
% Instructions: Go over every example, find its closest centroid, and store
% the index inside idx at the appropriate location.
% Concretely, idx(i) should contain the index of the centroid
% closest to example i. Hence, it should be a value in the
% range 1..K
%
% Note: You can use a for-loop over the examples to compute this.
%
for i=1:size(X,1)
replica=repmat(X(i,:),[size(centroids,1),1]);
distance=sum((replica-centroids).^2,2);
[~,idx(i)]=min(distance);
end

% =============================================================

end



computeCentroids.m

利用logical类型索引矩阵，找出属于第k个聚类的点（受ex8 starter code 启发修改）function centroids = computeCentroids(X, idx, K)
%COMPUTECENTROIDS returns the new centroids by computing the means of the
%data points assigned to each centroid.
% centroids = COMPUTECENTROIDS(X, idx, K) returns the new centroids by
% computing the means of the data points assigned to each centroid. It is
% given a dataset X where each row is a single data point, a vector
% idx of centroid assignments (i.e. each entry in range [1..K]) for each
% example, and K, the number of centroi
4000
ds. You should return a matrix
% centroids, where each row of centroids is the mean of the data points
% assigned to it.
%

% Useful variables
[m n] = size(X);

% You need to return the following variables correctly.
centroids = zeros(K, n);

% ====================== YOUR CODE HERE ======================
% Instructions: Go over every centroid and compute mean of all points that
% belong to it. Concretely, the row vector centroids(i, :)
% should contain the mean of the data points assigned to
% centroid i.
%
% Note: You can use a for-loop over the centroids to compute this.
%
for k=1:K
idx_k = (idx==k);
%centroids(k,:)= sum(X .* repmat(idx_k,[1,n]) )./length(find(idx_k==1)) ;
centroids(k,:)=mean(X(idx_k,:));
end
% =============================================================

end


runkMeans.m
循环：
{
assign centroids
画图
计算centroids

}

设置previous centroids的目的是画出centroids移动轨迹



function [centroids, idx] = runkMeans(X, initial_centroids, ...
max_iters, plot_progress)
%RUNKMEANS runs the K-Means algorithm on data matrix X, where each row of X
%is a single example
% [centroids, idx] = RUNKMEANS(X, initial_centroids, max_iters, ...
% plot_progress) runs the K-Means algorithm on data matrix X, where each
% row of X is a single example. It uses initial_centroids used as the
% initial centroids. max_iters specifies the total number of interactions
% of K-Means to execute. plot_progress is a true/false flag that
% indicates if the function should also plot its progress as the
% learning happens. This is set to false by default. runkMeans returns
% centroids, a Kxn matrix of the computed centroids and idx, a m x 1
% vector of centroid assignments (i.e. each entry in range [1..K])
%

% Set default value for plot progress
if ~exist('plot_progress', 'var') || isempty(plot_progress)
plot_progress = false;
end

% Plot the data if we are plotting progress
if plot_progress
figure;
hold on;
end

% Initialize values
[m n] = size(X);
K = size(initial_centroids, 1);
centroids = initial_centroids;
previous_centroids = centroids;
idx = zeros(m, 1);

% Run K-Means
for i=1:max_iters

% Output progress
fprintf('K-Means iteration %d/%d...\n', i, max_iters);
if exist('OCTAVE_VERSION')
fflush(stdout);
end

% For each example in X, assign it to the closest centroid
idx = findClosestCentroids(X, centroids);

% Optionally, plot progress here
if plot_progress
plotProgresskMeans(X, centroids, previous_centroids, idx, K, i);
previous_centroids = centroids;
fprintf('Press enter to continue.\n');
pause;
end

% Given the memberships, compute new centroids
centroids = computeCentroids(X, idx, K);
end

% Hold off if we are plotting progress
if plot_progress
hold off;
end

end



plotProgresskMeans.m
用不同颜色画点 -> 在聚类中心画叉 -> 在本轮中心与上一轮中心之间连线，形成centroids移动轨迹function plotProgresskMeans(X, centroids, previous, idx, K, i)
%PLOTPROGRESSKMEANS is a helper function that displays the progress of
%k-Means as it is running. It is intended for use only with 2D data.
% PLOTPROGRESSKMEANS(X, centroids, previous, idx, K, i) plots the data
% points with colors assigned to each centroid. With the previous
% centroids, it also plots a line between the previous locations and
% current locations of the centroids.
%

% Plot the examples
plotDataPoints(X, idx, K);

% Plot the centroids as black x's
plot(centroids(:,1), centroids(:,2), 'x', ...
'MarkerEdgeColor','k', ...
'MarkerSize', 10, 'LineWidth', 3);

% Plot the history of the centroids with lines
for j=1:size(centroids,1)
drawLine(centroids(j, :), previous(j, :));
end

% Title
title(sprintf('Iteration number %d', i))

end



plotDataPoints.m
形成聚类号k到第k种颜色的映射palette -> 按照样本的顺序形成长度为m的矢量color（样本到颜色的映射）->传入作图函数
（参数“15”是画笔大小）function plotDataPoints(X, idx, K)
%PLOTDATAPOINTS plots data points in X, coloring them so that those with the same
%index assignments in idx have the same color
% PLOTDATAPOINTS(X, idx, K) plots data points in X, coloring them so that those
% with the same index assignments in idx have the same color

% Create palette
palette = hsv(K + 1);
colors = palette(idx, :);

% Plot the data
scatter(X(:,1), X(:,2), 15, colors);

end

kMeansInitCentroids.m
随机取K个样本点。打乱、取前K。注意打乱方法：一般的，a(randperm(length(a)))打乱a向量function centroids = kMeansInitCentroids(X, K)
%KMEANSINITCENTROIDS This function initializes K centroids that are to be
%used in K-Means on the dataset X
% centroids = KMEANSINITCENTROIDS(X, K) returns K initial centroids to be
% used with the K-Means on the dataset X
%

% You should return this values correctly
centroids = zeros(K, size(X, 2));

% ====================== YOUR CODE HERE ======================
% Instructions: You should set centroids to randomly chosen examples from
% the dataset X
%

% Initialize the centroids to be random examples
% Randomly reorder the indices of examples
randidx = randperm(size(X, 1));
% Take the first K examples as centroids
centroids = X(randidx(1:K), :);

% =============================================================

end

【第二部分】
ex7_pca.m
主成分分析 Principle Component Analysis
PCA(SVD) -> 投影 -> 恢复

1、一个简单例子 2D->1D
归一化 -> 运行PCA -> 作出PCA结果：特征向量
画出归一化后的数据点 -> 投影、恢复 -> 画出恢复后的数据点
并与相应原数据点连线，体现投影效果

2、图像PCA：人脸
-> 对图片进行PCA，展示出特征向量 
-> 原本为32×32图像1024维，现降到100维（投影）
-> 恢复，展示压缩后恢复的图像 

3、数据可视化
读取鸟图 -> 预处理（缩放为0~1小数、reshape为3*XXX，RGB3D）

Kmeans -> 三维可视化 -> 降维、二维可视化

【第一部分】与【第二部分】都对“鸟图”进行了Kmeans。【第一部分】是以鸟图方式直观展示：深黄、浅黄...统一以一种平均色黄替代，展示了图片压缩后色彩较少的效果。
【第二部分】则是在RGB三维空间上画出像素点的分布，可以见到Kmeans的效果是把RGB空间上靠近的点归为一类了。这里再用PCA降维（投影），把3D图变为2D
【第一部分】对鸟图进行了压缩，方法是利用Kmeans把原255*255*255种色彩用K种替代，每个像素只需存储0~K-1的索引。

【第二部分】对人像进行了压缩，方法是PCA，分布在高维空间的数据从某些角度看上去比较“薄”，把它“拍扁”损失相对较小，同时达到了降维的目的。
其它：

1、作出PCA结果（特征向量）







2、在立体图中可以发现：某些角度来看，像素点的分布比较“薄”，“拍扁”造成的损失较小；而另一些角度来看数据分布很广，应当尽量保留。PCA投影可以看成一种旋转，得到一种最佳视角，使得数据分布得最广。





%% Machine Learning Online Class
% Exercise 7 | Principle Component Analysis and K-Means Clustering
%
% Instructions
% ------------
%
% This file contains code that helps you get started on the
% exercise. You will need to complete the following functions:
%
% pca.m
% projectData.m
% recoverData.m
% computeCentroids.m
% findClosestCentroids.m
% kMeansInitCentroids.m
%
% For this exercise, you will not need to change any code in this file,
% or any other files other than those mentioned above.
%

%% Initialization
clear ; close all; clc

%% ================== Part 1: Load Example Dataset ===================
% We start this exercise by using a small dataset that is easily to
% visualize
%
fprintf('Visualizing example dataset for PCA.\n\n');

% The following command loads the dataset. You should now have the
% variable X in your environment
load ('ex7data1.mat');

% Visualize the example dataset
plot(X(:, 1), X(:, 2), 'bo');
axis([0.5 6.5 2 8]); axis square;

fprintf('Program paused. Press enter to continue.\n');
pause;

%% =============== Part 2: Principal Component Analysis ===============
% You should now implement PCA, a dimension reduction technique. You
% should complete the code in pca.m
%
fprintf('\nRunning PCA on example dataset.\n\n');

% Before running PCA, it is important to first normalize X
[X_norm, mu, sigma] = featureNormalize(X);

% Run PCA
[U, S] = pca(X_norm);

% Compute mu, the mean of the each feature

% Draw the eigenvectors centered at mean of data. These lines show the
% directions of maximum variations in the dataset.
hold on;
drawLine(mu, mu + 1.5 * S(1,1) * U(:,1)', '-k', 'LineWidth', 2);
drawLine(mu, mu + 1.5 * S(2,2) * U(:,2)', '-k', 'LineWidth', 2);

hold off;

fprintf('Top eigenvector: \n');
fprintf(' U(:,1) = %f %f \n', U(1,1), U(2,1));
fprintf('\n(you should expect to see -0.707107 -0.707107)\n');

fprintf('Program paused. Press enter to continue.\n');
pause;

%% =================== Part 3: Dimension Reduction ===================
% You should now implement the projection step to map the data onto the
% first k eigenvectors. The code will then plot the data in this reduced
% dimensional space. This will show you what the data looks like when
% using only the corresponding eigenvectors to reconstruct it.
%
% You should complete the code in projectData.m
%
fprintf('\nDimension reduction on example dataset.\n\n');

% Plot the normalized dataset (returned from pca)
plot(X_norm(:, 1), X_norm(:, 2), 'bo');
axis([-4 3 -4 3]); axis square

% Project the data onto K = 1 dimension
K = 1;
Z = projectData(X_norm, U, K);
fprintf('Projection of the first example: %f\n', Z(1));
fprintf('\n(this value should be about 1.481274)\n\n');

X_rec = recoverData(Z, U, K);
fprintf('Approximation of the first example: %f %f\n', X_rec(1, 1), X_rec(1, 2));
fprintf('\n(this value should be about -1.047419 -1.047419)\n\n');

% Draw lines connecting the projected points to the original points
hold on;
plot(X_rec(:, 1), X_rec(:, 2), 'ro');
for i = 1:size(X_norm, 1)
drawLine(X_norm(i,:), X_rec(i,:), '--k', 'LineWidth', 1);
end
hold off

fprintf('Program paused. Press enter to continue.\n');
pause;

%% =============== Part 4: Loading and Visualizing Face Data =============
% We start the exercise by first loading and visualizing the dataset.
% The following code will load the dataset into your environment
%
fprintf('\nLoading face dataset.\n\n');

% Load Face dataset
load ('ex7faces.mat')

% Display the first 100 faces in the dataset
displayData(X(1:100, :));

fprintf('Program paused. Press enter to continue.\n');
pause;

%% =========== Part 5: PCA on Face Data: Eigenfaces ===================
% Run PCA and visualize the eigenvectors which are in this case eigenfaces
% We display the first 36 eigenfaces.
%
fprintf(['\nRunning PCA on face dataset.\n' ...
'(this might take a minute or two ...)\n\n']);

% Before running PCA, it is important to first normalize X by subtracting
% the mean value from each feature
[X_norm, mu, sigma] = featureNormalize(X);

% Run PCA
[U, S] = pca(X_norm);

% Visualize the top 36 eigenvectors found
displayData(U(:, 1:36)');

fprintf('Program paused. Press enter to continue.\n');
pause;

%% ============= Part 6: Dimension Reduction for Faces =================
% Project images to the eigen space using the top k eigenvectors
% If you are applying a machine learning algorithm
fprintf('\nDimension reduction for face dataset.\n\n');

K = 100;
Z = projectData(X_norm, U, K);

fprintf('The projected data Z has a size of: ')
fprintf('%d ', size(Z));

fprintf('\n\nProgram paused. Press enter to continue.\n');
pause;

%% ==== Part 7: Visualization of Faces after PCA Dimension Reduction ====
% Project images to the eigen space using the top K eigen vectors and
% visualize only using those K dimensions
% Compare to the original input, which is also displayed

fprintf('\nVisualizing the projected (reduced dimension) faces.\n\n');

K = 100;
X_rec = recoverData(Z, U, K);

% Display normalized data
subplot(1, 2, 1);
displayData(X_norm(1:100,:));
title('Original faces');
axis square;

% Display reconstructed data from only k eigenfaces
subplot(1, 2, 2);
displayData(X_rec(1:100,:));
title('Recovered faces');
axis square;

fprintf('Program paused. Press enter to continue.\n');
pause;

%% === Part 8(a): Optional (ungraded) Exercise: PCA for Visualization ===
% One useful application of PCA is to use it to visualize high-dimensional
% data. In the last K-Means exercise you ran K-Means on 3-dimensional
% pixel colors of an image. We first visualize this output in 3D, and then
% apply PCA to obtain a visualization in 2D.

close all; close all; clc

% Reload the image from the previous exercise and run K-Means on it
% For this to work, you need to complete the K-Means assignment first
A = double(imread('bird_small.png'));

% If imread does not work for you, you can try instead
% load ('bird_small.mat');

A = A / 255;
img_size = size(A);
X = reshape(A, img_size(1) * img_size(2), 3);
K = 16;
max_iters = 10;
initial_centroids = kMeansInitCentroids(X, K);
[centroids, idx] = runkMeans(X, initial_centroids, max_iters);

% Sample 1000 random indexes (since working with all the data is
% too expensive. If you have a fast computer, you may increase this.
sel = floor(rand(1000, 1) * size(X, 1)) + 1;

% Setup Color Palette
palette = hsv(K);
colors = palette(idx(sel), :);
% 不同聚类使用不同颜色标记，与plotDataPoints.m中方法相同

% Visualize the data and centroid memberships in 3D
figure;
scatter3(X(sel, 1), X(sel, 2), X(sel, 3), 10, colors);
% 参数“10”指明圆圈大小
title('Pixel dataset plotted in 3D. Color shows centroid memberships');
fprintf('Program paused. Press enter to continue.\n');
pause;

%% === Part 8(b): Optional (ungraded) Exercise: PCA for Visualization ===
% Use PCA to project this cloud to 2D for visualization

% Subtract the mean to use PCA
[X_norm, mu, sigma] = featureNormalize(X);

% PCA and project the data to 2D
[U, S] = pca(X_norm);
Z = projectData(X_norm, U, 2);

% Plot in 2D
figure;
plotDataPoints(Z(sel, :), idx(sel), K);
title('Pixel dataset plotted in 2D, using PCA for dimensionality r
b05a
eduction');
fprintf('Program paused. Press enter to continue.\n');
pause;

pca.m
套公式，求协方差矩阵∑，svd分解function [U, S] = pca(X)
%PCA Run principal component analysis on the dataset X
% [U, S, X] = pca(X) computes eigenvectors of the covariance matrix of X
% Returns the eigenvectors U, the eigenvalues (on diagonal) in S
%

% Useful values
[m, n] = size(X);

% You need to return the following variables correctly.
U = zeros(n);
S = zeros(n);

% ====================== YOUR CODE HERE ======================
% Instructions: You should first compute the covariance matrix. Then, you
% should use the "svd" function to compute the eigenvectors
% and eigenvalues of the covariance matrix.
%
% Note: When computing the covariance matrix, remember to divide by m (the
% number of examples).
%
Sigma=1/m*(X'*X);
[U,S,~]=svd(Sigma);
% =========================================================================

end
projectData.m
套公式function Z = projectData(X, U, K)
%PROJECTDATA Computes the reduced data representation when projecting only
%on to the top k eigenvectors
% Z = projectData(X, U, K) computes the projection of
% the normalized inputs X into the reduced dimensional space spanned by
% the first K columns of U. It returns the projected examples in Z.
%

% You need to return the following variables correctly.
Z = zeros(size(X, 1), K);

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the projection of the data using only the top K
% eigenvectors in U (first K columns).
% For the i-th example X(i,:), the projection on to the k-th
% eigenvector is given as follows:
% x = X(i, :)';
% projection_k = x' * U(:, k);
%
Z=X*U(:,1:K);
% =============================================================

endrecoverData.m
套公式function X_rec = recoverData(Z, U, K)
%RECOVERDATA Recovers an approximation of the original data when using the
%projected data
% X_rec = RECOVERDATA(Z, U, K) recovers an approximation the
% original data that has been reduced to K dimensions. It returns the
% approximate reconstruction in X_rec.
%

% You need to return the following variables correctly.

%X_rec = zeros(size(Z, 1), size(U, 1));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the approximation of the data by projecting back
% onto the original space using the top K eigenvectors in U.
%
% For the i-th example Z(i,:), the (approximate)
% recovered data for dimension j is given as follows:
% v = Z(i, :)';
% recovered_j = v' * U(j, 1:K)';
%
% Notice that U(j, 1:K) is a row vector.
%

X_rec=Z*(U(:,1:K))';
%(AB)^T=B^T*A^T

% =============================================================

end
displayData.m
一张图片——X的一行
m：图片数
n：一张图片的像素数
-> 图片横向尺寸(example_width)可以作为参数传入，未传入则认为是正方形，像素总数开平方求得；再求得纵向尺寸(example_height)
-> 每行展示图片张数(display_rows)由图片总数开平方求得；再求得纵向张数(display_cols)

-> 根据以上数据生成空“画板”display_array
-> 逐行从X取出图片、reshape、粘贴至display_array
-> imagesc函数展示function [h, display_array] = displayData(X, example_width)
%DISPLAYDATA Display 2D data in a nice grid
% [h, display_array] = DISPLAYDATA(X, example_width) displays 2D data
% stored in X in a nice grid. It returns the figure handle h and the
% displayed array if requested.

% Set example_width automatically if not passed in
if ~exist('example_width', 'var') || isempty(example_width)
example_width = round(sqrt(size(X, 2)));
end

% Gray Image
colormap(gray);

% Compute rows, cols
[m n] = size(X);
example_height = (n / example_width);

% Compute number of items to display
display_rows = floor(sqrt(m));
display_cols = ceil(m / display_rows);

% Between images padding
pad = 1;

% Setup blank display
display_array = - ones(pad + display_rows * (example_height + pad), ...
pad + display_cols * (example_width + pad));

% Copy each example into a patch on the display array
curr_ex = 1;
for j = 1:display_rows
for i = 1:display_cols
if curr_ex > m,
break;
end
% Copy the patch

% Get the max value of the patch
max_val = max(abs(X(curr_ex, :)));
display_array(pad + (j - 1) * (example_height + pad) + (1:example_height), ...
pad + (i - 1) * (example_width + pad) + (1:example_width)) = ...
reshape(X(curr_ex, :), example_height, example_width) / max_val;
curr_ex = curr_ex + 1;
end
if curr_ex > m,
break;
end
end

% Display Image
h = imagesc(display_array, [-1 1]);

% Do not show axis
axis image off

drawnow;

end
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