clear
clf
%%
% We make an image from some text, then reload it as a matrix.
%
tobj = text(0,0,'Hello world','fontsize',44);
ex = get(tobj,'extent');
axis([ex(1) ex(1)+ex(3) ex(2) ex(2)+ex(4)]), 
axis off

saveas(gcf,'hello.png')

A = imread('hello.png');
A = double(rgb2gray(A));
figure(1)
imagesc(A)
colormap gray
[m, n] = size(A)

%%
% Next we show that the singular values decrease exponentially, until they
% reach zero (more precisely, are about (sigma_1 * macheps$).
[U, S, V] = svd(A);
sigma = diag(S);
np = 100;
figure(2)
semilogy(sigma(1:np), 'r.')
title('singular values')
xlabel('i')
ylabel('\sigma_i')
grid on

%%
% The rapid decrease suggests that we can get fairly good low-rank
% approximations. 
figure(3)
for i = 1:4
    subplot(2, 2, i)
    k = 2*i;
    Ak = U(:,1:k)*S(1:k,1:k)*V(:,1:k)';
    imshow(Ak, [0 255])
    title(sprintf('rank = %d',k))
end

%%
% Consider how little data is needed to reconstruct these images. For rank
% 8, for instance, we have 8 left and right singular vectors plus 8
% singular values, for a compression ratio of better than 25:1.  
compression = 8*(m+n+1) / (m*n)