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'); % ignore this line
axis([ex(1) ex(1)+ex(3) ex(2) ex(2)+ex(4)]), axis off % ignore this line
saveas(gcf,'hello.png')
A = imread('hello.png');
A = double(rgb2gray(A));
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 
). For all numerical purposes, this determines the rank of the matrix.
). For all numerical purposes, this determines the rank of the matrix.[U,S,V] = svd(A);
sigma = diag(S);
semilogy(sigma,'.')
title('singular values'), axis tight % ignore this line
xlabel('i'), ylabel('\sigma_i') % ignore this line
r = find(sigma/sigma(1) > 10*eps,1,'last')
The rapid decrease suggests that we can get fairly good low-rank approximations.
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)