A = imread(['Oxford-University-square-logo.jpg']); % load image 
A = im2double(rgb2gray(A)); % image to matrix 

clf,figure(1), 
subplot(2,2,1)
imshow(A), title('original image')
subplot(2,2,3:4)
semilogy(svd(A),'.-'),ylim([1e-15 norm(A)]),grid on
xlabel('rank r')
title('singular values')

%% now do low-rank approximation 
r = 40; % rank
[U,S,V] = svd(A);
Ar = U(:,1:r)*S(1:r,1:r)*V(:,1:r)'; % truncated SVD
subplot(2,2,2)
imshow(Ar);
title(['rank ',num2str(r),' approximation'])