Recently learned about compressed sensing - my mind has just been blown away.
Here I share you a few example images to demonstrate what is possible by using advanced applied math algorithms and also a few links to dive into the subject. The pictures above are the original images and the next pictures bellow are the reconstructed versions by the magic algorithms. These samples were taken from Sapiro's paper.
There's an interesting article on the subject in Wired mag and here's a good glossary page.