Here is a more detailed explanation of how this works.<p>Imagine you have a pixel image, with missing data, and with noise. Imagine you know <i>a-priori</i> that the image was generated simply by placing a few white circles against a black background. Then data was lost/discarded, and white noise was added to the remaining data.<p>(The white circles are cross sections of the patient's blood vessels or some such feature (bile ducts, in this case), and contain a contrast agent. The contrast agent appears white on the MRI, everything else appears black. )<p>Now that you know the pixel image has such a simple form, you don't need to do fancy noise removal to clean it up. All you need to do is figure out where the white circles are. Once you know where the are, you can just redraw the image based on your calculated locations of circles instead of reprocessing the old one.<p>You can play the same trick in k-space, which gives you a potent MRI reconstruction algorithm.<p>In practice, this doesn't work as well for more complicated images such as brains or abdomens. There is just much more to draw, and the images cease to be sparse.