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When you compress something you represent it in such a way that you can reconstruct the original based on less data. Effectively you're "discovering" data that wasn't sampled (stored) in the first place. Except, with lossless compression at least, you're not really doing this. The compression process discards only redundant data.

Compressed sensing works in much the same way except that you effectively treat your acquisition and display process as you would your reading-from-disk-and-decompressing process. Instead of sampling everything you skip sampling points that are likely to be redundant.

If you go too far then yes, you get image degradation. If you keep things reasonable (and reasonable depends to some extent on what kind of image you're acquiring), you can get a perfect reconstruction, just like you do when you reconstruct a gzip compressed image.

Regarding your example: it's easier if you consider the process as it actually occurs in MR. The image is actually acquired in the Fourier domain so every point you acquire has information about every pixel in the image. So you've got a dark spot indicating a tumor. Yes, that information has to be in your acquired data but that doesn't mean you're going to be able to actually distinguish that tumor when you reconstruct the image. Noise and artifacts may obscure it.

Compressed sensing effectively tells you what you need to sample in order to retain the information about that tumor in your reduced data set (with high likelihood), and how to reconstruct the data so that the tumor is actually evident in the image (mostly be reducing artifacts).

Here's an example (sorry, I can't post an image with a tumor due to privacy).

Image C is the original, where features are clearly visible. D is undersampled in a way you might do to make an MR scan faster. What looks like noise in the image isn't really noise, it's decoherent aliasing artifacts due to the undersampling. The information to reconstruct the image is still there, but the naive reconstruction technique can't reveal it. CS reconstruction (E) reveals it.

"Image Simulation" likely means that they simulated the acquisition. The recovery of the "after" image from the "before" image is probably as shown, it's just that the "before" image was not acquired from an actual camera. Those results don't look particularly amazing for compressed sensing. See this for example.

Your AC wish is my command.

The figure on the right comes from the same kind of research. The similarity is pretty cool:

http://www.robbtech.com/~robb/Science/ReceptiveField.shtml

Oh, I'm in the OTHER Denver, called Calgary, Canada.

Same deal with the mountains though:

http://www.robbtech.com/~robb/Gallery/Mountain%20Seasons/large-14.shtml

http://www.robbtech.com/~robb/Gallery/cochrane/cochrane.shtml

Oh, I'm in the OTHER Denver, called Calgary, Canada.

Same deal with the mountains though:

http://www.robbtech.com/~robb/Gallery/Mountain%20Seasons/large-14.shtml

http://www.robbtech.com/~robb/Gallery/cochrane/cochrane.shtml

It hasn't been published yet... e-mail me and I'll see what I can do. There's an e-mail link on the website I linked to for the Python version, here.

Mine is an implementation of the algorithm, based on the original paper, not this particular toolkit. Anisotropic diffusion is a well known algorithm. (so don't try to bully me with licenses)

If you're interested, the video card version needs to be modified to work with non-MRI images, but here's a fairly general purpose Python implementation. It's not long, so it's easy to see what's going on. It also happens to be faster (last I checked) than the C++ version included in the ITK medical image processing library. Link.