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Besides MRI research, has any useful applications come out of this?

It's potentially huge, because of the potential for sub-nyquist signal reconstruction. Imagine if the only extant photo of a historical event was a shitty jpeg or comparatively low-resolution picture. Compressed sensing yields techniques for reconstructing a higher quality version of the image than the original hardware or codec was capable of recording.

It's frustrating to me as I work a lot with audio and video but the math is much harder than anything else I've encountered in DSP. I feel stupid every time I dig into it.

Yeah ... but what type of real world data is sparse?

Many, many kinds. A typical photo is quite sparse. Oh sure, there is a lot of variation in the individual pixels etc., but it's usually a picture of something with shape and fairly strongly defined visual characteristics (which is what made it interesting enough to record in the first place). Random chroma noise, by contrast, is not sparse at all. In that sense, it has a much higher information content than a picture.

That Terry Tao blog linked above is by far the best entry point I've found into the subject but I'm really not qualified to explain or simplify it well, I'm afraid.

Have you seen this: http://www.acm.caltech.edu/l1magic/

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