"...Anyway, in seismology it was known since at least 1973 that l_1 minimization was promoting sparsity but nobody really knew why and what to make of it in the sense that none of the sensors were modified as a result of this finding. What the papers of Tao, Candes, Romberg  and Donoho  did was give a sense of the kind of acquisition that would be acceptable ( measurement matrix satisfying RIP, KGG....) which is the reason we can contemplate building hardware..."
I added the emphasis after a commenter stated that
"but nobody really knew why and what to make of it in the sense that none of the sensors were modified as a result of this finding"
I respectfully disagree
1) L1 in seismic was for deconvolution, nothing to do with CS.
2) The "Golden Oldies" section of Donoho's publications is worth reading, in particular "Superresolution via Sparsity Constraints", "Uncertainty Principles and Signal Recovery". It is quite clear that even in 1990 Donoho's understanding of L1 (and other sparsity promoting functional like entropy) and his relation with L0 was almost perfect.
To what I responded:
You seem to be arguing about the first part of the sentence, but are you sure you are arguing about the second part of that sentence (which is really the point I was making) ? Namely:
"..in the sense that none of the sensors were modified as a result of this finding.."
I realize that most of you in the signal processing community will feel strongly about what I am about to say but this is mostly because you have had a very raw deal all these years. Other people take data and then tell you : please clean it up I'll come back in the morning to pick it up when you're done. Would you want your kids to have that kind of a job ? It reminds me of being a thermal engineer in the space business, they would always lament being the last one to add anything in the design as it generally amounts to put wrinkled cloth on top of something. At dinner parties, you have a dry martini in one hand and you look sharp, and /or gorgeous while people laugh at your cynical view of what you are doing, you are so funny being the center of attention but deep down you feel dirty just explaining it.
Since this is the second time I see this type of blockage (the first one was here), let me be forthright about what I think on the matter:
Wavelets alone are not important
l_1 solvers promoting sparsity alone are not important
However, these unimportant pieces of the puzzle crystallize into something important because of these two papers  . I realize that l_1 promoting sparsity was discovered before empirically, I realize that wavelets were developed in the late 1980's and I realize that they both made the work of signal processing people tremendously easier, but none of these two findings amount to much unless you can change the sensors and the way we (literally) see the world. I could go further:
Fractals alone are not important
Structured sparsity solvers alone are not important
 E. J. Candès, J. Romberg and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math., 59 1207-1223. (pdf)
 D. Donoho, Compressed Sensing