We always hear about the need to justify mathematics as regards to applied work. Remember this "Donoho-Tao moment"? (from this 2008 newletter)
I've recently started following Nuit Blanche, and find your site to be a great reference for compressive sensing topics.
I believe I have a possible example of CS leading to "discoveries" in my field: seismic data acquisition, processing, and imaging. I have worked with Rich Baraniuk at Rice University on many topics over the past 20 years or so. Rich recognized that seismic data acquisition might be a good candidate for compressive sensing innovation, and began coaching me in this direction starting around 2008 when CS was gathering steam. In 2010, I started a formal research project at my company (ConocoPhillips) to investigate applications of CS to petroleum exploration and production. In 2012, we acquired our first field trials using CS designs for seismic data acquistion, and finally in 2014 we deployed our first full scale CS based system for seismic data acquistion. We call our framework Compressive Seismic Imaging, or CSI. Gotta have a good acronym ;-)
Seismic data acquisition for a single geologic prospect can cost anywhere between $10-$100MM USD, and involves the use of the largest moving objects ever created by man (6 x 18 km sensor arrays with upwards of 64,000 channels). Compressive Sensing allows us to increase what we call "acquisition efficiency" by a factor of 2 or more in each coordinate direction that is employed. For seismic data acquisition, we have 4 coordinate directions (source x, source y , receiver x, receiver y), so efficiencies on the order of 2**4 or 16x are possible. In addition, CS can be used to enable the use of multiple simultaneous sources, providing another factor of 4 or so. Just as in parallel computing, a significant portion of these projects is "serial", so the efficiencies might only have a 2-10x impact on cost. You do the math - impacting cost with a factor of 2 on a $100 million dollar project will get your attention.
Capital programs for seismic data acquistion exceed $1 billion dollars for many of the large oil companies, so there is a lot of upside for CS in our business. We have only started to get this technology into use, but we expect that in a few years the seismic acquistion business will fully embrace CSI.
Here is a link to a feature article on our CSI program in a popular news magazine for the exploration geophysics industry, "The Leading Edge", published by the Society of Exploration Geophysicists:
Geoscience Fellow, ConocoPhillips
So this is the applied part. What about the pure math connection ? Dustin Mixon recently wrote about Alexander Grothendieck's influence on his work:
(I say that his is not my field of study, and yet I have still seen his influence. For example, the Grothendieck inequality is a beautiful result in functional analysis that he proved as a graduate student before changing fields, and it has since found applications in hardness of approximation. Also, his development of Grothendieck groups provided a starting point for K-theory, which is the source of the best known lower bound for the 4M-4 conjecture.)
see also Phase Retrieval from masked Fourier transforms. In summary, you've got one of Grothendieck's theory providing bounds on the number of sensors and enabling a billion dollar industry, what do you want ?
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