A Statistics Problem from Spectroscopy that Hints of Compressive Sensing
The group of Dor Ben-Amotz, a chemist at Purdue, has built a number of Raman spectrometers, each of which incorporates what chemists call a Spatial Light Modulator, which may be a micro-mirror array, and a single, high-quality, photon receptor; the setup is similar to the the ``Single Pixel Camera'' developed by Rich Baraniuk's group at Rice. Ben-Amotz calls his approach ``Compressive Detection''. They started with a simple question: If a sample consists of one of a collection of known chemicals, what is the best way to set up the machine to detect which chemical it is? One can go on to ask, if a sample consists of a mixture of known chemicals, what is the best way to set up the machine to determine the concentration of each chemical in the mixture? What if there is fluorescence from the sample? Etc. These questions can be formulated as (nonstandard) problems in Optimal Design, a field in Statistics. We have partial results that characterize near-optimal settings in the machines; these results hint of techniques in Compressive Sensing. In simple cases, we can distinguish between two chemicals in 30 microseconds, in which time the sample has emitted only about 30 photons. The technique is so fast that we can conduct true ``chemical imaging'' in a reasonable time (the order of seconds to minutes for a 300x300-pixel image). This is joint work with Greg Buzzard (Math, Purdue), Dor Ben-Amotz (Chemistry, Purdue), and his students David Wilcox (graduated), Owen Rehrauer, Bharat Mankani, and Sarah Marie Matt.