Anna Gilbert, in her presentation entitled "Applications of Compressed Sensing to Biology", shows some of most recent results obtained with colleagues (one is Raghu Kainkaryam mentioned previously) performing Group Testing using Compressed Sensing ideas. The idea is to reduce the number of chips (SNP) for testing. The results are currently so so. The video is here.
When watching Assaf Naor's video at the intractability center, I did not realize that there were that many negative results in the Johnson-Lindenstrauss approach in other spaces.
Stefano Marchesini points out that the following reference: Increasing FTIR spectromicroscopy speed and resolution through compressive imaging by Julien Gallet, Michael Riley, Zhao Hao and Michael C. Martin. The abstract reads
At the Advanced Light Source at Lawrence Berkeley National Laboratory, we are investigating how to increase both the speed and resolution of synchrotron infrared imaging. Synchrotron infrared beamlines have diffraction-limited spot sizes and high signal to noise, however spectral images must be obtained one point at a time and the spatial resolution is limited by the effects of diffraction. One technique to assist in speeding up spectral image acquisition is described here and uses compressive imaging algorithms. Compressive imaging can potentially attain resolutions higher than allowed by diffraction and/or can acquire spectral images without having to measure every spatial point individually thus increasing the speed of such maps. Here we present and discuss initial tests of compressive imaging techniques performed with ALS Beamline 1.4.3?s Nic-Plan infrared microscope, Beamline 1.4.4 Continuum XL IR microscope, and also with a stand-alone Nicolet Nexus 470 FTIR spectrometer.
They are devising coded aperture with micrometer masks. The reconstruction is a traditional linear transform. I am sure they will eventually look at a nonlinear reconstruction technique to get a better spatial resolution as did Roummel Marcia and Rebecca Willett in Compressive Coded Aperture Superresolution Image Reconstruction ( the slides are here). I am also told by Ori Katz, one of the folks at Weizman performing Ghost Imaging (as mentioned here) that their reconstruction scheme is also linear. This is also the case of the coded aperture in the IR range that could be implemented in the LACOSTE program (as can be seen here and here). I think one of the main challenge of CS in the near future will be to characterize more straightforwardly how, in imaging (i.e. positive functions in 2D), nonlinear reconstruction techniques provide additional information than simpler and older linear algebra base reconstruction techniques.