Friday, November 29, 2013

Compressed Sensing in Imaging Mass Spectrometry

I know that for some of you, you are still digesting that turkey but this is one of the Donoho-Tao moment. Those moments happen when mathematics has a clear and direct impact on hardware or very applied instances. Here is one from Andreas Bartels who just sent me the following:
Hi Igor,

I would like to tell that our paper concerning „Compressed Sensing in Imaging Mass Spectrometry“ has just been published in the „Inverse Problems“ Journal! The final version is downloadable here:

The abstract reads:

Imaging mass spectrometry (IMS) is a technique of analytical chemistry for spatially resolved, label-free and multipurpose analysis of biological samples that is able to detect the spatial distribution of hundreds of molecules in one experiment. The hyperspectral IMS data is typically generated by a mass spectrometer analyzing the surface of the sample. In this paper, we propose a compressed sensing approach to IMS which potentially allows for faster data acquisition by collecting only a part of the pixels in the hyperspectral image and reconstructing the full image from this data. We present an integrative approach to perform both peak-picking spectra and denoising m/z-images simultaneously, whereas the state of the art data analysis methods solve these problems separately. We provide a proof of the robustness of the recovery of both the spectra and individual channels of the hyperspectral image and propose an algorithm to solve our optimization problem which is based on proximal mappings. The paper concludes with the numerical reconstruction results for an IMS dataset of a rat brain coronal section.

Thank you!

Here is the kicker in the conclusion:

Currently there are no mass spectrometers which allow for the acquisition of data in such manner

Yes, you read this right. A math paper describes how hardware makers should do their jobs. You don't see this that often. It continues:

However, considering the recent developments of the single pixel camera [8, 7], one could theoretically implement such a mass spectrometry by splitting the laser into several beams analogously as it is done in the digital micromirror device used in the single pixel camera. Then, instead of analyzing each pixel separately, one could analyze several pixels simultaneously and accumulate a measurement-mean spectrum for such a measurement. Note that modern mass spectrometers indeed use complex optics to achieve non-flat structured laser profile as in Bruker Daltonics smartbeam mass spectrometers [42], although the current optics does not allow to change the profile during an experiment.
We have theoretically proven that both the reconstruction of the spectra and the reconstruction of the m{z-images are robust. Further research might investigate the analysis of how the additional measurements of the gradients in theorem 3.6 could be omitted. Also, the actual bound in (3.23) on the number of measurements to take for robust recovery could be improved. The numerical results presented in this paper suggest that it is too pessimistic.
Wow. I can even imagine better measurement matrices!


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