Approaching a Random Radiation Detector

We are coming closer to the interaction between integral equation solving and compressed sensing. Case in point the new preprint from Lawrence Carin, Dehong Liu, and Ya Xue at Duke University entitled In Situ Compressive Sensing.

They use an additional known heterogenous medium in order to project the unknown original signal onto a new basis. This is not unlike the random lens imager of Rob Fergus, Antonio Torralba, and William T. Freeman at MIT where that heterogenous material is the random lens. Much work goes into calibrating that random lens using known images. Similarly here, the known heterogenous medium has a transfer function that is very well known (after calibration) and one uses this transfer function to produce "new" measurements or "new random projections" from the original unknown signal. These new signals are the CS measurements. Most of the work performed rests in the calibration/knowledge of this heterogenous medium in order to eventually extract information from the compressed measurements. The work pays off because, in the end, not only do you image the source, but you can also invert the whole field, i.e. you see everything between the source and your sensor.

What is interesting is the use of the Green's function for the Helmhotz equation because it clearly is a path toward the solving of the Linear Boltzmann Equation (which has a Helmhotz equation component for the discrete eigenvalues of the Case Eigenfunctions). [For more reference on Caseology, look up articles in TTSP and JQRST. This opens the door to radiation sensors (not just electromagnetic radations, but particle detection as well).]

Returning to the In-situ compressive sensing paper, the authors state:

Rather than measuring an antenna pattern in the far zone and in free space as a function of angle, the antenna may be placed within two large plates (or other multi-path-generating medium) and a relatively small number of measurements may be performed, with the angle and frequency dependent radiation pattern inferred using CS techniques like those considered here.

.... This suggests that the in situ CS technique may be used to perform measurements of evanescent fields, with this also supported by the theory presented in Sec. III. Note that such measurements require one to place a well-understood heterogeneous medium in the presence of evanescent fields, and therefore it is of interest to explore linkages to similar measurements performed with negative-index artificial materials. In particular, it may be worthwhile to investigate whether such metamaterials constitute a particularly attractive complex medium through which to perform in situ CS measurements, for sensing evanescent fields.

One can begin to see how Compressed Sensing is going to be implemented in other radiation fields. In Compressive Radar Imaging by Richard Baraniuk and Philippe Steeghs, the idea is to play on the dissimilarity between the signal and the target thereby making explicit the assumptions made on the target of interest. Whereas here, the expectation is for an additional hardware (a random but well known material) to produce the compressed measurements. This is done irrespective to any information on the target because in this case, it is difficult to modulate the signal.

In the radiation business, sources are difficult to modulate so it is likely that the in-situ approach will likely be favored even though the radar approach may be more suitable for radiation interrogation techniques. One can imagine that instead of using the spatial decomposition of the signal, it might be more relevant to use the energy spectrum make-up of the target of interest. A good knowledge of the heterogenous material could be performed using Monte-Carlo techniques (MCNP) and detection of the element of interest could be performed using the smashed filter approach. With regards to the solution technique of the deterministic Botlzmann equation, the Green's function [1] has been known for quite some time, so it may be a matter of coupling it to "random sources" in order to produce a new solution technique.

[1] G. J. Mitsis,Transport Solutions to the Monoenergetic Critical Problem. ANL-6787, Argonne. National Laboratory (1963)