Compressive Sensing as a way to solve Integral Equations ?

While reading Compressive Radar Imaging by Richard Baraniuk and Philippe Steeghs, I realized that something very different is happening with Compressed Sensing. I mentioned earlier that in the past, the solving of integral equations implies the use of weak formulations in order to obtain a moment based description of that integral equation. Hence, the general consensus with using an L_2 norm approach (or a least square reduction of the approximation error) is to use trial functions that are the same of the basis functions (in neutron transport, the situation is in fact a little different but nobody knows really why. ) This fact leads most researchers into projecting kernels onto sets of functions for which they become sparse like the wavelets. In other words, there is an expectation that the projected kernel will be very sparse and that matrix-vector operations will be reduced so that the overall method becomes very competitive with moment based method using non-sparse kernels. However, when the kernel reflects a geometry, it is difficult to know in advance which of the component of that kernel will have to be retained and one is therefore led to the nonlinear operation of computing them all first and then getting rid of the ones that are too small. This is a little bit what compressed sensing is avoiding when sampling a signal.
As it turns out, the integral equation that is being solved by a CS-Radar is different, the trial functions are specifically tailored to be incoherent with the basis functions. The big question would be to figure out how to do several iterations instead of just one. In neutron transport, we call some of these methods "source iteration". If we are to push the parallel further, on the one hand the Green's function is generally a description of the medium in which the neutron lives whereas the source is really the neutron distribution, hence there is no reason to believe that both types of functions should come from the same family and bear the same decomposition.

The neutron example goes further. Emmanuel Candes made a presentation last year on compressed sensing and showcased an example using neutron radiography at NIST on a fuel cell. It turns out, this is an experiment I did a long time ago at the Nuclear Science Center. The main issue being that in order to see exactly what is going on inside the fuell cell (which has a lot of graphite and copper), neutrons are the only way to see through and detect water or water vapor at the same time.
The reason for this type of investigation is simple: If water gets stuck at one place in the groves of the fuel cell, it will stop the fuel cell from working. It is therefore important to know exactly the hydrodynamics of the water in the cell while the cell is working. What is interesting is that the neutron scatters when in presence of water, so that the color on the graph of his presentation represents roughly the density of water in the cell. The neutron beams are used as in CT tomography with the fuel cell being hit from different angles. The compressed sensing formalism allows for fewer number of angles. What is interesting though is that many neutrons are deflected or diffused and therefore the problem at hand cannot be rigorously be seen as a CT scan. It is good enough for observation, but it would not be appropriate for neutron flux measurements for instance. The scattering issue should be solved using an integral equation called the linear Boltzmann equation!