This is a very nice example.
Every once in while, people ask me whether compressive sensing is worth it. The answer is invariably: It depends. In particular it depends on what you are already using and why you are doing it. In the case of through wall imaging radars, people will tell you that given enough samples, they can get a linear system with square matrix and use least squares or other techniques to get a solution. Here is what 100% samples will get you, with a back propagation operator:
It's beautiful only if you have been abusing substances. Don't do drugs it clouds your mind.
But this is the answer you get in most journals, the real insight here is "what was the question ?" In other words, what is conventional back propagation solving ? Most of us do not know the answer to that, rather what we do know is that back propagation is merely a convenient means of getting a solution. Quite simply, we don't know the type of constraint that is imposed on the solution even if the matrix is square!
If, like in compressive sensing, you are interested in a certain type of solution then you need to use the right sampling and reconstruction solvers that will allow you to impose a constraint. Without further due here is a compressive sensing approach to this problem:
Determining building interior structures using compressive sensing by Lagunas Targarona, Eva, Amin, Moeness, Fauzia Ahmad, Nájar Martón, Montserrat. The abstract reads:
We consider imaging of the building interior structures using Compressive Sensing (CS) with applications to through-the-wall imaging and urban sensing. We consider a monostatic synthetic aperture radar imaging system employing stepped frequency waveform. The proposed approach exploits prior information of building construction practices to form an appropriate sparse representation of the building interior layout. We devise a dictionary of possible wall locations, which is consistent with the fact that interior walls are typically parallel or perpendicular to the front wall. The dictionary accounts for the dominant normal angle reflections from exterior and interior walls for the monostatic imaging system. Compressive sensing is applied to a reduced set of observations to recover the true positions of the walls. Additional information about interior walls can be obtained using a dictionary of possible corner reflectors, which is the response of the junction of two walls. Supporting results based on simulation and laboratory experiments are provided. It is shown that the proposed sparsifying basis outperforms the conventional through-the-wall CS model, the wavelet sparsifying basis, and the block sparse model for building interior layout detection.