Remote sensing via $\ell_1$ minimization

We consider the problem of detecting the locations of targets in the far field by sending probing signals from an antenna array and recording the reflected echoes. Drawing on key concepts from the area of compressive sensing, we use an $\ell_1$-based regularization approach to solve this, in general ill-posed, inverse scattering problem. As common in compressed sensing, we exploit randomness, which in this context comes from choosing the antenna locations at random. With $n$ antennas we obtain $n^2$ measurements of a vector $x \in \C^{N}$ representing the target locations and reflectivities on a discretized grid. It is common to assume that the scene $x$ is sparse due to a limited number of targets. Under a natural condition on the mesh size of the grid, we show that an $s$-sparse scene can be recovered via $\ell_1$-minimization with high probability if $n^2 \geq C s \log^2(N)$. The reconstruction is stable under noise and under passing from sparse to approximately sparse vectors. Our theoretical findings are confirmed by numerical simulations.