Compressive inverse scattering: II. Multi-shot SISO measurements with born scatterers

Inverse scattering methods capable of compressive imaging are proposed and analyzed. The methods employ randomly and repeatedly (multiple-shot) the single-input–single-output (SISO) measurements in which the probe frequencies, the incident, and the sampling directions are related in a precise way and are capable of recovering exactly scatterers of sufficiently low sparsity. For point targets, various sampling techniques are proposed to transform the scattering matrix into the random Fourier matrix. Two schemes are particularly interesting: the first one employs multiple frequencies with the sampling angle always in the back-scattering direction resembling the synthetic aperture (SA) imaging; the second employs only single frequency with the sampling angle in the (nearly) forward-scattering direction in the high-frequency limit, resembling the setting of x-ray tomography. The results for point targets are then extended to the case of localized extended targets by interpolating from grid points. In particular, an explicit error bound is derived for the piece-wise constant interpolation which is shown to be a practical way of discretizing localized extended targets and enabling the compressed sensing techniques. For distributed extended targets, the Littlewood–Paley basis is used in analysis. A specially designed sampling scheme then transforms the scattering matrix into a block-diagonal matrix with each block being the random Fourier matrix corresponding to one of the multiple dyadic scales of the extended target. In other words, by the Littlewood–Paley basis and the proposed sampling scheme the different dyadic scales of the target are decoupled and therefore can be reconstructed scale-by-scale by the proposed method. Moreover, with probes of any single frequency ω the coefficients in the Littlewood–Paley expansion for scales up to ω/(2π) can be exactly recovered.