Compressive Inverse Scattering Ii. 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 frequency, the chosen incidence and the sampling angle are related in a precise way and are capable of recovering exactly (point of extended) 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 that of the X-ray tomography. For 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.