Compressive Inverse Scattering I. High Frequency Simo Measurements

The inverse scattering problem with point scatterers and the single-input-multipleoutput (SIMO) measurements is analyzed by the compressed sensing techniques with and without the Born approximation. Three main results about (probabilistic) recoverability of sparse target by the L-optimization technique called Basis Pursuit (BP) are obtained in the high frequency limit. In the absence of noise, it is shown that BP can recover exactly the target of sparsity up to the dimension of the measurement either with the multi-shot SIMO measurement for the Born scattering or with the single-shot SIMO measurement for the exact scattering. The stability with respect to noisy data is proved for weak or widely separated scatterers under a stronger sparsity constraint.