Qualitative inverse scattering for sparse-aperture data collections using a phase-delay frequency variation constraint

We present a formulation of the linear sampling method (LSM) for reconstructing target shape from spatially sparse data sets. The technique compensates for a lack of spatial data by incorporating $a$ priori propagation information into the LSM inversion. A constraint is placed on the LSM solution that enforces a phase relationship across frequency that is determined by the electrical path length between the array transmitters and the imaging scene pixels. The effect of the constraint is that image artifacts that do not evince the expected phase relationship according to their position relative to the array are suppressed. We apply the proposed technique to simulated and experimental sparse-aperture data sets and show that the resulting imagery is significantly more faithful to the true target shape as compared to the imagery created with the standard LSM.