Targets are detected in a scene behind a wall by first transmitting a pulse through the wall. Then, a primary impulse response is detected by a sparse regularized least squares inversion applied to received signals corresponding to the reflected pulse. A delay operator that matches the primary impulse response to similar impulse responses in the received signals is also determined. A distortion of the pulse after the pulse passes through the wall but before the pulse is reflected by the target can also be determined. The distortion is used in an iterative process to refine the detection of the target and to suppress ghosting artifacts.