Acquiring perfect and uniform sampling seismic data requires an enormous and costly field effort. Often due to lack of available field equipment and cost constrains, seismic survey acquisitions need to compromise in terms of surface and/or subsurface coverage. In addition, existing surface facilities and natural obstacles can cause acquisition artifacts and footprints. Under these circumstances the recorded data need to be reconstructed or interpolated so that subsequent processing will be free from processing artifacts. In this paper we present a reconstruction algorithm along three spatial dimensions that is based on sparse inversion and parameter optimization of a combined linear and parabolic Radon transforms. We show a number of applications and examples on data reconstruction and regularization ranging from one to three spatial dimensions.