Compressive tomography consists of estimation of high dimensional objects from measurements distributed over lower dimensions. Examples include reconstruction of 3D spectral data cubes from 2D focal planes and reconstruction of 3D volumes from 2D x-ray projections or holograms. Compressive tomographic estimation is improved if projections are structured to randomize the sampling phase space. To illustrate this principle, we show that structured x-ray illumination enables improvements in reconstructed image quality for compressed measurements relative to full Radon sampling and that structured millimeter wave illumination improves estimation of 3D surfaces. © 2014 OSA.