This paper reports a novel cone-ray model of background-oriented schlieren (BOS) imaging that accounts for depthof-field effects. Reconstructions of the density field performed with this model are far more robust to the blur associated with a finite aperture than conventional reconstructions, which presume a thin-ray pinhole camera. Our model is characterized and validated using forward evaluations of simulated buoyancy-driven flow and both simulated and experimental BOS measurements of hypersonic flow over a sphere. Moreover, the model is embedded in a neural reconstruction algorithm, which is demonstrated with a total variation penalty and the compressible Euler equations. Our cone-ray technique dramatically improves the accuracy of BOS reconstructions: the shock interface is well-resolved in all our tests, irrespective of the camera's aperture setting, which spans f-numbers from 22 down to 4.