We show that the average intensity spectrum for an object illuminated and viewed through the same random screen contains diffraction-limited information on the Fourier modulus of the object amplitude. For physical clarity, the theory is first presented for a two-pinhole, Michelson aperture and then extended to include the nonredundant and full-aperture cases. Theoretical predictions are confirmed by detailed computer simulations. We propose and implement a method for the recovery of the Fourier modulus of the object.