• coherent diffraction imaging;
  • lensless imaging;
  • oversampling;
  • phase retrieval;
  • image reconstruction;
  • X-ray free-electron lasers

Coherent diffraction imaging (CDI) is high-resolution lensless microscopy that has been applied to image a wide range of specimens using synchrotron radiation, X-ray free-electron lasers, high harmonic generation, soft X-ray lasers and electrons. Despite recent rapid advances, it remains a challenge to reconstruct fine features in weakly scattering objects such as biological specimens from noisy data. Here an effective iterative algorithm, termed oversampling smoothness (OSS), for phase retrieval of noisy diffraction intensities is presented. OSS exploits the correlation information among the pixels or voxels in the region outside of a support in real space. By properly applying spatial frequency filters to the pixels or voxels outside the support at different stages of the iterative process (i.e. a smoothness constraint), OSS finds a balance between the hybrid input–output (HIO) and error reduction (ER) algorithms to search for a global minimum in solution space, while reducing the oscillations in the reconstruction. Both numerical simulations with Poisson noise and experimental data from a biological cell indicate that OSS consistently outperforms the HIO, ER–HIO and noise robust (NR)–HIO algorithms at all noise levels in terms of accuracy and consistency of the reconstructions. It is expected that OSS will find application in the rapidly growing CDI field, as well as other disciplines where phase retrieval from noisy Fourier magnitudes is needed. The MATLAB (The MathWorks Inc., Natick, MA, USA) source code of the OSS algorithm is freely available from