Image Reconstruction from the Oversampled Diffraction Pattern

Abstract In recording the intensity of the diffraction pattern from an arbitrary object, the phase is lost. This gives rise to the “phase problem”in reconstructing the object in crystallography, where the pattern is sampled at the Bragg peaks. It was shown by Bates and by Hayes that sampling at twice crystallographic density (2x oversampling) in each dimension is sufficient in principle to obtain a full reconstruction of the object. It was pointed out by Sayre in 1991 that the diffraction pattern of a non-crystalline object is continuous, therefore the pattern may be sampled on a finer scale, providing the additional information for the image reconstruction. The loss of phase corresponds to a loss of half the information. It seems therefore that 2x oversampling in each dimension (8x in three dimension) may be unnecessary. We find that oversampling 1.67x in each dimension is sufficient to obtain a reconstruction, even in the presence of noise. However our algorithm fails when we reduce the oversampling factor to 1.58x.