Phase extension in crystallography using the iterative Fienup-Gerchberg-Saxton algorithm and Hilbert transforms

A procedure for phase extension in electron crystallography is proposed based on the iterative Fienup-Gerchberg-Saxton algorithm in combination with the use of discrete Hilbert transforms. This transform is used to provide oversampling in reciprocal space, thus satisfying the Shannon sampling requirement and introducing reflections with fractional indices. When the procedure is combined with the knowledge of a small set of strong phased Bragg reflections from electron-microscope images (or direct methods), the magnitudes of many non-Bragg reflections can be calculated with useful accuracy, thus enhancing the performance of the iterative algorithm for phase extension. The effects of various constraints used in the iterative algorithm are discussed. In this way, it is shown that the iterative algorithm conventionally used for phasing diffuse scattering from non-periodic objects can also be applied to problems in conventional crystallography to find the phases of high-order (resolution) beams from a known set of low-order (resolution) phases.