The phase problem with structured sampling

The phase problem is a main limitation in diffraction imaging. Uniqueness properties of the phase problem are important, and are well characterised for well-sampled Fourier amplitude data. Here we consider uniqueness of the phase problem is cases where the Fourier amplitude data are sampled at a sufficient density, but the sample locations are structured in a particular way that arises in imaging of 1D and 2D crystals. Uniqueness is characterised by a suitably defined constraint ratio. Simulations of phase retrieval using an iterative projection algorithm show the influence of the sampling structure both in terms of difficulty of reconstruction and the resulting reconstruction error, and its dependence on the constraint ratio. The results show that, compared to random sampling of the Fourier intensity, structured sampling leads to more convergence difficulties and increased reconstruction error at equivalent constraint ratios. The results have implications for ab initio phasing in imaging 1D and 2D crystals using x-ray free-electron lasers.