Discrete Hilbert Transforms and the Phase Problem in Crystallography

With analogy to optical-image reconstruction theory, the role of `in-between' structure factors (i.e. those having half-integral Miller indices) has been investigated in the context of the crystallographic phase problem. It is shown that in-between structure factors can be incorporated into the phasing process by means of the autocorrelation function (the Patterson for a single unit cell). Three-dimensional discrete Hilbert transforms (DHTs) that are obtained express the in-between structure factors in terms of standard ones. When the casual Fourier transform condition is satisfied, a DHT applied to an intensity function provides twofold intensity oversampling in the reciprocal space.