Hybrid input-output algorithm

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An algorithm proposed by R. Fienup in 1982. The original application of the algorithm was the phase retrieval of diffraction patterns of non-periodic objects. However, the algorithm or its components have been used several times also in crystallography.

Let ρ(n) be a trial scattering density in the nth iteration cycle. Let ρ'(n) be a density obtained from ρ(n) by Fourier-transforming ρ(n), replacing all Fourier amplitudes by the experimentally observed amplitudes, and applying inverse Fourier transform. Then the density ρ(n + 1) is defined pixel-wise by the following scheme:

CF 4.png

In crystallography, the support is usually not known \textit{a~priori}, and a dynamical support must be used, i.e. the support is newly defined in each iteration cycle based on a predefined criterion.