Hybrid input-output algorithm

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.

Be [math]\rho^{(n)}[/math] a trial scattering density in [math]n^{th}[/math] iteration cycle. Be [math]\rho'^{(n)}[/math] a density obtained from [math]\rho^{(n)}[/math] by Fourier-transforming [math]\rho^{(n)}[/math], replacing all Fourier amplitudes by the experimentally observed amplitudes, and applying inverse Fourier transform. Then the density [math]\rho^{(n+1)}[/math] is defined pixel-wise by the following scheme:

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