Iterative reconstruction of three-dimensional objects from averaged Fourier-transform magnitude: solution and fiber x-ray scattering problems

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Abstract
Reconstruction of objects based on solution x-ray scattering and fiber x-ray diffraction both involve measurements that are averages of the magnitude squared of the Fourier transform of the electron density and involve electron densities with symmetries. We describe an iterative algorithm for such problems that is a generalization of the Gerchberg—Saxton algorithm from two dimensions (where the measurement is the magnitude squared of an unknown complex number) to n dimensions (where the measurement is the norm squared of an unknown n vector). An example of the reconstruction, for solution x-ray scattering from cowpea mosaic virus, is described.