An image reconstruction problem motivated by X-ray fiber diffraction analysis is considered. The experimental data are sums of the squares of the amplitudes of particular sets of Fourier coefficients of the electron density, and a part of the electron density is known. The image reconstruction problem is to estimate the unknown part of the electron density, the "image." A Bayesian approach is taken in which a prior model for the image is based on the fact that it consists of atoms, i.e., the unknown electron density consists of separated, sharp peaks. Currently used heuristic methods are shown to correspond to certain maximum a posteriori estimates of the Fourier coefficients. An analytical solution for the Bayesian minimum mean-square-error estimate is derived. Simulations show that the minimum mean-square-error estimate gives good results, even when there is considerable data loss, and out-performs the maximum a posteriori estimates.

The restoration of an incomplete image from a known part and experimental data in the form of the Fourier amplitude squared sums is formulated as a Bayesian estimation problem. This problem is motivated by the structure completion problem in X-ray fiber diffraction analysis. An appropriate prior of uniformly distributed impulses is used. The Bayesian MMSE and MAP estimates are obtained. Simulations are used to compare the performance of the estimates. The results show that the MMSE estimate significantly outperforms the other estimates. The restored images exhibit some bias towards the known part of the image. This can be partly reduced by an unbiasing procedure.