An improved prior for image reconstruction in X-ray fiber diffraction

The structure completion problem in fiber diffraction is addressed from a Bayesian perspective. The experimental data are sums of the squares of the amplitudes of particular sets of Fourier coefficients of the electron density. In addition, a part of the electron density is known. The image reconstruction problem is to estimate the missing part of the electron density. A Bayesian approach is taken in which the 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. The conventional prior assumes that the positions of the unknown atoms are uniformly distributed. We improve this prior by treating the positions of the known atoms as containing normally distributed coordinate errors. 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 better results when the new prior is used.