Bayesian image reconstruction with a hyperellipsoidal posterior in x-ray fiber diffraction

The. structure completion problem in x-ray 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 posterior for the Fourier coefficients typically takes the form of an independent and identically distributed multivariate normal density restricted to the surface of a hypersphere. However, the electron density often exhibits symmetry, in which case the Fourier coefficient components are no longer independent or identically distributed. A diagonalization process results in a independent (non-identical) multivariate normal probability density function, restricted to a hyperspherical surface. The analytical form for the mean of the posterior density function is derived. The mean can be expressed as a weighting function on the Fourier coefficients of the known part of the electron density. The weighting function for the hyperellipsoidal and hyperspherical cases are compared.