Bayesian image reconstruction from partial image and aliased spectral intensity data

An image reconstruction problem motivated by xray fiber diffraction analysi s is considered. The experimental data are sums of the squares of the ampli tudes of particular sets of Fourier coefficients of the electron density, a nd a part of the electron density is known. The image reconstruction proble m is to estimate the unknown part of the electron density, the "image." A B ayesian 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 cons ists of separated, sharp peaks. Currently used heuristic methods are shown to correspond to certain maximum a posteriori estimates of the Fourier coef ficients. An analytical solution for the Bayesian minimum mean-square-error estimate is derived. Simulations show that the minimum mean-square-error e stimate gives good results, even when there is considerable data loss, and out-performs the maximum a posteriori estimates.