Bayesian image reconstruction from partial image and spectral amplitude data

We address a problem of reconstruction of a periodic image from information in image space and Fourier space. The real space information consists of knowledge of part of the image, while the Fourier space information is data in the form of sums of the squares of the amplitudes of sets of particular Fourier coefficients of the image. Such a problem occurs in the determination of polymer structures from x-ray fiber diffraction data. We present a Bayesian approach to this problem, incorporating a prior model for the image based on the structure being made up of ''atoms''. The Bayesian minimum mean-square-error estimate for the missing part of the image is derived. Currently used heuristic estimates are the maxima of certain posterior densities. Simulations are performed for varying amounts of real and Fourier space information to assess the performance of the different estimators. The performance of the minimum mean-square-error estimate is superior to that of the other estimates.