Model bias in Bayesian image reconstruction from x-ray fiber diffraction data

Abstract

An image-reconstruction problem in x-ray fiber diffraction analysis (which is used to determine the atomic structure of biopolymers) is considered. The problem is to reconstruct an image (the electron density function) given data that are squared sums of the Fourier coefficients of the image as well as partial information (the model) on the image. A Bayesian estimation approach based on a prior for the missing part of the image is considered. Current (heuristic) approaches to this problem correspond to certain maximum a posteriori estimates. These estimates exhibit bias toward the model, and current methods to reduce the bias are based on scaling of the Fourier coefficients. A new procedure to remove bias, based on orthogonalization, is derived and shown by simulations to be superior to scaling. Bias and unbiasing are compared for the different maximum a posteriori estimates, for different amounts of missing information. These results are also compared with a new minimum mean-square-error estimate for this problem that has the form of weighted maximum a posteriori Fourier coefficients. The minimum mean-square-error estimate is free from bias and gives results superior to the unbiased maximum a posteriori estimates.