Electron density images from imperfect data by iterative entropy maximization

Abstract

Information theory provides a uniquely powerful apparatus for reconstructing an image from imperfect data in its dual space. An iterative procedure based on constrained entropy maximization is presented here for reconstruction of electron density from imperfect single-crystal X-ray diffraction data. In the ideal situation, continuous electron density, its periodicity given by the lattice, is the Fourier transform of an infinite set of structure factors whose squared moduli are observables. An ideal electron density is unattainable from experiment because of the common experimental inadequacies of incomplete and noisy data discussed by Gull and Daniell¹. These inadequacies are partially overcome in the present procedure which yields true super-resolution and is economical even for protein crystal structures as illustrated.