The use of non-local constraints in maximum-entropy electron density reconstruction

Different expressions of the maximum-entropy estimates of the electron density function, corresponding to different prior information are obtained. They show that no general-purpose configurational entropy of density maps exists. Some universal properties of the modellings are discussed. In particular, the meaning of super-resolution is clarified. The information of lower and upper bounds of the electron density is not in general strong enough to produce atomic maps. Atomicity is then introduced as non-local constraints and applied to the problem of phase extension using experimental data and low-resolution model phases. In all cases, the knowledge of phases up to 3.5-3 Å and observed moduli up to 1.5-1 Å allows an estimate of the electron density of roughly the same quality as the 1 Å map obtained from a Fourier summation to be produced.