Statistical geometry. I. A self-consistent approach to the crystallographic inversion problem based on information theory

The problem of inverting crystallographic diffraction data to obtain structural information is examined within the maximum-entropy formulation of information theory. The principal features of the present method (termed statistical geometry) are: (i) all predictions of the method are consistent with the given information (constraints) and least biased with respect to missing information, (ii) the adoption of weak (typically non-linear) constraints for incorporating the major part of the structural information guarantees that a solution exists in practice and leads to filtering of the structure maps consistent with the accuracy of the data, (iii) general conditions are established which lead to unique solutions for the structure map, (iv) atomicity is not a prerequisite, (v) other methods of crystallographic inversion may be incorporated via the adoption of appropriate constraint relations, and (vi) the task of numerical solution is roughly linear in the number of reflexions and in the number of pixels in the structure, and involves only straightforward numerical techniques. These features suggest that the method is especially well suited to problems such as the structure determination of biological macromolecules, and the determination of high-resolution electron-density maps, although it manifestly provides a general framework for treating a wide class of image-processing problems.