Statistical geometry. II. Numerical solution via the single pixel equation

A simple single pixel equation (SPE) is presented which, when solved self-consistently for each pixel, can yield exact solutions to the statistical inversion problem for diffraction data outlined in paper I of this series [Wilkins, Varghese & Lehmann (1983). Acta Cryst. A39, 47-60]. The SPE approach was used to obtain the results presented in I and is shown here to have both practical and heuristic advantages in that it: (i) provides a very transparent approach to the task of solving the fundamental equations of the statistical geometric problem, (ii) can greatly improve the rate of convergence and (iii) readily allows the convexity of the constraint contributions to be monitored and, if desired, controlled. For the important case of 'phase refinement' via constraint (1) of I and the assumption of: (i) a complete data set of E_k up to the resolution limit of the data and (ii) uniform errors (i.e. σ_{k, 1} = σ), it is shown that the maximum-entropy structure (MES) can be fully refined via the SPE in only one Fourier transform cycle, and so should be extremely efficient for biological macromolecules.