The methods generally exploit constraints or statistical correlations between the phases of different Fourier components that result from the fact that the scattering density must be a positive real number.
The key step was taken by Hauptman and Karle, who developed a practical method to employ the Sayre equation for which they were awarded the 1985 Nobel prize in Chemistry.
[1] The Nobel Prize citation was "for their outstanding achievements in the development of direct methods for the determination of crystal structures.
However, they are generally not feasible by themselves for larger molecules such as proteins.
[2][3] Several software packages implement direct methods.