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Karle-Hauptman method In x-ray crystallography, the diffraction pattern is essentially the Fourier transform of the periodic crystal structure which has both an amplitude and a phase. In practice, the phase information cannot be obtained; hence, one has to invert the Fourier transform without the phase and, hence, the algorithm is not unique. J. Karle and H. Hauptman developed a system of inequalities which the amplitudes have to satisfy. Using these inequalities as constraints, one can uniquely determine the inverse Fourier transform if the number of atoms in a unit cell is sufficiently small. The inequalities are based on the fact that the electron density is a positive function of which the diffraction is the Fourier transform (cf. J. Karle, J. Chem. Inf. Comput. Sci., 34, 381, 1994).