Multidimensional phase problems

Uniqueness properties of phase problems in three or more dimensions are investigated. It is shown that an N-dimensional image is overdetermined by its continuous Fourier amplitude. A sampling of the Fourier amplitude, with a density approximately 22−N times the Nyquist density, is derived that is sufficient to uniquely determine an N-dimensional image. Both continuous and discrete images are considered. Practical implications for phase retrieval in multidimensional imaging, particularly in crystallography where the amplitude data are undersampled, are described. Simulations of phase retrieval for two- and three-dimensional images illustrate the practical implications of the theoretical results.