PHASE RETRIEVAL FROM THE MAGNITUDE OF THE FOURIER-TRANSFORMS OF NONPERIODIC OBJECTS

It is suggested that, given the magnitude of Fourier transforms sample d at the Bragg density, the phase problem is underdetermined by a fact or of 2 for 1D, 2D, and 3D objects. It is therefore unnecessary to ove rsample the magnitude of Fourier transforms by 2 X in each dimension ( i.e., oversampling by 4 X for 2D and 8 x for 3D) in retrieving the pha se of 2D and 3D objects. Our computer phasing experiments accurately r etrieved the phase from the magnitude of the Fourier transforms of 2D and 3D complex-valued objects by using positivity constraints on the i maginary part of the objects and loose supports, with the oversampling factor much less than 4 for 2D and 8 for 3D objects. Under the same c onditions we also obtained reasonably good reconstructions of 2D and 3 D complex-valued objects from the magnitude of their Fourier transform s with added noise and a central stop. (C) 1998 Optical Society of Ame rica.