Phase retrieval

Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex signal , of amplitude , and phase :

where x is an M-dimensional spatial coordinate and k is an M-dimensional spatial frequency coordinate, phase retrieval consists in finding the phase that for a measured amplitude satisfies a set of constraints. Important applications of phase retrieval include X-ray crystallography, transmission electron microscopy and coherent diffractive imaging, for which . (Fienup 1982:2759) Uniqueness theorems for both 1-D and 2-D cases of the phase retrieval problem, including the phaseless 1-D inverse scattering problem, were proved by Klibanov and his collaborators (see References).