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Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex signal F {\displaystyle F} , of amplitude | F | {\displaystyle |F|} , and phase ψ {\displaystyle \psi } :
where x is an M-dimensional spatial coordinate and k is an M-dimensional spatial frequency coordinate. Phase retrieval consists of finding the phase that satisfies a set of constraints for a measured amplitude. Important applications of phase retrieval include X-ray crystallography, transmission electron microscopy and coherent diffractive imaging, for which M = 2 {\displaystyle M=2}. Uniqueness theorems for both 1-D and 2-D cases of the phase retrieval problem, including the phaseless 1-D inverse scattering problem, were proven by Klibanov and his collaborators.