Phase retrieval

Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex spectrum ${\displaystyle F(k)}$, of amplitude ${\displaystyle |F(k)|}$, and phase ${\displaystyle \psi (k)}$:

${\displaystyle F(k)=|F(k)|e^{i\psi (k)}=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ik\cdot x}\,dx}$

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 ${\displaystyle M=2}$.^[1] 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 (see References).