Phase retrieval

Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex signal F ( k ) {displaystyle F(k)} , of amplitude | F ( k ) | {displaystyle |F(k)|} , and phase ψ ( k ) {displaystyle psi (k)} : Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex signal F ( k ) {displaystyle F(k)} , of amplitude | F ( k ) | {displaystyle |F(k)|} , and phase ψ ( k ) {displaystyle psi (k)} : 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} . (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). The error reduction is a generalisation of the Gerchberg–Saxton algorithm. It solves for f ( x ) {displaystyle f(x)} from measurements of | F ( x ) | {displaystyle |F(x)|} . It uses iteration of a four-step process. For the k {displaystyle k} 'th iteration the steps are as follows: Step (1): G k ( u ) {displaystyle G_{k}(u)} , ϕ k {displaystyle phi _{k}} , and g k ( x ) {displaystyle g_{k}(x)} are estimates of, respectively, F ( u ) {displaystyle F(u)} , ψ {displaystyle psi } and f ( x ) {displaystyle f(x)} . In the first step g k ( x ) {displaystyle g_{k}(x)} undergoes Fourier transformation: Step (2): The experimental value of | F ( u ) | {displaystyle |F(u)|} , calculated from the diffraction pattern via the signal equation, is then substituted for | G k ( u ) | {displaystyle |G_{k}(u)|} , giving an estimate of the Fourier transformation: