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 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 . 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). Here we consider 1-D discrete Fourier transform (DFT) phase retrieval problem. The DFT of a complex signal is given by and the oversampled DFT of is given by where . Since the DFT operator is bijective, this is equivalent to recovering the phase . It is common recovering a signal from its autocorrelation sequence instead of its Fourier magnitude. That is, denote by the vector after padding with zeros. The autocorrelation sequence of is then defined as and the DFT of , denoted by , satisfies . The error reduction is a generalization of the Gerchberg–Saxton algorithm. It solves for from measurements of by iterating a four-step process. For the th iteration the steps are as follows: Step (1): , , and are estimates of, respectively, , and . In the first step we calculate the Fourier transform of : Step (2): The experimental value of , calculated from the diffraction pattern via the signal equation, is then substituted for , giving an estimate of the Fourier transform: where the ' denotes an intermediate result that will be discarded later on. Step (3): the estimate of the Fourier transform is then inverse Fourier transformed: Step (4): then must be changed so that the new estimate of the object, , satisfies the object constraints. is therefore defined piecewise as: where is the domain in which does not satisfy the object constraints.