Reconstruction of an object from the modulus of its Fourier transform

Abstract

We present a digital method for solving the phase-retrieval problem of optical-coherence theory: the reconstruction of a general object from the modulus of its Fourier transform. This technique should be useful for obtaining high-resolution imagery from interferometer data.

Coherent X-ray diffractive imaging on the single-cell-level of microbial samples

“…The rst practical phase retrieval algorithm (GS) for isolated non-crystalline specimens was developed by Gerchberg and Saxton in the 1970s [92]. Since then many di erent algorithms such as the prominent Error-Reduction (ER) [82] and the Hybrid-Input-Output (HIO) [86] have been presented. More recent algorithms are the Di erence-Map (DM) [73], the Hybrid Projection Re ection algorithm (HPR) [18] and the Relaxed Averaged Alternating Re ection algorithm (RAAR) [177].…”

Time-resolved imaging using x-ray free electron lasers

“…The pattern extends to a diffraction angle of 15 • at the midpoint of its edge. On the basis of low-fluence optical parameters 24 , we estimate 3,25 that the absorbed energy The FEL beam is incident from the left and is focused to a 20µm spot on the sample, which consisted of a 20-nm-thick transmissive silicon nitride membrane with a picture milled through its entire thickness using an focussed ion beam (FIB) (the sample is enlarged in the inset, and the scale bar indicates 1 µm). The direct beam passes through the sample window and exits the camera through a hole in a graded multilayer planar mirror.…”

Surface structure solution by X-ray diffraction: structure completion with positivity and atomicity constraints

“…The idea of an input-output feedback loop for phasing that iteratively satisfies conditions in real and reciprocal space has been suggested by Fienup [13] for problems where a non-negative distribution is sought, and where only the amplitudes of the Fourier transform of that quantity are accessible by experiment. The aim is to obtain increasingly better estimates of the phases of this Fourier transform by iteratively satisfying the reciprocal-space constraints and the real-space requirement of the positivity of the sought distribution.…”

“…at the last step within the right-hand box gives rise to the output electron distribution, ft ðnÞ j g: Thus, in such an input-output scheme [13], the box on the right of the flow chart transforms an input electron distribution fu ðnÞ j g to an output one ft ðnÞ j g at iteration n by combining experimental information about the measured amplitudes jF q j with the current estimates of the phases f ðnÞ q associated with those amplitudes. The boxes on …”

“…The iterative algorithm recovers the phases of the functions in both spaces. An extension of this algorithm when amplitude data is available only in reciprocal space, and where some general constraints may be imposed in real space, such as a spatial localization, or some bounds on the realspace distribution, was subsequently proposed by Fienup and applied to phase determination problems in astronomy and optics [13]. We have earlier proposed an adaptation of this method [9,10] for the problem of structure completion in surface X-ray diffraction (SXRD), where initial phases of the CTRs are taken to be those calculated for the known underlying bulk structure.…”