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.
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Coherent diffractive imaging (CDI) is a "lensless" technique for 2D or 3D reconstruction of the image of nanoscale structures such as nanotubes, nanocrystals, porous nanocrystalline layers, defects, potentially proteins, and more. In CDI, a highly coherent beam of X-rays, electrons or other wavelike particle or photon is incident on an object. The beam scattered by the object produces a diffraction pattern downstream which is then collected by a detector. This recorded pattern is then used to reconstruct an image via an iterative feedback algorithm.
Ptychography (/t(ʌ)ɪˈkogræfi/ t(a)i-KO-graf-ee) is a computational method of microscopic imaging. It generates images by processing many coherent interference patterns that have been scattered from an object of interest. Its defining characteristic is translational invariance, which means that the interference patterns are generated by one constant function (e.g. a field of illumination or an aperture stop) moving laterally by a known amount with respect to another constant function (the specimen itself or a wave field).
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles and intensities of these diffracted beams, a crystallographer can produce a three-dimensional picture of the density of electrons within the crystal. From this electron density, the mean positions of the atoms in the crystal can be determined, as well as their chemical bonds, their crystallographic disorder, and various other information.
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Phase retrieval consists in the recovery of a complex-valued signal from intensity-only measurements. As it pervades a broad variety of applications, many researchers have striven to develop phase-retrieval algorithms. Classical approaches involve techniqu ...
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