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The Gerchberg–Saxton (GS) algorithm is an iterative
Applications and publications on phase retrieval from the University of Rochester, Institute of Optics
A Python-Script of the GS by Dominik Doellerer
* MATLAB GS algorithm
{{DEFAULTSORT:Gerchberg-Saxton algorithm Digital signal processing Physical optics Articles with example pseudocode
The Gerchberg–Saxton (GS) algorithm is an iterative phase retrieval
Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex signal F(k), of amplitude , F (k), , and phase \psi(k):
::F(k) = , F(k), e^ =\int_^ f(x)\ e^\,dx
where ''x'' is an ''M''-dimensional spatia ...
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
for retrieving the phase of a complex-valued wavefront from two intensity measurements acquired in two different planes. Typically, the two planes are the image plane and the far field (diffraction) plane, and the wavefront propagation between these two planes is given by the Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
. The original paper by Gerchberg and Saxton considered image and diffraction pattern of a sample acquired in an electron microscope.
It is often necessary to know only the phase distribution from one of the planes, since the phase distribution on the other plane can be obtained by performing a Fourier transform on the plane whose phase is known. Although often used for two-dimensional signals, the GS algorithm is also valid for one-dimensional signals.
The pseudocode
In computer science, pseudocode is a plain language description of the steps in an algorithm or another system. Pseudocode often uses structural conventions of a normal programming language, but is intended for human reading rather than machine re ...
below performs the GS algorithm to obtain a phase distribution for the plane "Source", such that its Fourier transform would have the amplitude distribution of the plane "Target".
Pseudocode algorithm
Let: FT – forward Fourier transform IFT – inverse Fourier transform ''i'' – the imaginary unit, √−1 (square root of −1) exp – exponential function (exp(x) = ''e''''x'') Target and Source be the Target and Source Amplitude planes respectively A, B, C & D be complex planes with the same dimension as Target and Source Amplitude – Amplitude-extracting function: e.g. for complex ''z'' = ''x'' + ''iy'', amplitude(''z'') = sqrt(''x''·''x'' + ''y''·''y'') for real ''x'', amplitude(''x'') = , ''x'', Phase – Phase extracting function: e.g. Phase(z) = arctan(y / x) end Let algorithm Gerchberg–Saxton(Source, Target, Retrieved_Phase) is A := IFT(Target) while error criterion is not satisfied B := Amplitude(Source) × exp(i × Phase(A)) C := FT(B) D := Amplitude(Target) × exp(i × Phase(C)) A := IFT(D) end while Retrieved_Phase = Phase(A) This is just one of the many ways to implement the GS algorithm. Aside from optimizations, others may start by performing a forward Fourier transform to the source distribution.See also
*Phase retrieval
Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex signal F(k), of amplitude , F (k), , and phase \psi(k):
::F(k) = , F(k), e^ =\int_^ f(x)\ e^\,dx
where ''x'' is an ''M''-dimensional spatia ...
* Fourier optics Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or '' superposition'', of plane waves. It has some parallels to the Huygens–Fresnel pr ...
* Holography
Holography is a technique that enables a wavefront to be recorded and later re-constructed. Holography is best known as a method of generating real three-dimensional images, but it also has a wide range of other applications. In principle, i ...
* Computer-generated holography Computer-generated holography (CGH) is the method of digitally generating holographic interference patterns. A holographic image can be generated e.g. by digitally computing a holographic interference pattern and printing it onto a mask or film for ...
* Adaptive-additive algorithm In the studies of Fourier optics, sound synthesis, stellar interferometry, optical tweezers, and diffractive optical elements (DOEs) it is often important to know the spatial frequency phase of an observed wave source. In order to reconstruct thi ...
References
External links
* Dr W. Owen Saxton's pageApplications and publications on phase retrieval from the University of Rochester, Institute of Optics
A Python-Script of the GS by Dominik Doellerer
* MATLAB GS algorithm
{{DEFAULTSORT:Gerchberg-Saxton algorithm Digital signal processing Physical optics Articles with example pseudocode