- Adaptive-additive algorithm
In the studies of
Fourier optics ,sound synthesis , stellarinterferometry ,optical tweezers , and diffractive optical elements (DOEs) it is often important to know thespatial frequency phase of an observed wave source. In order to reconstruct this phase the Adaptive-Additive Algorithm (or AA algorithm), which derives from a group of adaptive (input-output) algorithms, can be used. The AA algorithm is aniterative algorithm that utilizes theFourier Transform to calculate an unknown part of a propagating wave, normally thespatial frequency phase (k space). This can be done when given the phase’s known counterparts, usually an observedamplitude (position space) and an assumed startingamplitude (k space). To find the correct phase thealgorithm uses error conversion, or the error between the desired and the theoretical intensities. The AA algorithm is currently being implemented by [http://hillslab.umd.edu/ Dr. Wendell Hill III] , Alex Robel, V. Kotlyar Soifer, and [http://physics.nyu.edu/grierlab/cgh2b/node6.html David G Grier] .The algorithm
History
The adaptive-additive algorithm was originally created to reconstruct the
spatial frequency phase of light intensity in the study of stellarinterferometry . Since then, the AA algorithm has been adapted to work in the fields ofFourier Optics by Soifer and [http://hillslab.umd.edu/ Dr. Hill] ,soft matter andoptical tweezers by [http://physics.nyu.edu/grierlab/cgh2b/node6.html Dr. Grier] , andsound synthesis by Robel.Pseudo-code algorithm
1. Define input amplitude and random phase 2. Forward Fourier Transform 3. Separate transformed amplitude and phase 4. Compare transformed amplitude/intensity to desired output amplitude/intensity 5. Check convergence conditions 6. Mix transformed amplitude with desired output amplitude and combine with transformed phase 7. Inverse Fourier Transform 8. Separate new amplitude and new phase 9. Combine new phase with original input amplitude 10. Loop back to Forward Fourier Transform
Example
For the problem of reconstructing the
spatial frequency phase ("k"-space) for a desired intensity in the image plane ("x"-space). Assume theamplitude and the starting phase of the wave in "k"-space is a A_0 and phi_n^{k} respectively.Fourier transform the wave in "k"-space to "x" space.: A_0e^{iphi_n^{k xrightarrow{FFT} A_n^fe^{iphi_n^{f
Then compare the transformed
intensity I_n^f with the desiredintensity I_0^f, where: I_n^f = left(A_n^f ight)^2,
: varepsilon = sqrt{left(I_n^f ight)^2 - left(I_0 ight)^2}.
Check varepsilon against the convergence requirements. If the requirements are not met then mix the transformed
amplitude A_n^f with desiredamplitude A^f.: ar{A}^f_n = left [a A^f + (1-a) A_n^f ight] ,
where "a" is mixing ratio and
: A^f = sqrt{I_0}.
"Note that "a" is a percentage, defined on the interval 0 ≤ "a" ≤ 1.
Combine mixed
amplitude with the "x"-space phase andinverse Fourier transform .: ar{A}^{f}e^{iphi_n^f} xrightarrow{iFFT} ar{A}_n^ke^{iphi_n^k}.
Separate ar{A}_n^k and phi^k_n and combine A_0 with phi^k_n. Increase loop by one n o n + 1 and repeat.
Limits
*If a = 1 then the AA algorithm becomes the
Gerchberg–Saxton algorithm .
*If a = 0 then ar{A}^k_n = A_0.ee also
*
Gerchberg–Saxton algorithm
*Fourier optics
*Holography
*Interferometry
* Sound SynthesisReferences
*citation
last=Dufresne
first=Eric
last2=Grier
first2=David G
last3=Spalding
title=Computer-Generated Holographic Optical Tweezer Arrays
journal=Review of Scientific Instruments
volume=72 | issue=3
month=December | year=2000.
*citation
last=Grier
first=David G
title=Adaptive-Additive Algorithm
date=October 10 ,2000 .
url=http://www.physics.nyu.edu/~dg86/cgh2b/node6.html.
*citation
last=Robel
first=Axel
title=Adaptive Additive Modeling With Continuous Parameter Trajectories
url=http://ieeexplore.ieee.org/iel5/10376/32978/101109TSA2005858529.pdf?arnumber=101109TSA2005858529.
*citation
last=Robel
first=Axel
title=Adaptive-Additive Synthesis of Sound Technical
place=University of Berlin Germany
publisher=Einsteinufer 17, 10587
location=Berlin, Germany.
url=http://i2pi.com/PAPERS/music-dsp/adaptive-additive-synthesis-of.pdf.
*cite book
last=Soifer
first=V. Kotlyar
last2=Doskolovich,
first2=L.
title=Iterative Methods for Diffractive Optical Elements Computation
year=1997
publisher=Taylor & Francis
location=Bristol, PA
isbn=978-0748406340External links
* [http://staff.chess.cornell.edu/~shen/workshop2003/presentations/Talk_DiFabrizio.pdf A PDF/Power Point Presentation that describes the uses and variations of the AA algorithm] "Berkeley, Ca".
* [http://physics.nyu.edu/grierlab/cgh2b/node6.html David Grier's Lab] Presentation on optical tweezers and fabrication of AA algorithm.
* [http://www-ccrma.stanford.edu/~roebel/addsyn/index.html Adaptive Additive Synthesis for Non Stationary Sound] Dr. Axel Robel.
* [http://hillslab.umd.edu/ Hill Labs] "University of Maryland College Park".
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