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In the studies of [[Fourier optics]], [[sound synthesis]], [[Stellar astronomy|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 this [[phase (waves)|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 an [[iterative]] [[algorithm]] that utilizes the [[Fourier Transform]] to calculate an unknown part of a propagating wave, normally the [[spatial frequency]] [[phase (waves)|phase]] (k space). This can be done when given the phase’s known counterparts, usually an observed [[amplitude]] (position space) and an assumed starting [[amplitude]] (k space). To find the correct [[phase (waves)|phase]] the [[algorithm]] uses error conversion, or the error between the desired and the theoretical [[intensity (physics)|intensities]].

==The algorithm==

===History===

The adaptive-additive algorithm was originally created to reconstruct the [[spatial frequency]] [[phase (waves)|phase]] of light intensity in the study of stellar [[interferometry]]. Since then, the AA algorithm has been adapted to work in the fields of [[Fourier Optics]] by Soifer and Dr. Hill, [[soft matter]] and [[optical tweezers]] by Dr. Grier, and [[sound synthesis]] by Röbel.

===Algorithm===
# Define input amplitude and random phase
# Forward Fourier Transform
# Separate transformed amplitude and phase
# Compare transformed amplitude/intensity to desired output amplitude/intensity
# Check convergence conditions
# Mix transformed amplitude with desired output amplitude and combine with transformed phase
# Inverse Fourier Transform
# Separate new amplitude and new phase
# Combine new phase with original input amplitude
# Loop back to Forward Fourier Transform

===Example===

For the problem of reconstructing the [[spatial frequency]] phase (''k''-space) for a desired [[intensity (physics)|intensity]] in the image plane (''x''-space). Assume the [[amplitude]] and the starting phase of the wave in ''k''-space is <math>A_0</math> and <math>\phi_n^{k}</math> respectively. [[Fourier transform]] the wave in ''k''-space to ''x'' space.

: <math>A_0e^{i\phi_n^{k}} \xrightarrow{FFT} A_n^fe^{i\phi_n^{f}}</math>

Then compare the transformed [[intensity (physics)|intensity]] <math>I_n^f</math> with the desired intensity <math>I_0^f</math>, where

: <math>
I_n^f = \left(A_n^f\right)^2,
</math>

: <math>
\varepsilon = \sqrt{\left(I_n^f\right)^2 - \left(I_0\right)^2}.
</math>

Check <math>\varepsilon</math> against the convergence requirements. If the requirements are not met then mix the transformed [[amplitude]] <math>A_n^f</math> with desired amplitude <math>A^f</math>.

: <math>\bar{A}^f_n = \left[a A^f + (1-a) A_n^f\right],</math>

where ''a'' is mixing ratio and

: <math>A^f = \sqrt{I_0}</math>.

Note that ''a'' is a percentage, defined on the interval 0 ≤ ''a'' ≤ 1.

Combine mixed amplitude with the ''x''-space phase and [[inverse Fourier transform]].

: <math>\bar{A}^{f}e^{i\phi_n^f} \xrightarrow{iFFT} \bar{A}_n^ke^{i\phi_n^k}.</math>

Separate <math>\bar{A}_n^k</math> and <math>\phi^k_n</math> and combine <math>A_0</math> with <math>\phi^k_n</math>. Increase loop by one <math> n \to n + 1</math> and repeat.

====Limits====
* If <math>a = 1</math> then the AA algorithm becomes the [[Gerchberg–Saxton algorithm]].
* If <math>a = 0</math> then <math>\bar{A}^k_n = A_0</math>.

==See also==

* [[Gerchberg–Saxton algorithm]]
* [[Fourier optics]]
* [[Holography]]
* [[Interferometry]]
* [[Synthesizer|Sound Synthesis]]

==References==

* {{citation
| last1=Dufresne
| first1=Eric
| last2=Grier
| first2=David G
| last3=Spalding
| title=Computer-Generated Holographic Optical Tweezer Arrays
| journal=Review of Scientific Instruments
| volume=72 | issue=3
| pages=1810
| doi=10.1063/1.1344176
|date=December 2000| arxiv=cond-mat/0008414| bibcode=2001RScI...72.1810D| s2cid=14064547
}}.
* {{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=Röbel
| first=Axel
| title=Adaptive Additive Modeling With Continuous Parameter Trajectories
| journal=IEEE Transactions on Audio, Speech, and Language Processing
| volume=14
| issue=4
| pages=1440–1453
| doi=10.1109/TSA.2005.858529
| year=2006
| s2cid=73476
}}.
* {{citation
| last=Röbel
| first=Axel
| title=Adaptive-Additive Synthesis of Sound
| place= ICMC 1999
| citeseerx=10.1.1.27.7602
}}
* {{Citation
| last1=Soifer
| first1=V. Kotlyar
| last2=Doskolovich
| first2=L.
| title=Iterative Methods for Diffractive Optical Elements Computation
| year=1997
| publisher=Taylor & Francis
| location=Bristol, PA
| isbn=978-0-7484-0634-0}}

==External links==
* [http://physics.nyu.edu/grierlab/cgh2b/node6.html David Grier's Lab] Presentation on optical tweezers and fabrication of AA algorithm.
* [https://web.archive.org/web/20070609193556/http://www-ccrma.stanford.edu/~roebel/addsyn/index.html Adaptive Additive Synthesis for Non Stationary Sound] Dr. Axel Röbel.
* [http://hillslab.umd.edu/ Hill Labs] ''University of Maryland College Park''.]

{{DEFAULTSORT:Adaptive-Additive Algorithm}}
[[Category:Digital signal processing]]
[[Category:Physical optics]]

Adaptive-additive algorithm - Revision history

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