- 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 Synthesis**References***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=978-0748406340**External 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".

*Wikimedia Foundation.
2010.*

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