- 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 frequencyphase 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 an iterative algorithmthat utilizes the Fourier Transformto calculate an unknown part of a propagating wave, normally the spatial frequencyphase (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 the algorithmuses 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 adaptive-additive algorithm was originally created to reconstruct the
spatial frequencyphase of light intensity in the study of stellar interferometry. Since then, the AA algorithm has been adapted to work in the fields of Fourier Opticsby Soifer and [http://hillslab.umd.edu/ Dr. Hill] , soft matterand optical tweezersby [http://physics.nyu.edu/grierlab/cgh2b/node6.html Dr. Grier] , and sound synthesisby Robel.
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
For the problem of reconstructing the
spatial frequencyphase ("k"-space) for a desired intensity in the image plane ("x"-space). Assume the amplitudeand the starting phase of the wave in "k"-space is a and respectively. Fourier transformthe wave in "k"-space to "x" space.
Then compare the transformed
intensitywith the desired intensity, where
Check against the convergence requirements. If the requirements are not met then mix the transformed
amplitudewith desired amplitude.
where "a" is mixing ratio and
"Note that "a" is a percentage, defined on the interval 0 ≤ "a" ≤ 1.
amplitudewith the "x"-space phase and inverse Fourier transform.
Separate and and combine with . Increase loop by one and repeat.
*If then the AA algorithm becomes the
*If then .
* Sound Synthesis
title=Computer-Generated Holographic Optical Tweezer Arrays
journal=Review of Scientific Instruments
volume=72 | issue=3
month=December | year=2000.
October 10, 2000.
title=Adaptive Additive Modeling With Continuous Parameter Trajectories
title=Adaptive-Additive Synthesis of Sound Technical
place=University of Berlin Germany
publisher=Einsteinufer 17, 10587
title=Iterative Methods for Diffractive Optical Elements Computation
publisher=Taylor & Francis
* [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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