Adaptive-additive algorithm

<h2 id="the-algorithm">The algorithm</h2>
<h3>History</h3>
<p>The adaptive-additive algorithm was originally created to reconstruct the <a href="/facts/Spatial_frequency/b0un3CNv">spatial frequency</a> <a href="/facts/Phase_(waves)/cpGtuVll">phase</a> of light intensity in the study of stellar <a href="/facts/Interferometry/X3E1SsQT">interferometry</a>. Since then, the AA algorithm has been adapted to work in the fields of <a href="/facts/Fourier_Optics/H4bVRGYx">Fourier Optics</a> by Soifer and Dr. Hill, <a href="/facts/Soft_matter/62NTslbq">soft matter</a> and <a href="/facts/Optical_tweezers/wsPxwmX6">optical tweezers</a> by Dr. Grier, and <a href="/facts/Sound_synthesis/NrmdVrPa">sound synthesis</a> by Röbel.
</p>
<h3>Algorithm</h3>
<ol><li>Define input amplitude and random phase</li>
<li>Forward Fourier Transform</li>
<li>Separate transformed amplitude and phase</li>
<li>Compare transformed amplitude/intensity to desired output amplitude/intensity</li>
<li>Check convergence conditions</li>
<li>Mix transformed amplitude with desired output amplitude and combine with transformed phase</li>
<li>Inverse Fourier Transform</li>
<li>Separate new amplitude and new phase</li>
<li>Combine new phase with original input amplitude</li>
<li>Loop back to Forward Fourier Transform</li></ol>
<h3>Example</h3>
<p>For the problem of reconstructing the <a href="/facts/Spatial_frequency/b0un3CNv">spatial frequency</a> phase (<i>k</i>-space) for a desired <a href="/facts/Intensity_(physics)/KFWkJkJI">intensity</a> in the image plane (<i>x</i>-space). Assume the <a href="/facts/Amplitude/aEckuXJM">amplitude</a> and the starting phase of the wave in <i>k</i>-space is 
  
    
      
        
          A
          
            0
          
        
      
    
    {\displaystyle A_{0}}
  
 and 
  
    
      
        
          ϕ
          
            n
          
          
            k
          
        
      
    
    {\displaystyle \phi _{n}^{k}}
  
 respectively. <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> the wave in <i>k</i>-space to <i>x</i> space.
</p>

A
          
            0
          
        
        
          e
          
            i
            
              ϕ
              
                n
              
              
                k
              
            
          
        
        
          
            →
            
              F
              F
              T
            
          
        
        
          A
          
            n
          
          
            f
          
        
        
          e
          
            i
            
              ϕ
              
                n
              
              
                f
              
            
          
        
      
    
    {\displaystyle A_{0}e^{i\phi _{n}^{k}}{\xrightarrow {FFT}}A_{n}^{f}e^{i\phi _{n}^{f}}}

<p>Then compare the transformed <a href="/facts/Intensity_(physics)/KFWkJkJI">intensity</a> 
  
    
      
        
          I
          
            n
          
          
            f
          
        
      
    
    {\displaystyle I_{n}^{f}}
  
 with the desired intensity 
  
    
      
        
          I
          
            0
          
          
            f
          
        
      
    
    {\displaystyle I_{0}^{f}}
  
, where
</p>

I
          
            n
          
          
            f
          
        
        =
        
          
            (
            
              A
              
                n
              
              
                f
              
            
            )
          
          
            2
          
        
        ,
      
    
    {\displaystyle I_{n}^{f}=\left(A_{n}^{f}\right)^{2},}

ε
        =
        
          
            
              
                (
                
                  I
                  
                    n
                  
                  
                    f
                  
                
                )
              
              
                2
              
            
            −
            
              
                (
                
                  I
                  
                    0
                  
                
                )
              
              
                2
              
            
          
        
        .
      
    
    {\displaystyle \varepsilon ={\sqrt {\left(I_{n}^{f}\right)^{2}-\left(I_{0}\right)^{2}}}.}

<p>Check 
  
    
      
        ε
      
    
    {\displaystyle \varepsilon }
  
 against the convergence requirements. If the requirements are not met then mix the transformed <a href="/facts/Amplitude/aEckuXJM">amplitude</a> 
  
    
      
        
          A
          
            n
          
          
            f
          
        
      
    
    {\displaystyle A_{n}^{f}}
  
 with desired amplitude 
  
    
      
        
          A
          
            f
          
        
      
    
    {\displaystyle A^{f}}
  
.
</p>

A
                ¯
              
            
          
          
            n
          
          
            f
          
        
        =
        
          [
          
            a
            
              A
              
                f
              
            
            +
            (
            1
            −
            a
            )
            
              A
              
                n
              
              
                f
              
            
          
          ]
        
        ,
      
    
    {\displaystyle {\bar {A}}_{n}^{f}=\left[aA^{f}+(1-a)A_{n}^{f}\right],}

<p>where <i>a</i> is mixing ratio and
</p>

A
          
            f
          
        
        =
        
          
            
              I
              
                0
              
            
          
        
      
    
    {\displaystyle A^{f}={\sqrt {I_{0}}}}
  
.
<p>Note that <i>a</i> is a percentage, defined on the interval 0 ≤ <i>a</i> ≤ 1.
</p><p>Combine mixed amplitude with the <i>x</i>-space phase and <a href="/facts/Inverse_Fourier_transform/aM1aMkf0">inverse Fourier transform</a>.
</p>

A
                ¯
              
            
          
          
            f
          
        
        
          e
          
            i
            
              ϕ
              
                n
              
              
                f
              
            
          
        
        
          
            →
            
              i
              F
              F
              T
            
          
        
        
          
            
              
                A
                ¯
              
            
          
          
            n
          
          
            k
          
        
        
          e
          
            i
            
              ϕ
              
                n
              
              
                k
              
            
          
        
        .
      
    
    {\displaystyle {\bar {A}}^{f}e^{i\phi _{n}^{f}}{\xrightarrow {iFFT}}{\bar {A}}_{n}^{k}e^{i\phi _{n}^{k}}.}

<p>Separate 
  
    
      
        
          
            
              
                A
                ¯
              
            
          
          
            n
          
          
            k
          
        
      
    
    {\displaystyle {\bar {A}}_{n}^{k}}
  
 and 
  
    
      
        
          ϕ
          
            n
          
          
            k
          
        
      
    
    {\displaystyle \phi _{n}^{k}}
  
 and combine 
  
    
      
        
          A
          
            0
          
        
      
    
    {\displaystyle A_{0}}
  
 with 
  
    
      
        
          ϕ
          
            n
          
          
            k
          
        
      
    
    {\displaystyle \phi _{n}^{k}}
  
. Increase loop by one 
  
    
      
        n
        →
        n
        +
        1
      
    
    {\displaystyle n\to n+1}
  
 and repeat.
</p>
<h4>Limits</h4>
<ul><li>If 
  
    
      
        a
        =
        1
      
    
    {\displaystyle a=1}
  
 then the AA algorithm becomes the <a href="/facts/Gerchberg%25E2%2580%2593Saxton_algorithm/Ghlyj0a2">Gerchberg–Saxton algorithm</a>.</li>
<li>If 
  
    
      
        a
        =
        0
      
    
    {\displaystyle a=0}
  
 then 
  
    
      
        
          
            
              
                A
                ¯
              
            
          
          
            n
          
          
            k
          
        
        =
        
          A
          
            0
          
        
      
    
    {\displaystyle {\bar {A}}_{n}^{k}=A_{0}}
  
.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Gerchberg%25E2%2580%2593Saxton_algorithm/Ghlyj0a2">Gerchberg–Saxton algorithm</a></li>
<li><a href="/facts/Fourier_optics/H4bVRGYx">Fourier optics</a></li>
<li><a href="/facts/Holography/4J1gW0M6">Holography</a></li>
<li><a href="/facts/Interferometry/X3E1SsQT">Interferometry</a></li>
<li><a href="/facts/Synthesizer/NrmdVrPa">Sound Synthesis</a></li></ul>

<ul><li>Dufresne, Eric; Grier, David G; Spalding (December 2000), "Computer-Generated Holographic Optical Tweezer Arrays", <i>Review of Scientific Instruments</i>, 72 (3): 1810, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/cond-mat/0008414">cond-mat/0008414</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2001RScI...72.1810D">2001RScI...72.1810D</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1063%2F1.1344176">10.1063/1.1344176</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:14064547">14064547</a>.</li>
<li>Grier, David G (October 10, 2000), <a href="http://www.physics.nyu.edu/~dg86/cgh2b/node6.html"><i>Adaptive-Additive Algorithm</i></a>.</li>
<li>Röbel, Axel (2006), "Adaptive Additive Modeling With Continuous Parameter Trajectories", <i>IEEE Transactions on Audio, Speech, and Language Processing</i>, 14 (4): 1440–1453, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2FTSA.2005.858529">10.1109/TSA.2005.858529</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:73476">73476</a>.</li>
<li>Röbel, Axel, <i>Adaptive-Additive Synthesis of Sound</i>, ICMC 1999, <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.7602">10.1.1.27.7602</a>{{citation}}:  CS1 maint: location (link)</li>
<li>Soifer, V. Kotlyar; Doskolovich, L. (1997), <i>Iterative Methods for Diffractive Optical Elements Computation</i>, Bristol, PA: Taylor & Francis, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7484-0634-0</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://physics.nyu.edu/grierlab/cgh2b/node6.html">David Grier's Lab</a> Presentation on optical tweezers and fabrication of AA algorithm.</li>
<li><a href="https://web.archive.org/web/20070609193556/http://www-ccrma.stanford.edu/~roebel/addsyn/index.html">Adaptive Additive Synthesis for Non Stationary Sound</a> Dr. Axel Röbel.</li>
<li><a href="http://hillslab.umd.edu/">Hill Labs</a> <i>University of Maryland College Park</i>.]</li></ul>

Adaptive-additive algorithm open-in-new

Adaptive-additive algorithm