Optical correlator

<h2 id="introduction">Introduction</h2>
<p>The correlator has an input signal which is multiplied by some filter in the Fourier domain. An example filter is the <a href="/facts/Matched_filter/Tl8Bm687">matched filter</a> which uses the <a href="/facts/Cross_correlation/Fs1hY3br">cross correlation</a> of the two signals.
</p><p>The cross correlation or correlation plane, 
  
    
      
        c
        (
        x
        ,
        y
        )
      
    
    {\displaystyle c(x,y)}
  
 of a 2D signal 
  
    
      
        i
        (
        x
        ,
        y
        )
      
    
    {\displaystyle i(x,y)}
  
 with 
  
    
      
        h
        (
        x
        ,
        y
        )
      
    
    {\displaystyle h(x,y)}
  
 is
</p>

c
        (
        x
        ,
        y
        )
        =
        i
        (
        x
        ,
        y
        )
        ⊗
        
          h
          
            ∗
          
        
        (
        −
        x
        ,
        −
        y
        )
      
    
    {\displaystyle c(x,y)=i(x,y)\otimes h^{*}(-x,-y)}

<p>This can be re-expressed in Fourier space as
</p>

C
        (
        ξ
        ,
        η
        )
        =
        I
        (
        ξ
        ,
        η
        )
        
          H
          
            ∗
          
        
        (
        −
        ξ
        ,
        −
        η
        )
      
    
    {\displaystyle C(\xi ,\eta )=I(\xi ,\eta )H^{*}(-\xi ,-\eta )}

<p>where the capital letters denote the Fourier transform of what the lower case letter denotes. So the correlation can then be calculated by inverse Fourier transforming the result.
</p>
<h2 id="implementation">Implementation</h2>
<p>According to <a href="/facts/Fresnel_Diffraction/lxyq42M4">Fresnel Diffraction</a> theory a <a href="/facts/Convex_lens/ejlH6n5c">convex lens</a> of <a href="/facts/Focal_length/brGIUkEY">focal length</a> 
  
    
      
        f
      
    
    {\displaystyle f}
  
 will produce the exact Fourier transform at a distance 
  
    
      
        f
      
    
    {\displaystyle f}
  
 behind the lens of an object placed 
  
    
      
        f
      
    
    {\displaystyle f}
  
 distance in front of the lens. So that <a href="/facts/Complex_amplitudes/LyjSSUDo">complex amplitudes</a> are multiplied, the light source must be <a href="/facts/Coherence_(physics)/NF1Rgdoa">coherent</a> and is typically from a <a href="/facts/Laser/jk6PTvvT">laser</a>. The input signal and filter are commonly written onto a <a href="/facts/Spatial_light_modulator/PjnfVdzb">spatial light modulator</a> (SLM).
</p><p>A typical arrangement is the <a href="/facts/Fourier_optics/H4bVRGYx">4f correlator</a>. The input signal is written to an SLM which is illuminated with a laser. This is Fourier transformed with a lens and this is then modulated with a second SLM containing the filter. The resultant is again Fourier transformed with a second lens and the correlation result is captured on a camera.
</p>
<h2 id="filter-design">Filter design</h2>
<p>Many filters have been designed to be used with an optical correlator. Some have been proposed to address hardware limitations, others were developed to optimize a merit function or to be invariant under a certain transformation.
</p>
<h3>Matched filter</h3>
<p>The matched filter maximizes the signal-to-noise ratio and is simply obtained by using as a filter the Fourier transform of the reference signal 
  
    
      
        r
        (
        x
        ,
        y
        )
      
    
    {\displaystyle r(x,y)}
  
.
</p>

H
        (
        ξ
        ,
        η
        )
        =
        R
        (
        ξ
        ,
        η
        )
      
    
    {\displaystyle H(\xi ,\eta )=R(\xi ,\eta )}

<h3>Phase-only filter</h3>
<p>The phase-only filter<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> is easier to implement due to limitation of many SLMs and has been shown to be more discriminant than the matched filter.
</p>

H
        (
        ξ
        ,
        η
        )
        =
        
          
            
              R
              (
              ξ
              ,
              η
              )
            
            
              |
              
                R
                (
                ξ
                ,
                η
                )
              
              |
            
          
        
      
    
    {\displaystyle H(\xi ,\eta )={\frac {R(\xi ,\eta )}{\left\vert R(\xi ,\eta )\right\vert }}}

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>A. VanderLugt, Signal detection by complex spatial filtering, IEEE Transactions on Information Theory, vol. 10, 1964, pp. 139–145. <a href="http://deepblue.lib.umich.edu/bitstream/2027.42/8080/5/bad4575.0001.001.pdf" target="_blank">http://deepblue.lib.umich.edu/bitstream/2027.42/8080/5/bad4575.0001.001.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>J. L. Horner and P. D. Gianino, Phase-only matched filtering, Appl. Opt. 23, 1984, 812–816 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Optical correlator open-in-new

Optical correlator